We review our approach to the second law of thermodynamics as a theorem asserting growth mean (Gibbs–von Neumann) entropy quantum spin systems undergoing automorphic (unitary) adiabatic transformations. Non-automorphic interactions with environment, although known produce on average strict reduction finite number degrees freedom, are proved conserve average. The results depend crucially two pro...

<p style='text-indent:20px;'>Consider the second order nonautonomous lattice systemswith singular perturbations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \epsilon \ddot{u}_{m}+\dot{u}_{m}+(Au)_{m}+\lambda_{m}u_{m}+f_{m}(u_{j}|j\in I_{mq}) = g_{m}(t),\; \; m\in \mathbb{Z}^{k},\; \epsilon&amp...

We present some sufficient conditions ensuring the upper semicontinuity and the continuity of the Bregman projection operator ΠgC and the relative projection operator P g C in terms of the D-approximate (weak) compactness for a nonempty closed set C in a Banach space X. We next present certain sufficient conditions as well as equivalent conditions for the convexity of a Chebyshev subset of a Ba...

In this paper, locally Lipschitz functions acting between infinite dimensional normed spaces are considered. When the range is a dual space and satisfies the Radon–Nikod´ym property, Clarke's generalized Jacobian will be extended to this setting. Characterization and fundamental properties of the extended generalized Jacobian are established including the nonemptiness, the β-compactness, the β-...

We will study the lattice dynamical system of a nonlinear Boussinesq equation. Our objective is to explore the existence of the global attractor for the solution semiflow of the introduced lattice system and to investigate its upper semicontinuity with respect to a sequence of finite-dimensional approximate systems. As far as we are aware, our result here is the first concerning the lattice dyn...

We consider the Cauchy problem for a non-autonomous nonclassical diffusion equation of the form ut − ε∆ut −∆u+ f(u) + λu = g(t) on Rn. Under an arbitrary polynomial growth order of the nonlinearity f and a suitable exponent growth of the external force g, using the method of tail-estimates and the asymptotic a priori estimate method, we prove the existence of an (H1(Rn) ∩ Lp(Rn), H1(Rn) ∩ Lp(Rn...