A note on dual prehomomorphisms from a group into the Margolis–Meakin expansion of a group

Semigroup Forum - We give a category-free order theoretic variant of a key result in Auinger and Szendrei (J Pure Appl Algebra 204(3):493–506, 2006) and illustrate how it might be used to...     

A unifying approach to the Margolis–Meakin and Birget–Rhodes group expansion

Let G be a group. We show that the Birget–Rhodes prefix expansion \(G^{Pr}\) and the Margolis–Meakin expansion M(X; f) of G with respect to \(f:X\rightarrow G\) can be regarded as inverse subsemigroups of a common E-unitary inverse semigroup P. We construct P as an inverse subsemigroup of an E-unitary inverse monoid \(U/\zeta \) which is a homomorphic image of the free product U of the free semigroup \(X^+\) on X and G. The semigroup P satisfies a universal property with respect to homomorphisms into the permissible hull C(S) of a suitable E-unitary inverse semigroup S, with \(S/\sigma _S=G\), from which the characterizing universal properties of \(G^{Pr}\) and M(X; f) can be recaptured easily.     

A note on the Chermak–Delgado lattice of a finite group

In this note, we describe the structure of finite groups G whose Chermak–Delgado lattice is the interval [G∕Z(G)] = {H∈L(G)∣Z(G)≤H≤G}.     

A note on the global existence of a two-dimensional chemotaxis-Navier–Stokes system

In this short note, we establish the global existence of weak solutions and classical solutions to the two-dimensional chemotaxis-Navier–Stokes system for both the Cauchy problem and the initial-boundary value problem under some suitable small conditions on the initial data. In particular, we improve the recent results obtained by Duan–Li–Xiang (J. Differential Equations, 2017).     

A note on the exact discretization for a Cauchy–Euler equation: application to the Black–Scholes equation

We construct the exact finite difference representation for a second-order, linear, Cauchy–Euler ordinary differential equation. This result is then used to construct new non-standard finite difference schemes for the Black–Scholes partial differential equation.     

A note on the zeros of Freud–Sobolev orthogonal polynomials

We prove that the zeros of a certain family of Sobolev orthogonal polynomials involving the Freud weight function e^-x4${\mathrm{e}}^{-x^4}$ on R$\mathbb{R}$ are real, simple, and interlace with the zeros of the Freud polynomials, i.e., those polynomials orthogonal with respect to the weight function e^-x4${\mathrm{e}}^{-x^4}$. Some numerical examples are shown.     

A note on the Fuglede–Putnam theorem

We prove the following generalization of the Fuglede–Puntam theorem. Let N be an unbounded normal operator in the Hilbert space, and let A be an unbounded self-adjoint operator such that $D(N)\subseteq D(A)$ . Then, $ AN\subseteq N^*A \Rightarrow AN^*\subseteq NA.$      

A note on solutions of the Korteweg–de Vries hierarchy

olutions of the Korteweg–de Vries hierarchy are discussed. It is shown that results by Wazwaz [Wazwaz AM. Multiple-soliton solutions of the perturbed KdV equation. Commun Nonlinear Sci Simul 2010;15911:3270–73] are the well-known consequences of the full integrability for the Korteweg–de Vries hierarchy.     

Erratum to: A note on the Brück conjecture

In the article “A note on the Brück conjecture” (Arch. Math. 95 (2010), 257–268), the proofs of our results Theorem 1.4 and Theorem 1.6 mainly depend on Lemma 2.6, which is wrong. In this corrigendum, we give a correct proof of Theorem 1.4. Theorem 1.6 can be proved similarly; so we omit its proof.     

Solitary-wave solutions to a dual equation of the Kaup–Boussinesq system

In this paper, we employ the bifurcation theory of planar dynamical systems to investigate the travelling-wave solutions to a dual equation of the Kaup–Boussinesq system. The expressions for smooth solitary-wave solutions are obtained.     

A note on regularity of the pressure in the Navier–Stokes system

In this note we improve the standard regularity of the dynamic part of the pressure in the Navier–Stokes system. Using the theory of elliptic equations with \(L^1\) right-hand side we prove that, in addition to be in \(L^2\), the dynamic pressure belongs to \(W^{1,\alpha }_{loc} \) with \(1<\alpha <\frac{n}{n-1}\), in case of Dirichlet boundary condition. For pressure boundary condition the dynamic pressure is proved to be in \(W^{1,\alpha } \). As a consequence, for the force \(\mathbf{f} \in L^q (\Omega )^n \) and \(q>n /2 \) the pressure turns out to be continuous.     

A note on the regularity criterion of the two-dimensional Newton–Boussinesq equations

In this paper, we consider the two-dimensional Newton–Boussinesq equations with the incompressibility condition. We obtain a regularity criterion for the Newton–Boussinesq equations by virtue of the commutator estimate.