I need help with this exercise from Algebra II, I'm stuck.

Find GCD $(a_n:a_{n+1})$ for each $n \in \mathbb{N}$

$(a_n)_{n\in\mathbb{N}} :\left\{\begin{matrix} a_1 = 2 \\ a_2 = 4 \\ a_{n+2} = 4a_{n+1}+3a_n \end{matrix}\right.$

I am working on a proof of correctness for an algorithm I came up with. I encountered the following problem en route. I would appreciate if anyone had some idea or could point me to the relevant literature.

Consider a random variable $X$ distributed hypergeometrically with parameters $(n,m,i)$, i.e.,

$$p_X(x)=\frac{{i \choose x}{n-i \choose m-x}}{{n \choose m}},$$ where $\max(0,m+i-n)\leq x \leq \min(i,m)$.

*The simple question I am asking is are there any good lower bounds for $$\sum_x (p_X(x))^2$$ as a function of $n$ specifically?*

**My attempt and observations:**

Using tail bounds (see 1) for the hypergeometric distribution gives $$\sum_{x:|x-\mathbb E X|\leq\sqrt n}p_X(x)\geq 1-2e^{-2}=c.$$ Now we can use Cauchy-Schwarz to get something along the lines of: \begin{align*} \sum_x p_X(x)^2 &\geq \sum_{x:|x-\mathbb E X|\leq\sqrt n}p_X(x)^2\\ &\geq \frac{1}{2\sqrt n} c^2=\frac{c'}{\sqrt n}. \end{align*}

This bound turns out to be weak for my purposes. I also computed the quantity for $n\in \{3,4,5,...,1000\}$ and for all relevant $i,m$ and it looks like $\sum_x p_X(x)^2 \geq \frac{1}{\log^2 n}$. This solves my problem if it is indeed true for all $n$.

Any help/leads would be appreciated. Thanks!

This pic below is exploded view of a cone and I'm trying to calculate the Euler characteristic of the surface made from the segment *M*.

At first I thought Euler characteristic is 0 but they say it is 1.

exploded view of a cone (Sorry, I have added a link)

Sorry for my bad English. I'm Russian user. I am engaged in the decision of the homework, but I fell into a stupor when I could not solve this "elementary" mathematical example. Help me decide please. I can not do it for two days. Me remains 5 hours, did not even go to bed, before the assignment.enter image description here

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that

$$ 0<\lim_{\beta\rightarrow 1}(1-\beta)\sum_{n=0}^{\infty}\beta^nX_n <\infty. $$

I have already shown that the limit is finite if $\mathbb{E}\log_+(X_n))<\infty$, but I am having trouble showing the nonzero part. Does anybody have a reference or idea on how to prove this result?

Thank you in advance!

The first theorem of section 5.3. in Evan's PDE discusses approximating a function in $W^{p, k}$ by it's mollifications.

Suppose $k$ is a positive integer, $1\leq p <\infty$ and $U$ is an open subset of $\mathbb{R}^n$. We define $$ U_\epsilon = \{x\in U : d(x, \partial U) > \epsilon\} $$ Assume also that $u\in W^{k,p}$ and define $$ u^\epsilon = \eta_\epsilon \ast u \qquad \text{in } U_\epsilon $$ Then the claim is that $$ u^\epsilon \to u \quad\text{in }W^{k,p}_{loc}(U) $$

I can follow the entire proof except for one part: the author fixes an open set $V \subset \subset U$ and argues that $$ u^\epsilon \to u \quad \text{in } W^{k,p}(V). $$ For this statement to even make sense, we would need each $u^\epsilon \in W^{k,p}(V)$. However, I do not see why this is true. How do we know that the function $u^\epsilon$ has weak derivatives on $U_\epsilon$? I know that $u^\epsilon$ is smooth on $U_\epsilon$, but how does this guarantee the existence of a weak derivative?

I know that I am supposed to integrate by parts, but not all the assumptions required to do so are met. For instance, the boundary of $V$ (or $U_\epsilon$) might not be $C^1$.

Let $S$ be some base scheme, $H$ a finite flat group scheme over $S$, and $\alpha: \mu_p \to H$ a homomorphism of group schemes ($p$ a prime). Is the kernel of $\alpha$ necessarily flat over $S$?

(I know that kernels of general homomorphisms of FFGS $G \to H$ need not be flat, but I don't know of a counterexample when $G = \mu_p$.)

**Definition.** A closed subset $S$ of a topological space $X$ is called a *separator* between points $x,y\in X\setminus S$ if the points $x$ and $y$ belong to different connected components of $X\setminus S$. A separator $S$ is called an *irreducible* separator between $x$ and $y$ is $S$ coincides with each closed separator between $x$ and $y$ that is contained in $S$.

