Flocking With Short-Range Interactions

Journal of Statistical Physics - We study the large-time behavior of continuum alignment dynamics based on Cucker–Smale (CS)-type interactions which involve short-range kernels, that is,...     

Existence of piecewise weak solutions of a discrete Cucker–Smale's flocking model with a singular communication weight

We prove existence of global C¹$C^1$ piecewise weak solutions for the discrete Cucker–Smale's flocking model with a non-Lipschitz communication weight ψ(s)=s^−α$\psi (s)=s^{-\alpha }$, 0<α<1$0<\alpha <1$. We also discuss the possibility of finite in time alignment of the velocities of the particles.     

PDE Models for Pricing Stocks and Options With Memory Feedback

This paper describes partial differential equation (PDE) models for pricing stocks and options in the presence of memory feedback. Of interest are economic situations in which the stock (option) value at time T depends on some type of average of its past values. Derived PDEs resemble viscous Burgers' equations.     

On certain generalizations of the Gagliardo–Nirenberg inequality and their applications to capacitary estimates and isoperimetric inequalities

We derive the inequality $$\int_\mathbb{R}M(|f'(x)|h(f(x))) dx\leq C(M,h)\int_\mathbb{R}M\left({\sqrt{|f''(x)\tau_h(f(x))|}\cdot h(f(x))}\right)dx$$ with a constant C(M, h) independent of f, where f belongs locally to the Sobolev space ${W^{2,1}(\mathbb{R})}$ and f′ has compact support. Here M is an arbitrary N-function satisfying certain assumptions, h is a given function and ${\tau_h(\cdot)}$ is its given transform independent of M. When M(λ) =  λ ^p and ${h \equiv 1}$ we retrieve the well-known inequality ${\int_\mathbb{R}|f'(x)|^{p}dx \leq (\sqrt{p - 1})^{p}\int_\mathbb{R}(\sqrt{|f''(x) f(x)|})^{p}dx}$ . We apply our inequality to obtain some generalizations of capacitary estimates and isoperimetric inequalities due to Maz’ya (1985).