lemma: For every list of powers of consecutive numbers, prove that at least one of the numbers is co=prime to their sum.

Please prove this lemma.

So we are given an infinite supply of bulbs, following Exp($\lambda$)

You arrive at time t, note the bulb that is burning at time t.

$L_t$ = Lifetime of the bulb we noted

$A_t$= Lifetime of the noted bulb spent until time t

$B_t$ = Remaining lifetime of the noted bulb

I was able to calculate the distribution of the $L_t$, which came out as

P[$L_t \leq x$] = $1-((x\lambda + 1)* e^{-\lambda x}$ for $0<x\leq t$ and 1-(($t\lambda + 1)e^{-\lambda x}$ for $x \geq t$

I know $B_t$ follows Exp($\lambda$), which is the paradox.

How to calculate the distribution of $A_t$?

I was trying to use change of variable but because of dependent variable I was unable to proceed

Find each of the following quantities exactly using csc(a) = -3 and tan(b) = -7 such that b is a quadrant 11 angle and that -pi < a < -pi/2

Is it true that every tournament is contained in some vertex-transitive tournament? If not, is it known which tournaments satisfy this property? This seems like a basic question, but I have not been able to find a relevant reference.

The following extension of the Jordan Curve Theorem is well known: every closed connected hypersurface of the sphere $\mathbb S^N$ separates $S^N$ into exactly two connected components. As a consequence, every compact connected hypersurface of the euclidean space $\mathbb R^N$ separates $\mathbb R^N$ into exactly two connected components.

Does this result keep its validity if the word `compact' is replaced by *closed*? With other words, is it true that every closed connected hypersurface of the euclidean space $\mathbb R^N$ separates $\mathbb R^N$ into exactly two connected components?

I am interested in (as explicit as possible) descriptions of non-Gorenstein integral projective curves. Most of the literature on singular curves appears to be focused around the Gorenstein case, with a notable exception being the paper of Kleiman, *The Canonical Model of a Singular Curve*, and a handful of others. Any other references would be appreciated. Specifically I am interested in knowing what has been understood about their derived categories of coherent sheaves (or the subcategory of perfect complexes), so I would very interested in any developments along those lines.

There are some estimates about initial value problem for example Kato's $H$-smoothness and Strichartz's estimate. These estimates roughly say the solution $u$ of some PDE is bounded for initial value, say $u_0$, in the $L^p$-norm sense. That is, $$\||\nabla|^s u\|_{L^p} \le \|u_0\|_{L^q} \quad and \quad \|u\|_{L^p}\le \|u_0\|_{L^q} $$ for some $s>0$ and $1/p +1/q = 1$ for $p>1$.

I hope know that meaning of bounding for norm of initial value. In addition, meaning of regularity, $|\nabla|^s$ also.

I saw this result in *A Model Category Structure for Differential Graded Coalgebras* by Getzler-Goerss, but when the coalgebra is non-negatively graded, is this property also satisfied when the dg coalgebra is $\mathbb{Z}$-graded?.

Thanks.

Let $(X,\mathcal T)$ be a topological space. About the subsets $A,B,C$ of $X$ it is known that $$\mathrm {cl} (C)= A \cup B\,, \quad \mathrm {cl} (A) = A\,, \quad \mathrm {cl} (B) = B\,.$$

Does it always imply that there exist two subsets of $X$, $C_A$ and $C_B$ such that $$C = C_A \cup C_B\,, \quad \mathrm {cl} (C_A) = A\,, \quad \mathrm {cl} (C_B) = B \; ?$$

If this is not true in general, is it true at least for Euclidean spaces?

**Preliminaries**

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a complete probability space.

Let $D$ be a complete, separable, metrizable topological space with Borel $\sigma$-algebra $\mathcal{B}(D)$ (such as $D = \mathbb{R}^q$ with $\sigma$-algebra $\mathcal{B}(D) = \mathcal{B}(\mathbb{R}^d)$).

Let $\mathbb{R}$ be equipped with its canonical Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$.

Let $g: \Omega \times D \rightarrow \mathbb{R}$ be a bounded $(\mathcal{G} \otimes \mathcal{B}(D) ) / \mathcal{B}(\mathbb{R})$-measurable function.

Let $\Pi: \Omega \rightarrow D$ be a $\mathcal{G}/\mathcal{B}(D)$-measurable random variable.