Using Kuratowski-Zorn Lemma it is easy to prove that each closed separator in a locally path-connected (or more generally locally continuum-connected) space contains a closed irreducible separator. Is the case result true without the local path connectedness?

**Question.** Does each closed separator between two points $a,b$ of a metrizable continuum contain an irreducible closed separator between $a$ and $b$?

I hope that the answer to this question should be know but somehow I cannot find in the books of Kuratowski and Nadler.

I am trying to use the Weibull Kernel (rather then Gaussian Kernel) in NW estimator, where Weibull Kernel is(image of Weibull kernel).After using this kernel in NW estimator, I've made the following expression. Is it correct??? Please provide your views about this, and point out any mistake.

$${\frac{\Gamma(1+h)}{h(\frac{x-x_i}{h })}}\times\left({\frac{{(\frac{x-x_i}{h}) \Gamma(1+h)}}{(\frac{x-x_i}{h})}}\right)^{(\frac{1}{h}-1)}\times \exp\left[-\left({\frac{{(\frac{x-x_i}{h}) \Gamma(1+h)}}{(\frac{x-x_i}{h})}}\right)^{\left(\frac{1}{h}\right)}\right]$$

Let $DM_{\text{gm}}$ be the category of Voevodsky´s geometric motives. Let $p,q\in \mathbb{Z}$ be integers with $p<0$.

Is it true that $$\text{Hom}_{DM_{\text{gm}}}(M_{\text{gm}}(X),\mathbb{Z}(p)[q])=0,$$ where $M(X)$ is the motive of a smooth scheme $X$ over a field $k$ and $\mathbb{Z}(p)$ is the Tate motive?

Let $n\ge 3$ and $X$ be a compact connected $n$-manifold (without boundary).

I need a reference to the following facts (which I believe are true at least in dimension $n=3$):

**Fact 1.** For every closed connected subset $A\subset X$ that can be embedded to $\mathbb R^{n-1}$ the complement $X\setminus A$ is connected.

**Fact 2.** For any closed subset $B\subset X$ whose connected components can be embedded to $\mathbb R$, the identity embedding $X\setminus B\to X$ induces an injective homomorphism $H_1(X\setminus B;G)\to H_1(X;G)$ in singular homologies with coefficients in some group $G$ (for example $\mathbb Z$ or $\mathbb Z/2\mathbb Z$).

**Remark.** The Alexander-Pontryagin Duality Theorem implies that Facts 1 and 2 are true if $X$ is the $n$-sphere. So, I need these facts for an arbitrary compact connnected $n$-manifold without boundary.

For $n\in N_+$, define f(n) to be that for any n-vertice graph G, if any two triangle in G don't have a common edge, then G has at most f(n) triangles.

Do we have some good estimates for f(n)?

By triangle removal lemma, we can prove for any $\varepsilon>0$, while n is large, we have $f(n)<\varepsilon n^2$.

(Intuitively, for a regular partition of G, if any triangle contributes an edge to a "low density part" (or some ignored part), then the "low density part" can not suffer so much edge. So there's some triangle which every edge contained in a "high density part", and then we'll get $cn^3$ triangles, which can't independent in edge.)

If we can prove that $f(n)<\frac{\varepsilon n^2}{ln\ n}$ for large n, then we can prove that there exists infinitely many triples of primes which forms an arithmetic sequence, in a combinatorial way.

Let $A\in\mathbb{R}^{n\times n}$ be a matrix having eigenvalues with strictly negative real part (in other words, $A$ is supposed to be Hurwitz stable). Let $\mathrm{tr}(\cdot)$ denote the trace operator and $I$ the identity matrix. Let $>$ be the standard partial order in the cone of positive definite matrices.

Consider the following optimization problem $$\tag{$\star$}\label{prob} \max_{\substack{X\in\mathbb{R}^{n\times n}\\ X>0,\ \mathrm{tr}(X)=1}} \mathrm{tr}\left(X(I+P)^{-1}\right), $$ where $P$ is the (unique) positive definite solution of the following Lyapunov equation $AP+PA^\top=-X$.

**My question.** Does the solution of \eqref{prob} admit a closed-form expression?

In particular, after running some numerical simulations, it seems that the solution of \eqref{prob} depends only on the eigenvalues of $A$ and not on its particular structure. In fact, in all the simulations that I carried out so far, I ended up with the same value of \eqref{prob} in case of matrices $A$ that share the same spectrum. However, this fact does not seem trivial to prove (and perhaps is not even true); so any help in clarifying this conjecture is greatly appreciated. Thanks!

