How is Grönwall’s inequality applied here?

Let $ h\in C^1(\mathbb R)$ such that $ h’$ is Lipschitz continuous and $ $ L\varphi:=-h’\varphi’+\varphi”\;\;\;\text{for }\varphi\in C^2(\mathbb R).$ $ Now, let $ (X_t)_{t\ge0}$ be the unique strong solution of $ $ {\rm d}X_t=-h'(X_t){\rm d}t+\sqrt2{\rm d}W_t\tag1,$ $ where $ (W_t)_{t\ge0}$ is a Brownian motion.

I’ve read (in this paper, below Assumption 2.4) that if $ V:\mathbb R\to[0,\infty)$ with $ $ V(x)\xrightarrow{x\to\infty}\infty\tag2$ $ and $ a,d>0$ with $ $ LV\le-aV+d\tag3,$ $ then $ $ \operatorname E\left[V(X_t)\mid\mathcal F_s^X\right]\le e^{-a(t-s)}V(X_s)+\frac da(1-e^{-a(t-s)})\tag4.$ $ Why does $ (4)$ hold?

Obviously, $ (4)$ is an application of (the Itō formula and) a Gronwall-type lemma. Actually, it’s precisely Theorem 6 here. However, in order to apply that theorem, we should need that the process $ \left(\operatorname E\left[V(X_t)\mid\mathcal F_s^X\right]\right)_{t\ge0}$ is continuous. This shouldn’t hold, unless $ V$ is (continuous and) bounded (which would allow an application of Lebesgue’s dominated convergence theorem). But $ V$ is clearly assumed to be unbounded by $ (2)$ . So, what am I missing?

(Clearly, the authors of the paper are missing assumptions on $ V$ anyway. In order for $ (3)$ to make sense, $ V$ needs to be twice differentiable (at least in some weak sense).

Computational type theorists: how do you compare terms for equality here?

I am attempting to implement Simple Type Theory in the language D. How do you compare subterms to a term $ R$ for the sake of computing the covering abstractors of $ R$ in $ M$ ?

By reference (class objects), syntactically by string value, or by some defined equality function?

Definition. (Body, scope, covering abstractors) Let $ \lambda x.P$ be a component of a term $ M$ . The displayed component $ P$ is called the body of or the scope of the abstractor $ \lambda x$ .

The covering abstractors of a component $ R$ of $ M$ are the abstractors in $ M$ whose scopes contain $ R$ .

This is from the book “Basic Simple Type Theory”.

FCC Repeals Net Neutrality! (click here for no more rights)

FCC wants to remove some regulations for ISP providers…
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what does this mean?

more rights for ISPs….
less rights for consumers…

I would imagine these rules were put into place to protect
consumers to begin with, now the FCC…

FCC Repeals Net Neutrality! (click here for no more rights)

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Why does the base class need to have a virtual destructor here if the derived class allocates no raw dynamic memory?

The following code causes a memory leak:

#include <iostream> #include <memory> #include <vector>  using namespace std;  class base {     void virtual initialize_vector() = 0; };  class derived : public base { private:     vector<int> vec;  public:     derived()     {         initialize_vector();     }      void initialize_vector()     {         for (int i = 0; i < 1000000; i++)         {             vec.push_back(i);         }     } };  int main() {     for (int i = 0; i < 100000; i++)     {         unique_ptr<base> pt = make_unique<derived>();     } } 

It didn’t make much sense to me, since class derived allocates no raw dynamic memory, and unique_ptr deallocates itself. I get that class base’s implicit destructor is being called instead of derived’s, but I don’t get why that’s a problem here. If I were to write an explicit destructor for derived, I wouldn’t write anything for vec.