DSolve does not return a solution when initial condition is added

Consider the following equation $ $ y”(x) = -2e^{-y}. $ $

The following code

DSolve[y''[x] == -2 Exp[-y[x]], y[x], x] //FullSimplify 

returns

{{y[x] -> Log[(2 (-1 + Cosh[Sqrt[C[1]] (x + C[2])]))/C[1]]},   {y[x] -> Log[(2 (-1 + Cosh[Sqrt[C[1]] (x + C[2])]))/C[1]]}} 

(They the same solution. Let’s ignore that first.) If I impose an initial condition $ y(0) = 0$ then Mathematica fails to return a solution

DSolve[{y''[x] == -2 Exp[-y[x]], y[0] == 0}, y[x], x] 

with the error message

DSolve::bvfail: For some branches of the general solution, unable to solve the conditions.

But a solution does exist. One can choose $ C[1] = 1$ and $ C[2] = \cosh^{-1}(3/2)$ in the solution and verify that $ y(0) = 0$ .

Any idea why is this the case?

Sufficient condition for the absolute convergence of Fourier series of a function on the adele quotient $\mathbb A_k/k$

Let $ G$ be a compact abelian group. The unitary characters of $ G$ form an orthonormal basis of $ L^2(G)$ , so every square integrable function $ f: G \rightarrow \mathbb C$ admits a Fourier expansion

$ $ f(x) = \sum\limits_{\chi \in \hat{G}} c_{\chi} \chi(x) \tag{1}$ $

where the $ c_{\chi}$ are uniquely determined complex numbers satisfying $ \sum\limits |c_{\chi}|^2 < \infty$ , and the right hand side converges to $ f$ in the $ L^2$ -norm.

If moreover $ $ \sum\limits |c_{\chi}| < \infty \tag{2}$ $ then (1) is actually a pointwise limit (and in fact a uniform limit).

When $ G = \mathbb R/\mathbb Z$ , it is well known that a sufficient condition for (2) is that $ f$ be smooth (even just $ C^1$ ).

What about when $ G = \mathbb A_k/k$ for $ k$ a number field, and $ \mathbb A_k$ the adeles of $ k$ ? There is a notion of a smooth function on $ \mathbb A_k$ (being smooth in the archimedean argument, and locally constant in the nonarchimedean). Does the Fourier series of a smooth function $ f$ on $ \mathbb A_k/k$ satisfy (2)? Or if not, is there a well known sufficient condition on $ f$ for (2) to hold?

One-Sided Analyticity Condition Guarantees Analytic Function?

Let $ f \ \colon \ [0,\infty) \to \mathbb{R}$ be a function satisfying:

  • $ f$ is differentiable infinitely many times in $ (0,\infty)$ , and has a right-derivative of any order at $ 0$ .
  • $ f$ satifsfies the condition (condition 3 here) for analyticity: for every compact $ K \subset [0,\infty)$ there exists a constant $ C_K$ such that $ $ \forall x \in K:\forall n \geq 0:|f^{(n)}(x)|\leq C_K^{n+1}n!$ $ where in the last formula, if $ K$ contains $ 0$ and $ x=0$ , then the $ n$ -th derivative in the formula is the $ n$ -th right derivative in $ 0$ .

Is it true in this case that $ f$ is analytic in $ [0,\infty)$ and that for some $ \epsilon > 0$ ,

$ $ \forall x \in [0,\epsilon) \ \colon \ f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$ $

?

If in addition we have a constant $ C$ such that $ $ \forall x \in [0,\infty):\forall n \geq 0:|f^{(n)}(x)|\leq C^{n+1}n!$ $

does the following hold: $ $ \forall x \in [0,\infty) \ \colon \ f(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n$ $

?

Tip for autofocus in low light and low contrast condition

I have been shooting for several years in the north of Canada and I notice that I don’t get a really high percentage of keeper when using my Canon 100-400 Mk II or 70-200 F2.8 Mk II with either 5DIV or 1DX2.

Even after careful lens calibration, I often get a 50cm-100cm (20in – 40in) back focus which is problematic when shooting dog’s face for example. In this case they are usually coming at 10-15 miles per hour which I try to capture at something like :

  • 1/800sec – F5.6 – 400mm – ISO 1600 to 3200
  • Subject is usually within 50m – 20m (150ft-60ft) away
  • Speed 10-20mph
  • temperatures around -20c to -45c (0F to -45F) which makes the camera a tad slow to respond.
  • Subject with really low contrast

What would be the optimum way to achieve focus:

  • 1 Would you use only one focus point or use an extended area focus?
  • 2 One Shot AF to avoid using the tracking that doesn’t seem to work anyway or AI Servo?
  • 3 Which point would you use? I usually use the lowest points but looking at the picture below that might not be optimum?! a tad confused with that figure.

