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Q: Computing the expected number of people in a stampede An overcrowded stadium is so full that there are roughly 3.2 people per square meter. At some point, 20 people in a grid stampede in one direction, doing so on the square of area $12a$ meters. The stampede spreads out smoothly over one second; when it has reached its maximum density, the stampede is reversed. An extremely biased individual wanders into the stadium and builds some barrier at random at some point $(x,y)$ within the square, and at a density $\lambda$. If the barrier completely stops the stampede, what is the expected number of people in the stadium when the stampede is reversed? So far I’ve drawn the regions I’m thinking about. I’m thinking that the chance that the stampede proceeds further than the barrier, and thus stops at the barrier is $1-\exp(-\lambda 12a)$, but I’m not sure where to go from there. Thanks! A: Let’s call $S$ the stampede, and $I$ its initial distribution, then the probability of $S$ stopping at the barrier is given by $P_\lambda(S\subseteq \bar I) = P_\lambda(S\cup I \subseteq \bar I) = 1-e^{ -\lambda\mathcal A}$, where $\mathcal A$ is the area of the barrier. Now we want to find the expected number of people in the stadium when the stampede is reversed, so we have to find $E[S]$ (the expectation of the random variable $S$) as a function of $E[S]$, and then take the derivative. In general, $E[X]$ is computed as $$E[X] = \int_\Omega X(\omega)\,d\omega$$ where $\Omega$ is the sample space, so in this case the sample space is $\Omega=\{1,2,\ldots,\infty\}$ as we want the stampede to be infinite, and the probability of each outcome is determined by $P_\lambda(X=n) = \frac{\lambda^n}{n!}$ for $n\in\mathbb N$. From that we have \$E[X] =

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Engineered bifunctional imine-based disulfides for redox-induced biological response. [reaction: see text] The design and synthesis of new, engineered bifunctional disulfides is described. The parent bifunctional peptide, S-N(CH2)6-CO-C(=NH)-NH-C(=S)-S-N(CH2)6-CO-OMe, was found to promote phagocytosis by J774A.1 cells. The C-terminal S-N(CH2)6-CO-C(=NH)-NH-C(=S) motif was altered by the introduction of a variety of structurally unique cysteines to form S-N(CH2)6-CO-C(=NH)-NH-C(=S)-C(SH)(X), where X = F, Cl, Br, I, NCS, or COCF3. The absolute stereochemistry of peptidyl-S-prolin-X was altered to create conformationally restricted analogues. In addition, the Aib residue at the C-terminus was incorporated to form a highly constrained peptide analog (R)Phe-S-N(CH2)6-CO-C(=NH)-NH-C(=S)-C(S)Aib-S-N(CH2)6-CO-OMe. The most effective analogue of the above peptides, S-N(CH2)6-CO-C(=NH)-NH-C(=S)-S-N(CH2)6-CO-F, was designed to include the most favorable polarities for attachment to cellular ligands. S-N(CH2)6-CO-C(=NH)-NH-C(=S)-F was found to enhance the phagocytosis of J774A.1 cells by up to a factor of 5.6 relative to a control peptide (Boc-S-N(CH2)6-CO-C(=NH)-NH-CO-OMe) lacking cysteine. to the problem. —— exabrial I have a great idea to solve many of these problems, it’s called the internet. —— NKCSS I started my own SSO and from what I understand the problem is still unsolved.

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