Characterization of summable families

Could somenone help me with to prove the following result?

If $ C=\{\lambda\in{\Lambda}:x_\lambda\neq{0}\}$ is countable and for every bijective mapping $ \tau:\mathbb{N}\longrightarrow{C}$ $ \sum_{n\geq{1}}x_{\tau(n)}$ is a convergent serie, then $ \{x_\lambda:\lambda\in{\Lambda}\}$ is a summable family.

I have proved the contrary implication, but I dont’t know how to prove this one.

Thanks.

Connected components in flat families

Let $ f:X\rightarrow S$ be a proper flat family of schemes with non-empty geometrically reduced fibers and $ S$ Noetherian integral scheme. Is it true that $ X$ has finitely many subschemes such that the restriction of $ f$ to each of them is a proper and flat family of schemes over $ S$ with non-empty geometrically connected and geometrically reduced fibers?

Behavior of L-function in families of elliptic curves

I’m curious about how the L-functions of elliptic curves behave as the elliptic curves vary in families. In other words, if we regard the L-function $ L(E_{/K},s)$ of an elliptic curve $ E$ over a number field $ K$ as a function on the moduli space of elliptic curves over $ K$ , $ \mathcal{M}_{E/K}$ , in the first variable, what can be said about its properties, both for fixed values of the second variable $ s$ , and as a two-variable function on $ \mathcal{M}_{E/K} \times \mathbb{C}$ ?

The question has a natural extension to other moduli spaces of arithmetic varieties.

References to where it is discussed in detail would be appreciated.

Google Families Notifications

I am trying to turn off email notifications so that i dont get an email each time my son installs something. I have gone to all the steps guided in the support link:

https://support.google.com/families/answer/7184159?hl=en

Manage notifications When you manage your child’s Google Account, you can decide if you want to get notifications through email and the Family Link app.

Note: Certain notifications can’t be turned off.

Open the Family Link app Family Link. In the top left, tap Menu Menu and then Notifications settings. Tap the type of notification you would like to change. For each child, turn notifications on or off. You can also change your notification settings from the web:

Visit families.google.com. Click Menu Menu and then Notifications settings. For each child, turn notifications on or off.

The issue is when i go to the “Manage Notifications” there is no place to change these settings. only the below view: (only the “Learn more” is hyperlinked. Same in the actual app itself.)

enter image description here

Any help would be great thanks.

Eigenvalues of products of exponential families

I have a question about a close cousin of the multiplicative eigenvalue problem.

Let $ U$ be a special unitary matrix with diagonalization $ D = \operatorname{diag}(e^{2 \pi i a_1}, \ldots, e^{2 \pi i a_n})$ . The $ a_j$ may be normalized so as to satisfy $ a_1 \le a_2 \le \cdots \le a_n \le a_1 + 1$ and $ a_1 + \cdots + a_n = 0$ . These extra conditions have the advantage of producing a canonical sequence of logarithms: we may define a function $ \operatorname{LogSpec}$ by $ $ \operatorname{LogSpec} U = (a_1, \ldots, a_n).$ $

It also has the disadvantage of being not smooth. Given a point in $ \mathbb R^n$ satisfying only the equality $ a_1 + \cdots + a_n = k$ for $ k \in \mathbb{Z}$ , this point can be moved into by the region satisfying the family of inequalities (without modifying its image through $ t \mapsto e^{2 \pi i t}$ ) by repeated reflection. Let’s call this assignment $ R$ , as in $ $ R\colon\thinspace \left\{ a_* \in \mathbb R^n \mid a_1 + \cdots + a_n = 0\right\} \to \left\{a_* \in \mathbb R^n \middle| \begin{array}{c} a_1 + \cdots + a_n = 0, \ a_j \le a_{j+1}, \; a_n \le a_1 + 1 \end{array} \right\}.$ $ By consequence, curves like $ $ \gamma(t) = \operatorname{LogSpec} \exp\left(\begin{array}{cccc} it & 0 \ 0 & -it \end{array}\right),$ $ which are smooth in $ SU(2)$ before postcomposition with $ \operatorname{LogSpec}$ , become a kind of sawtooth function. For ease of reference below, I’ll call the image of a convex set through $ R$ folded-convex.

Question: I would like to know a reference for (or, indeed, a proof of) the following result:

$ \DeclareMathOperator{\LogSpec}{LogSpec}$ ??Theorem??: Let $ \xi_1, \ldots, \xi_m$ be a sequence of $ n \times n$ anti-Hermitian matrices, each exponentiating to a closed subgroup of $ U(n)$ . The assignment $ $ (t_1, \ldots, t_m) \mapsto \LogSpec \left( \prod_{j=1}^m \exp(\xi_j t_j) \right)$ $ sends convex sets in $ \mathbb R^m$ to folded-convex sets in $ \mathbb{R}^n$ .

In the classical version of the multiplicative eigenvalue problem, the set $ $ L_{m,n} = \left\{(\LogSpec U_j)_{j=1}^m \in \mathbb{R}^{n \cdot m} \middle| \begin{array}{c} \text{$ U_j$ unitary}, \ U_1 \cdots U_m = 1 \end{array}\right\} \subseteq \mathbb{R}^{n \cdot m}$ $ is shown to be convex by a clever application of symplectic reduction. The method of proof in Meinrenken and Woodward’s A symplectic proof of Verlinde factorization involves giving an explicit model for the moduli of flat connections on the trivial $ U(n)$ –bundle over a punctured Riemann sphere, then using its symplectic structure and a symplectic convexity theorem (suitably augmented to cope with loop groups) to deduce the convexity of $ L_{m,n}$ .

Their methods are especially well-suited to dealing with formulas like $ $ 1 = \operatorname{Ad}_{c_1}(t_1) \cdots \operatorname{Ad}_{c_m}(t_m),$ $ where $ t_j \in \mathfrak t_j \subseteq \mathfrak{su}(n)$ are anti-Hermitian diagonal and $ c_j \in SU(n)$ are special unitary. I’m new to this material and to symplectic geometry broadly, and so I’ve been unable to tweak these methods into saying something about this more restricted problem, where there are far fewer adjoint actions in play. Despite that, this seems like the kind of problem that would have attracted classical attention, and so I’m hopeful that there exists a resource that works this out. I’m also happy to hear about adjacent results—maybe I can make do with one of them.

Caveat lector: the theorem seems true in numerical experiment, but without a proof, there may well be edge cases unaccounted for. I’d be very, very happy to hear about those.

[ Christmas ] Open Question : Is it wrong to feel irritated that I bought all my families Christmas gifts and my gf wants to put her name on the tags with mine?

I got presents for my family and she didn’t help pay for any of them and is putting her name on the tag with mine.. I get we are together but she isn’t doing the same for me for her family and I said something and she said are you gonna help me buy my families gifts? I feel this is unfair and also she puts her name on my Christmas cards but won’t put mine on hers. I’m just a bit irritated that I’m paying for these gifts and she didn’t help and is making it seem like she did and I don’t want my family getting her something if she can’t even get them something. That’s just how I feel

How many families constitue a Major Clan in Legend of the Five Rings

In Legend of the Five Rings, a list of families is presented for each Major Clan. There are some in the source book (usually 4 or 5 for each Clan), and some others in additional books.

However, does that mean that every samurai in a Major Clan belong to those families? Or does that mean that those families are the “big ones”, but they are several other families in the Clan?

i.e. can I make up a NPC last name at the top of my head without having to check the validity of the name?

I have read through all the lore information in Legends of the Five Rings 4th Edition core book, but could not find the answer to this.