Modular Construction of Porous Hydrogen-Bonded Molecular Materials from Melams

Ordered materials with predictable structures and properties can be made by a modular approach, using molecules designed to interact with neighbors and hold them in predetermined positions. Incorporating 4,6-diamino-1,3,5-triazin-2-yl (DAT) groups in modules is an effective way to direct assembly because each DAT group can form multiple N−H⋅⋅⋅N hydrogen bonds according to established patterns. We have found that modules with high densities of N(DAT)₂ groups can be made by base-induced double triazinylations of readily available amines. The resulting modules can form structures held together by remarkably large numbers of hydrogen bonds per molecule. Even simple modules with only 1–3 N(DAT)₂ groups and fewer than 70 non-hydrogen atoms can crystallize to form highly open networks in which each molecule engages in over 20 N−H⋅⋅⋅N hydrogen bonds, and more than 70 % of the volume is available for accommodating guests. In favorable cases, guests can be removed to create rigorously porous crystalline solids analogous to zeolites and metal–organic frameworks.     

Regularization of a final value problem for a nonlinear biharmonic equation

In this paper, we consider the nonlinear biharmonic equation. The problem is ill‐posed in the sense of Hadamard. To obtain a stable numerical solution, we consider a regularization method. We show rigourously, with error estimates provided, that the corresponding regularized solutions converge to the true solution strongly in uniformly with respect to the space coordinate under some a priori assumptions on the solution.     

On the exponential stability of dynamic equations on time scales

In this paper, we deal with some theorems on the exponential stability of trivial solution of time-varying non-regressive dynamic equation on time scales with bounded graininess. In particular, well-known Perron's theorem is generalized on time scales. Under rather restrictive condition, that is, integral boundedness of coefficient operators, we obtain a characterization of the uniformly exponential stability.     

A Frank–Wolfe type theorem for nondegenerate polynomial programs

In this paper, we study the existence of optimal solutions to a constrained polynomial optimization problem. More precisely, let \(f_0\) and \(f_1, \ldots , f_p :{\mathbb {R}}^n \rightarrow {\mathbb {R}}\) be convenient polynomial functions, and let \(S := \{x \in {\mathbb {R}}^n \ : \ f_i(x) \le 0, i = 1, \ldots , p\} \ne \emptyset .\) Under the assumption that the map \((f_0, f_{1}, \ldots , f_{p}) :{\mathbb {R}}^n \rightarrow {\mathbb {R}}^{p + 1}\) is non-degenerate at infinity, we show that if \(f_0\) is bounded from below on \(S,\) then \(f_0\) attains its infimum on \(S.\)      

Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a half-line

Consider an evolution family U=(U(t,s))_t?s?0 on a half-line R₊ and a semi-linear integral equation . We prove the existence of stable manifolds of solutions to this equation in the case that (U(t,s))_t?s?0 has an exponential dichotomy and the nonlinear forcing term f(t,x) satisfies the non-uniform Lipschitz conditions: ‖f(t,x₁)−f(t,x₂)‖?φ(t)‖x₁−x₂‖ for φ being a real and positive function which belongs to admissible function spaces which contain wide classes of function spaces like function spaces of Lp type, the Lorentz spaces Lp,q and many other function spaces occurring in interpolation theory.     

The truncation method for a two-dimensional nonhomogeneous backward heat problem

We consider the backward heat problem

     

Regularity results for fractional diffusion equations involving fractional derivative with Mittag–Leffler kernel

This paper studies partial differential equation model with the new general fractional derivatives involving the kernels of the extended Mittag–Leffler type functions. An initial boundary value problem for the anomalous diffusion of fractional order is analyzed and considered. The fractional derivative with Mittag–Leffler kernel or also called Atangana and Baleanu fractional derivative in time is taken in the Caputo sense. We obtain results on the existence, uniqueness, and regularity of the solution.     

On a backward Cauchy problem associated with continuous spectrum operator

The nonlinear backward Cauchy problem

u_t+Au(t)=f(u(t)),u(T)=φ,