On images of locally finite derivations and E-derivations

Some cases of the LFED Conjecture, proposed by the second author [15], for certain integral domains are proved. In particular, the LFED Conjecture is completely established for the field of fractions $k(x)$ of the polynomial algebra $k[x]$, the formal power series algebra $k[[x]]$ and the Laurent formal power series algebra $k[[x]][x^{?1}]$, where $x=(x_1,x_2,\dots ,x_n)$ denotes n commutative free variables and k a field of characteristic zero. Furthermore, the relation between the LFED Conjecture and the Duistermaat–van der Kallen Theorem [3] is also discussed and emphasized.     

Indecomposable involutive set-theoretic solutions of the Yang–Baxter equation

We describe the indecomposable involutive non-degenerate set-theoretic solutions of the Yang–Baxter equation as dynamical extensions of non-degenerate left cycle sets. Moreover we characterize the indecomposable dynamical extensions and we produce several examples. As an application we construct a family of finite indecomposable solutions whose structure groups have not the unique product property.     

On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains

We construct a bounded C¹ domain Ω in for which the regularity for the Dirichlet and Neumann problems for the Laplacian cannot be improved, that is, there exists f in such that the solution of in Ω and either on or on is contained in but not in for any . An analogous result holds for Sobolev spaces with .     

‐Theory for Schrödinger systems

In this article, we study for the ‐realization of the vector‐valued Schrödinger operator . Using a noncommutative version of the Dore–Venni theorem due to Monniaux and Prüss, we prove that the ‐realization of , defined on the intersection of the natural domains of the differential and multiplication operators which form , generates a strongly continuous contraction semigroup on . We also study additional properties of the semigroup such as extension to L¹, positivity, ultracontractivity and prove that the generator has compact resolvent.     

Embedding Banach spaces into the space of bounded functions with countable support

We prove that a WLD subspace of the space consisting of all bounded, countably supported functions on a set Γ embeds isomorphically into if and only if it does not contain isometric copies of . Moreover, a subspace of is constructed that has an unconditional basis, does not embed into , and whose every weakly compact subset is separable (in particular, it cannot contain any isomorphic copies of ).     

The martingale problem for a class of nonlocal operators of diagonal type

We consider systems of stochastic differential equations of the form $$\begin{array}{c}\hfill d{X}_t^i=\sum _{j=1}^d{A}_{ij}\left(X_{t?}\right)d{Z}_t^j\end{array}$$ for $i=1,?,d$ with continuous, bounded and non‐degenerate coefficients. Here $Z_t^1,?,Z_t^d$ are independent one‐dimensional stable processes with ${\alpha }_1,?,{\alpha }_d\in (0,2)$. In this article we research on uniqueness of weak solutions to such systems by studying the corresponding martingale problem. We prove the uniqueness of weak solutions in the case of diagonal coefficient matrices.     

Synthesis,Properties, and Applications of 2-D Materials: A Comprehensive Review

Recent advances in atomically thin two-dimensional (2-D) materials have led to a variety of promising future technologies for post-CMOS nanoelectronics and energy generation. This review is an attempt to thoroughly illustrate the current status and future prospects for 2-D materials other than graphene (e.g., BN nanosheets, MoS₂, NbSe₂, WS₂, etc.), which have already been contemplated for both low-end and high-end technological applications. An overview of the different synthesis techniques for 2-D materials is presented here, with an exploration of the potential for developing methods of controllable large scale synthesis. Furthermore, we summarize the underlying theories which correlate the structural and physical properties of 2-D materials with their state-of-the-art applications. Finally, we show that utilizing the unprecedented properties arising from these materials would lead to innovative devices. Such devices would significantly reduce both device dimensions and power consumption, as necessary for the creation of tomorrow's sustainable technology.     

Change of electrostatic potential of mean force between two curved surfaces due to different salt composition,ion valence and size under certain ionic strength

Change of an electrostatic potential of mean force (EPMF) between two cylindrical rod surfaces with salt composition, ion valence, and ion size at a constant ionic strength of 0.3 M is studied by a classical density functional theory (CDFT) in a primitive model electrolyte solution. Several novel observations are made: (i) strength of a so-called like charge attraction (LCA) reduces in an invariable manner with the salt solution changing from single 2:1 electrolyte to mixture of 2:1 and 1:1 type electrolytes of varying concentration ratios; the change is even over entire range of the composition variation under low surface charge strength, and tends to be insensitive to the composition variation in the presence of the divalent counter-ion, and more and more drastic at a critical point the divalent counter-ion disappears, respectively, as the surface charge strength becomes big enough. (ii) Both monovalent counter-ion and co-ion diameters have only a marginal effect on both the LCA strength and equilibrium distance, and the former “abnormally” affects less than the latter. (iii) Depending on the surface charge strength considered, the divalent counter-ion diameter influences the LCA strength in solution comprised of 2:1 type and 1:1 type electrolytes, monotonously or non-monotonously. All of these findings provide forceful support for a recently proposed hydrogen-bonding style mechanism explaining the LCA.     

Quintic hyperbolic nonpolynomial spline and finite difference method for nonlinear second order differential equations and its application

An efficient numerical method based on quintic nonpolynomial spline basis and high order finite difference approximations has been presented. The scheme deals with the space containing hyperbolic and polynomial functions as spline basis. With the help of spline functions we derive consistency conditions and high order discretizations of the differential equation with the significant first order derivative. The error analysis of the new method is discussed briefly. The new method is analyzed for its efficiency using the physical problems. The order and accuracy of the proposed method have been analyzed in terms of maximum errors and root mean square errors.     

Nonuniform dichotomic behavior: Lipschitz invariant manifolds for ODEs

We obtain global and local theorems on the existence of invariant manifolds for perturbations of nonautonomous linear differential equations assuming a very general form of dichotomic behavior for the linear equation. Besides some new situations that are far from the hyperbolic setting, our results include, and sometimes improve, some known stable manifold theorems.