Type of focus point from https://snapshot.canon-asia.com/article/en/12-powerful-new-features-of-the-eos-5d-mark-iv

  • A: Cross-type focusing: f/4 horizontal + f/5.6 or f/8 vertical
  • B: f/5.6 or f/8 vertical focusing
  • C: Cross-type focusing: f/5.6 or f/8 vertical + f/5.6 or f/8 horizontal
  • D: Dual cross-type focusing: f/2.8 right diagonal + f/2.8 left diagonal f/5.6 or f/8 vertical + f/5.6 or f/8 horizontal

At that point I would take any advice! Maybe I am just not using the tracking properly… not sure! Also note that the cold condition makes it challenging to keep the tiny points on the subject which is why I usually end up using extended area focus which might be harder on the camera focusing wise.

Thanks for your input!

EntityQuery condition on translation status

Im currently using Drupal::entityQuery() to build a list of node.

I need to be able to query only contentEntities that have a specific translation not published. I struggle because I don’t know how to make a condition on an entity translated field/property.

to sum up: How to query all nodes that have ‘de’ translation AND which their ‘de’ translation are not published?

Is there a way to do it with Drupal::entityQuery() or should I directly go with altering the query with QueryInterface::addTag() where I would be able to customize the query?

Condition on a Fontaine Laffaille module which prescribes the image of the associated Galois representation

The Setup:

Let $ \mathbb{F}$ be a finite field of characteristic $ p$ and $ W(\mathbb{F})$ the ring of Witt-vectors with residue field $ \mathbb{F}$ . Recall that filtered Dieudonne’ $ W(\mathbb{F})$ -module also known as a Fontaine-Laffaille module is a $ W(\mathbb{F})$ -module furnished with a decreasing, exhaustive, separated filtration of submodules $ \{F^i M\}$ and for each integer $ i$ a $ \sigma$ -semilinear map $ \varphi^i=\varphi_M^i:F^i M\rightarrow M$ . These maps are required to satisfy two conditions

  1. the following compatibility relation is satisfied $ \varphi^{i+1}=p \varphi^i$ ,

  2. $ \sum_i \varphi^i(F^i M)=M$ .

Let $ \text{MF}_{tor}^{f}$ denote the category of Fontaine-Laffaille modules $ M$ with morphisms satisfying the conditions alluded to above. For $ a<b$ let $ \text{MF}_{tor}^{f,[a,b]}$ let the full subcategory of $ \text{MF}_{tor}^{f}$ whose underlying modules $ M$ satisfy $ F^0 M=M$ and $ F^p M=0$ .

The Fontaine-Laffaille functor $ \text{U}:\text{MF}_{tor}^{f,[0,p]}\rightarrow \text{Rep}_{W(\mathbb{F})}(\text{G}_{\mathbb{Q}_p})$ where $ \text{Rep}_{W(\mathbb{F})}(\text{G}_{\mathbb{Q}_p})$ is the category of continuous $ W(\mathbb{F})[\text{G}_{\mathbb{Q}_p}]$ modules that are finite-length $ W(\mathbb{F})$ -modules. It is a basic fact that if $ M$ has the structure of a free $ W(\mathbb{F})$ -module of rank $ n$ (in greater detail, $ F^j M$ are all free $ W(\mathbb{F})$ -modules and the maps $ \varphi^j$ and $ W(\mathbb{F})$ -module maps and may be viewed as matrices with entries in $ W(\mathbb{F})$ ) then $ \rho_M:=\text{U}(M)$ is a Galois representation $ \rho_M:\text{G}_{\mathbb{Q}_p}\rightarrow \text{GL}_n(W(\mathbb{F}))$ .

Question: Let $ G\subset \text{GL}_n$ be an algebraic subgroup of $ \text{GL}_n$ defined over $ \mathbb{Q}$ (I’m mainly interested in the exceptional groups, like for instance $ G_2$ ). What condition on the matrices $ \varphi^j_M$ ensures that the image of $ \rho_M:=\text{U}(M)$ lies in $ G(W(\mathbb{F}))$ so that it is a Galois representation $ \rho_M:\text{G}_{\mathbb{Q}_p}\rightarrow G(W(\mathbb{F}))$ .

Comment: For the classical groups like $ \text{GSp}_{2n}$ I’m aware how to do this. It involves making use of the alternating form and the functoriality of $ U$ .