## 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?

## How many word can we make with the given condition?

In a case, we have infinite times of the letter B, D, M and only one O. How many different word containing those letter can we make(can be meaningless in this term)?

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

?

## A necessary condition for linear dependence?

Given the vector space $$\mathbb{C}^n$$ over $$\mathbb{C}$$. A sufficient condition for the set of vectors

$$\{ (1, x_1, x_1^2, \dots, x_1^{n-1}), (1, x_2, \dots, x_2^{n-1}), \dots, (1, x_n, \dots x_n^{n-1}) \}$$

to be linearly dependent is that for exactly two scalars $$x_i = x_j$$, where $$i \neq j$$.

But is it also a necessary condition?

## 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.

• 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.

## 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 and action missing from Workflows in SharePoint designer 2010

I was trying to add conditions / Actions to my SharePoint workflow through SP Designer 2010 and realised they had gone missing. I re-installed the designer to no avail. Any idea what is wrong?

I am trying to add those to a List Workflow. Screenshot below. Thanks.

## 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 $$a let $$\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 $$U$$.

## What is the meaning and role of global hyperbolicity condition for semi-Riemannian manifolds

What is the heuristic meaning of the global hyperbolicity condition for semi-Riemannian manifolds?

Also what is the role of this condition in the study of geodesic connectedness?