Suppose $X=C_1\cup C_2\dots\cup C_N$, be a reduced but reducible curve, and $C_i$'s are $\mathbb{P}^1$. Then I think it is well known that every torsion free sheaf $\mathcal{F}$ with pure dimension one (which means that the support of all of the subsheaves of $\mathcal{F}$ has dimension 1), can be defined in a short exact sequence as,

$$0\rightarrow \mathcal{F}\rightarrow\mathcal{F}_{C_1}\oplus\mathcal{F}_{C_2}\oplus\dots\oplus\mathcal{F}_{C_N}\rightarrow T\rightarrow 0$$

where $T$ is a torsion sheaf supported on the intersection of $C_i$'s, and $\mathcal{F}_{C_i} = \mathcal{F}|_{C_i}/$torsion .

Then my question is if we have another pure dimension one sheaf $\mathcal{G}$, can I write the following short exact sequence?

$$0\rightarrow \mathcal{F}\otimes\mathcal{G} \rightarrow(\mathcal{F}\otimes\mathcal{G})_{C_1}\oplus(\mathcal{F}\otimes\mathcal{G})_{C_2}\oplus\dots\oplus(\mathcal{F}\otimes\mathcal{G})_{C_N}\rightarrow T\rightarrow 0$$

where $(\mathcal{F}\otimes\mathcal{G})_{C_i}$ is simply $\mathcal{F}_{C_i}\otimes\mathcal{G}_{C_i}$. In other words, is the following equality correct?

$$\mathcal{F}|_{C_i}/torsion \otimes \mathcal{G}|_{C_i}/torsion = (\mathcal{F}\otimes\mathcal{G})|_{C_i}/torsion$$

For $U_q(\frak{g})$ the Drinfeld--Jimbo quantum group, its category of representations is equivalent to the category of representations of $U(\frak{g})$, or equivalently the category of Lie algebra representations of $\frak{g}$. Both categories have an obvious monoidal structure, what is not obvious is if this is an equivalence of monoidal categories.

Edit: As Phil's comment below, this is not a monodial equivalence. As the linked answer says, the problem is that the associators in both cases are different. What is the easiest way to see that this is so? The discussion about 6j symbols and coordinates on stacks is unfortunately lost on me. In fact, for modules over a Hopf algebra the assoicators look easy to the naive untrained eye, why is it about the assoiators on either category that is non-trivial?

Many years ago, I was surprised to learn that Andrew Odlyzko does not consider the existing computational evidence for the Riemann hypothesis to be overwhelming. As I understand it, one reason is as follows. Define the Riemann–Siegel theta function by
$$\vartheta(t) := -\frac{t}{2}\log\pi + \arg \Gamma\left(\frac{2it+1}{4}\right)$$
and define the Hardy function
$Z(t) := e^{i\vartheta(t)} \zeta(1/2 + it)$. Then $Z(t)$ is real when $t$ is real, and the Riemann hypothesis implies that for $t$ sufficiently large, $Z(t)$ has no positive local minimum or negative local maximum, so its zeros are interlaced with its minima and maxima. On the other hand, as I believe Lehmer ("On the roots of the Riemann zeta-function," *Acta Mathematica* **95** (1956), 291–298) was the first to point out, there exist "Lehmer pairs" of zeros of $Z(t)$ that are unusually close together, which may be regarded as "near counterexamples" to the Riemann hypothesis. Harold Edwards has suggested that Lehmer pairs "must give pause to even the most convinced believer in the Riemann hypothesis."

There is a relationship between Lehmer pairs and large values of $Z(t)$. It is known that $Z(t)$ is unbounded, but it approaches its asymptotic growth rate very slowly. As Odlyzko has explained (see Section 2.9 in particular), there is reason to believe that current computations are not yet exhibiting the true asymptotic behavior of $Z(t)$. So one could argue that the existing computational data about Lehmer pairs is still in the realm of the "law of small numbers."

A related observation concerns $S(t) := \pi^{-1}\arg\zeta(1/2 + it)$. Let me quote from Chapter 22 of John Derbyshire's book *Prime Obsession*, where among other things he reports on a conversation he had with Odlyzko.

For the entire range for which zeta has so far been studied—which is to say, for arguments on the critical line up to a height of around $10^{23}$—$S$ mainly hovers between $-1$ and $+1$. The largest known value is around 3.2. There are strong reasons to think that if $S$ were ever to get up to around $100$, then the RH might be in trouble. The operative word there is "might"; $S$ attaining a value near $100$ is a *necessary* condition for the RH to be in trouble, but not a *sufficient* one.

Could values of the $S$ function ever get that big? Why, yes. As a matter of fact, Atle Selberg proved in 1946 that $S$ is unbounded; that is to say, it will *eventually*, if you go high enough up the critical line, exceed any number you name! The rate of growth of $S$ is so creepingly slow that the heights involved are beyond imagining; but certainly $S$ will eventually get up to $100$. Just how far would we have to explore up the critical line for $S$ to be that big? Andrew: "Probably around $T$ equals $10^{10^{10,000}}$." Way beyond the range of our current computational abilities, then? "Oh, yes. *Way* beyond."