Let $H : \Omega \rightarrow \mathbb{R}$ be a $\mathcal{G}/\mathcal{B}(\mathbb{R})$-measurable random variable, defined by $$ H(\omega) := g(\omega, \Pi(\omega)).$$ Note, that, since $g$ is bounded, we have $H \in \mathcal{L}^2(\Omega, \mathcal{G}, \mathbb{P})$.

Let $j: D \rightarrow \mathcal{L}^2(\Omega, \mathcal{G}, \mathbb{P}) $ be defined by $$ j(\pi)(\omega) := g(\omega, \pi) $$

For all $\pi \in D$, let $j(\pi)$ be independent of $\Pi$.

**Question**

I am interested in the conditional expectation $$\mathbb{E}[H \mid \Pi] :\Omega \rightarrow \mathbb{R}$$ of $H$ with respect to $\Pi$. More specifically, I suspect that (a $\mathbb{P}$-unique version of) this condititional expectation is given by

$$ \mathbb{E}[H \mid \Pi] (\omega) = \mathbb{E}[j(\Pi(\omega))], \quad (\dagger) $$ whereby $\mathbb{E}[j(\Pi(\omega))]$ can of course also be written as $$\mathbb{E}[j(\Pi(\omega))] = \int_{\Omega} j(\Pi(\omega))(\tilde{\omega}) d\mathbb{P}(\tilde{\omega}) . $$

How can I prove, that $(\dagger)$ is the case? I have tried, tracking the definition of conditional expectation and using Fubini, but with little success so far.

Thanks for any advice!

Consider the extension

$$1\to SU(2)\to X\to O\to1,$$

there are 4 possibilities for $X$: $X=O\times SU(2)$ or $E\times_{\mathbb{Z}_2}SU(2)$ or $Pin^+\times_{\mathbb{Z}_2}SU(2)$ or $Pin^-\times_{\mathbb{Z}_2}SU(2)$ where $E$ is defined in Freed-Hopkins's work1 as the colimit of the group $E(d)$, the group $E(d)$ is defined to be the subgroup of $O(d)\times\mathbb{Z}_4$ consisting of the pairs $(A,j)$ such that $\det A=j^2$, where $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ is the multiplicative group of order 4.

Here the notation $G_1\times_{\mathbb{Z}_2} G_2 :=\frac{G_1\times G_2}{\mathbb{Z}_2} $ is defined as mod out the common $\mathbb{Z}_2$ of $G_1\times G_2$.

The question is about computing $MT(E(d)\times_{\mathbb Z_2} SU(2))$ and the bordism group $\Omega_d^{E \times_{\mathbb Z_2}SU(2)}$.

(1) There is a short exact sequence of groups: $1\to SO(d)\to E(d)\to\mathbb{Z}_4\to 1$. So naively, people may suspect that $$MT(E(d)\times_{\mathbb Z_2} SU(2))=MT E(d)\wedge\Sigma^{-3}M SO(3)=MSO(d)\wedge\Sigma^{-2}M\mathbb Z_4\wedge\Sigma^{-3}M SO(3).$$ However, this is likely to be incorrect.

(2) The space $B(E \times_{\mathbb Z_2}SU(2))$ sits in a homotopy pullback square: a map $M \to B(E \times_{\mathbb Z_2}SU(2))$ is determined by two maps $M \to BO$ and $M\to BSO(3)$ which correspond to bundles $TM$ and $V_{SO(3)}$ such that $w_1(TM)^2=w_2(V_{SO(3)})$.

To compute the bordism group $\Omega_d^{E \times_{\mathbb Z_2}SU(2)}$, we need to know the Madsen-Tillmann spectrum $MT(E \times_{\mathbb Z_2}SU(2))$ and decompose it as the wedge sum or smash product of some familiar spectra.

The attachment is the author's attempt,

but the map $f$ is not a homotopy equivalence. I actually obtain an identification $$ \text{Thom$(B(E \times_{\mathbb Z_2}SU(2)),-2V)=MT(Pin^+ \times_{\mathbb Z_2}SU(2))$} $$ which is already known in 1604.06527 paper, but we need to know $$ \text{Thom$(B(E \times_{\mathbb Z_2}SU(2)),-V)=MT(E \times_{\mathbb Z_2}SU(2))$,}$$

where $V$ is the induced virtual bundle of dimension 0 by $B(E \times_{\mathbb Z_2}SU(2)) \to BO$.

