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…
What do you think giving more control to ISPs, they could
offer more, offer less, raise your ISP rates, etc…

https://www.nytimes.com/2017/12/14/technology/net-neutrality-repeal-vote.html

or maybe ISPs will control your life like the govt does….

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.