In light of what I learned from another MO question of mine, about fake integers for which the Riemann hypothesis fails, I got to wondering—If exploring $Z(t)$ and $S(t)$ for the actual Riemann zeta function is hitting our computational limits, could we perhaps gain some insight by computationally studying other zeta functions? More specifically:

Are there $L$-functions in the Selberg class for which there are analogues of $Z(t)$ and $S(t)$ which are computationally more tractable than the Riemann zeta function, for which we could computationally explore the analogue of the "$S\approx100$" regime? (Incidentally, I don't understand what is significant about the $S\approx100$ regime. Anybody know?)

Are there Beurling generalized number systems for which the analogue of RH fails but which can be shown computationally to mimic the empirically observed behavior of $Z(t)$ and $S(t)$ (including, I guess, the GUE phenomenon)?

If $(X,\tau)$ is a topological space, let $FH(X)$ denote the collection of $x\in X$ such that there is a non-identity homeomorphism $\varphi:X\to X$ with $\varphi(x) = x$.

What is an example of a $T_2$-space $(X,\tau)$ such that $FH(X)$ is dense in $X$, but $FH(X)\neq X$?

Can any mathematical model be represented as an algorithm? I think that for constructive mathematics it is possible, but for the classical it is impossible. As far as I understand, constructive mathematics makes it possible to construct a mathematical object. So represent the mathematical model (as the set of mathematical objects with relathionships) in the form of an algorithm. I mean model as a simplified description, especially a mathematical one, of a system or process, to assist calculations and predictions

"Mathematicians usually have no problems with infinite structures or non-constructive proofs (that show that something exists, bit do not show you how to find/construct it). Thus a model can be quite useless for algorithms but useful for explaining things. On the other hand, an algorithm always is some kind of model. So the "model" concept is probably wider."

I want to solve the following non linear matrix equation for $X\in\mathbb{R}^{N\times N}$:

\begin{equation} XX^{\top}+ABX^{\top}-A=0 \qquad (1) \end{equation}

For a given matrices $A\in\mathbb{R}^{N\times N}, B\in\mathbb{R}^{N\times N}$ and $A=A^{T}, A\succeq0$.

Is there a known numerical solution for (1)?

If not, assuming that the solution is symmetric, $X=X^{\top}$, we get the following matrix equation:

\begin{equation} X^{2}+ABX-A=0 \qquad (2)\\ X=X^{\top} \end{equation}

Is there a known numerical solution for (2)?

I know that eq.(1) and eq.(2) must have a solution (due to basic optimization considerations).

Thanks.

**EDIT:**

(1) Emitted stability requirement.

(2) B is a low rank matrix.

(3) My attempt: I used the necessary condition (As Robert suggested): \begin{equation} ABX^{\top}=XB^{\top}A \qquad (3) \end{equation}

to get the following equation:

\begin{equation} XX^{\top}+\frac{1}{2}ABX^{\top}+\frac{1}{2}XB^{\top}A-A=0 \qquad (4) \end{equation}

Now we can use Riccati to find a symmetric solution. However, not every solution of (4) is a solution of (1).

Let $\square=[0,1]\times[0,1]$ be the unit square and $f\colon\square\to \square$ is a continuous map that fixes the points on the boundary.

Assume $f$ is a limit of homeomorphisms $\square\to \square$. (By Moore's theorem it is equivalent to the condition that for any point $p\in \square$ the inverse image $f^{-1}\{p\}$ is connected and its complement is connected.)

Is it true that for most segments in $\square$, their inverse images are Jordan arcs?

Say, given $s\in [0,1]$ consider the vertical unit segment $I_s$ defined by $x=s$. Is it true that for a dense G-delta set of values $s\in [0,1]$ the inverse image $$J_s=f^{-1}I_s$$ is a Jordan arc?

**Comments:**

If $f$ is injective on $J_s$ then $J_s$ is a Jordan arc (evident). One may expect that $f$ is injective for most of values $s$, but this is not the case --- even if all $J_s$ are Jordan arcs, the map $f$ might map an arc in $J_s$ to one point for all $0<s<1$.

For sure $J_s$ has vanishing measure for all but countable set of values $s$.

The problem can be reformulated in terms of decomposition of $\square$ into the inverse images $f^{-1}\{x\}$. The main trouble comes from the sets $f^{-1}\{x\}$ that are not Jordan arcs nor single points. I do not know examples with uncountably many such sets. (I alos do not know if one can find uncountably may connected but not path connected compact disjoint sets in the plane.) These problems are related to the problem that there at most countable set of Y shapes in the plane, see "Ys in the plane" in "Mathematical Puzzles" of Peter Winker.