- Is Thom$(B(E \times_{\mathbb Z_2}SU(2)),-2V)$=smash product of Thom$(B(E \times_{\mathbb Z_2}SU(2)),-V)$ and Thom$(B(E \times_{\mathbb Z_2}SU(2)),-V)$? If so, how to obtain Thom$(B(E \times_{\mathbb Z_2}SU(2)),-V)$ as the "square root" of Thom$(B(E \times_{\mathbb Z_2}SU(2)),-2V)$?

1 Reflection positivity and invertible topological phases Daniel S. Freed, Michael J. Hopkins, arXiv:1604.06527

Supposing I have complex square matrices $B_i$ and $C_i$ ($i = 1,\dots,N$) of dimension $4 \times 4$.

- Is there an effective algorithm for solving the following problem?

$$\begin{align} A=&\underset{A\in\mathbb{C}^{4\times 4}}{\text{argmin}}\sum_{i=1}^{N}{\left\|AB_iA^{-1}-C_i\right\|_2}\\ \,\\ &\text{subject to }\left\{\begin{matrix}A_{12}=A_{13}=A_{42}=A_{43}=0\\A_{11}+A_{41}=1\\ A_{14}+A_{44}=1\end{matrix}\right. \end{align} $$

- How can I make sure that there exists only one unique solution for $A$?

Let $(X,d)$ be a metric space. Suppose that $\{A^n\}_{n \in \mathbb{N}}$ is a sequence of closed, non-empty subsets of $X$.

Is there a Hausdorff topology on the space of closed subsets of $X$, guaranteeing that if $A^n$ converges in this space to a $A\subseteq X$, then for any continuous function $f:X \rightarrow \mathbb{R}$, we have that $$ \sup_{x \in A}f(x)\leq \sup_{n \in \mathbb{N}}\sup_{x \in A^n}f(x) ? $$

Did someone develop ZFC by means of ZF plus axioms for a binary well-ordering constant, say $\blacktriangleleft$? Are there results that suggested accounts are conservative extensions of ZFC?

Let $F: C \to D$ be an infinity functor. Is it true that the homotopy fiber at $y$ can be described as $C \times_D D^{\simeq}_{/y}$? If not, is there a simple formula resembling this one?

Beside the infinity structure, its points are pairs $(x \in C, s: F(x) \to y \text{ equivalence})$. I heard this in a class but never seen a proof. I came now with a weird fact that seems a consequence of this fact and I would like to have a confirm.

Thanks, Andrea

Let $k$ be a field of characteristic zero and $R$ a local $k$-algebra. By Stacks \tag 00TX, if we assume that $R$ is of finite type, then the $R$ is smooth over $k$ if and only if $\Omega_{R/k}$ is free. I'm wondering if the analogous statement is true for formal smoothness, i.e is the formal smoothness of $R$ over $k$ equivalent to the freeness of $\Omega_{R/k}$?

My solution would be to use Stacks project 031I. We choose a polynomial ring P and a surjection $P\rightarrow R$ and denote by $J$ the kernel of $P\rightarrow R$. Then we have to show that the sequence $ (*) \ J/J^2 \rightarrow \Omega_{P/k}\otimes_P R \rightarrow \Omega_{R/k} \rightarrow 0$ is left exact and splits. If we can show that it is left exact, then by the projectivity of $\Omega_{R/k}$, we are done. By Stacks 00S2, we know that $ 0 \rightarrow H_1(L_{R/k}) \rightarrow J/J^2 \rightarrow \Omega_{P/k}\otimes_P R$. In http://www1.mat.uniroma1.it/people/manetti/dispense/defosing.pdf, it is writen (page 20, Def 9.2), that if $R$ is dg-algebra, and $M$ an $R$-dg-module ( and $k$ of characteristic zero!), then $H_i(L_{R/k}\otimes_R M)=\text{Tor}_i^R(L_{R/k},M)$. If we assume that $R$ is a dg-algebra, then for $i=1$ and $M=R$ we get that $H_1(L_{R/k})=0$. Hence the sequence (*) is left exact and we are done. However I'm not confident in this solution.

I have two question:

1) Are there any examples of complete manifold with strictly positive and bounded section curvature which has zero injectivity radius?

2) Is there a sequence of non-compact complete manifolds with strictly positive and bounded section curvature with injectivity radius approach to zero?

I think one may construct these examples from Beger's spheres, but I cannot do it rigorously.