Moment generating function of Maxima $E[t \exp(\max(X,Y))]$

Let $ X,Y$ be two standard normal random variables with correlation coefficient $ r\in[-1,1]$ .

I’m trying to compute the moment generating function of the maxima using Mathematica:

Expectation[Exp[t Max[X, r X + Sqrt[1 - r^2] Y]],             {X \[Distributed] NormalDistribution[],              Y \[Distributed] NormalDistribution[]}] 

Since the distribution of the maximum is known to follow the Skew Normal Distribution, I can easily compute the expectation myself to be

$ $ e^{t^2/2} (1 + \text{erf}(\sqrt{1-r}\,t/2),$ $

but Mathematica seems to get bogged down.

I have some slightly more complicated expectations I would like to evaluate, so I wonder if there is some way I may help Mathematica solve things like this? I’m using Mathematica 12.0.0, and I’ve tried adding $ Assumptions = -1 < r < 1.

For the Purpose of Detect Magic and Identify. Would a Magic item that ran out of power for the moment be considered a magic item?

If a magic item runs on "Magic Batteries" when the Item is out of power would it be considered a magic time for the purpose of the Identify and Detect Magic spell?

For example, I have a Mirror that shows through illusions, it has crystals around it that are destroyed as you used it, you can replace the crystals for more uses when it has no crystals, would it still be considered a magic item detectable by the spells?

Would the same work for items with charges that recharge at dawn? If the item has 0 charges and is not dawn yet, would it be still be considered a magic item?

Does the Bard Spell Fugue have any effect at the moment of cast?

On the Spell Compendium (p 86) there is the spell Fugue, which has interesting language to describe its effect. The spell description starts with:

Creatures that fail their save become affected by the haunting fugue in semirandom ways. On each affected creature’s turn (as long as it remains in the affected area), you make a Perform check….

Does that mean the spell doesn’t do anything on my turn when it is cast? If so, do my opponents get the opportunity to use their actions before this takes effect on their turn?

What happens at the moment I no longer meet multiclass prerequisites during gameplay?

Suppose I’ve got a multiclass fighter/cleric with both wisdom and strength of 16. Nice.

I fall victim to a shadow’s strength drain attack. Not nice. I roll a 4 and lose 4 points of strength.


I no longer meet the prerequisite for my multiclass. Do I lose my fighter class features until after my next rest, when that strength comes back?

life changing moment essay

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life changing moment essay

The norm squared of a moment map

I am studying the paper by E. Lerman:

Let $ (M,\sigma)$ be a connected symplectic manifold with an hamiltonien action of a compact Lie group $ G$ , so that there exist a moemnt map $ $ \mu : M\to\mathcal{G}^\ast$ $ $ \mathcal{G}^\ast$ being the dual of the Lie algebra of $ G$ . We assume that $ \mu$ is $ G$ -equivariant: $ $ \mu(g\cdot x)=\mathrm{Ad}_g^\ast\circ\mu(x)$ $ and that $ \mu$ is proper (the preimage of any compact is compact). Let $ f=\|\mu\|^2$ (for an Ad-invariant norm on $ \mathcal{G}^\ast$ ).

I know that the moment map is important by:

1- a convexity theorem of Atiyah and Guillemin-Sternberg.

2- symplectic reduction, where the quotient of the zero level of the moment map by the group makes it possible to construct new symplectic manifolds.

Hence my question:

What is the motivation to study the norm squared of a moment map? in particular, why is it important to know that the zero level set of the moment map is a retract by deformation of a piece of the manifold?

As I understand it, $ f$ behaves like a Morse-Bott function (Kirwan works) and that the stable manifold of a critical component of $ f$ is a submanifold. That the gradient flow of $ f$ is defined for all $ t\geq0$ . Here Lerman asserts that this is true because $ f$ is proper, but $ x^3$ is proper but its gradient $ -3x^2\partial_x$ is not defined for all $ t\geq0$ .

I think we have to show that $ \nabla_f$ is $ G$ -invariant and therefore complete.

That $ f$ is real analytic to show that the limit of a trajectory of any point $ \phi_t(x)$ is reduced to a point $ \phi_\infty(x)$ . That the applications $ t\to\phi_t(x)$ and $ x\to\phi_\infty(x)$ are continuous.

Properness of moment map

Suppose that a torus $ T$ acts on a non-compact manifold $ M$ . Assume that this action is Hamiltonian and that the fixed point set of $ T$ is compact. Let $ \mu:M\to\mathfrak{t}^{*}$ denote the moment map of the action, where $ \mathfrak{t}$ denotes the Lie algebra of $ T$ .

If there exists an $ X\in\mathfrak{t}$ such that $ \mu(X):M\to\mathbb{R}$ is a proper function that is bounded below, why is $ \mu:M\to\mathfrak{t}^{*}$ necessarily proper?

Why does Brussels airport not inform passengers about gates details? (until the last moment)

I was at Brussels airport yesterday and was surprised that gates information are not published until (exactly) 15 minutes before boarding time. The panel says

Relax, gate info at 16:20

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So I went to the gate as soon as it was known, and the aircraft was already there, connected to the passenger boarding bridge. Meaning the gate was definitely known for some time before it is published!

Then I queried some information from the staff ; and indeed, gate information is always only published 15 minutes before boarding time.

The only reason I could think of is BRU wants people to spend time in shops…

Is there another reason for not informing gate details earlier? (which is a bit annoying)

At which moment does the ‘Surprised’ state disappear?

I am trying to figure out if the surprised state ends on a creature after the first attack of a surprise round, or if the surprised state (which I would then see almost as an unofficial condition) ends only after the first round of combat.

In the second case, I would believe that a character (having the rogue’s Assassinate ability) attacking a surprised creature AND having multiple attacks would score a critical hit on all hits.

Example: Does a Monk 5/Rogue 3 surprising a creature get automatic critical hits on all hits if he does 2 attacks (with extra attack) and additional unarmed strike(s)?

Or does the critical hit only affect the first hit? However, that would surprise me as the rules mention any hit is a critical hit.