Droplet profiles under the influence of van der Waals forces

Euro. Jnl. of Applied Maths.12, 367–393

ELI MINKOV and AMY NOVICK-COHEN

Let $B(H)$ denote the algebra of all bounded linear operators on a separable, infinite-dimensional, complex Hilbert space $H$. Let $I$ be a two-sided ideal in $B(H)$. For operators $A, B$ and $X \in B(H)$, we say that $X$intertwines$A$and$B$modulo$I$ if $AX - XB \in I$. It is easy to see that if $X$ intertwines $A$ and $B$ modulo $I$, then it intertwines $A^{n}$ and $B^{n}$ modulo $I$ for every integer $n > $1. However, the converse is not true. In this paper, sufficient conditions on the operators $A$ and $B$ are given so that any operator $X$ which intertwines certain powers of $A$ and $B$ modulo $I$ also intertwines $A$ and $B$ modulo $J$ for some two-sided ideal $J \supseteq I$.

It is an open question whether every strongly locally $\varphi$-symmetric contact metric space is a $(\kappa,\mu)$-space. We show that the answer is positive for locally homogeneous contact metric manifolds.

In this paper, we construct a generalized degree theory of Browder-Petryshyn or Petryshyn type for a class of semilinear operator equations involving a Fredholm type mapping with infinite dimensional kernel.

We give a necessary and sufficient condition for $k$-step nilmanifolds associated with graphs $(k \geq 3)$ to admit Anosov automorphisms. We also prove the nonexistence of Anosov automorphisms on certain classes of 2-step and 3-step nilmanifolds.

In this note we answer two question posed by Berkani and Koliha [Acta Sci. Math.69 (2003), 359–376]. We show that generalized Browder's (resp. generalized $a$-Browder's) theorem holds for a Banach space operator if and only if Browder's (resp. $a$-Browder's) theorem does. We also give condition under which generalized Weyl's (resp. generalized $a$-Weyl's) theorem is equivalent to Weyl's (resp. $a$-Weyl's) theorem.

In this paper, we prove that, for any integer $n\ge 2,$ and any $\delta > 0$ there exists an $\epsilon(n,\delta) \ge 0$ such that if $M$ is an $n$-dimensional complete manifold with sectional curvature $K_M \ge 1$ and if $M$ has conjugate radius $\rho \ge\frac{\pi}{2}+\delta $ and contains a geodesic loop of length $2(\pi-\epsilon(n,\delta))$ then $M$ is diffeomorphic to the Euclidian unit sphere $\mathbb{S}^{n}.$

We prove that Mazur's functional characterization for one-sided estimates can be restricted to smaller classes of functionals in the case in which the functions under consideration are continuous. We apply this result to stability problems for dynamical systems in $l^\infty$, and in the Banach space of all selfadjoint operators on a Hilbert space.

For a closed topological $n$-manifold $X$, the surgery exact sequence contains the set of manifold structures and the set of tangential structures of $X$. In the case of a compact topological $n$-manifold with boundary $(X$, $\partial X)$, the classical surgery theory usually considers two different types of structures. The first one concerns structures whose restrictions are fixed on the boundary. The second one uses two similar structures on the manifold pair. In his classical book, Wall mentioned the possibility of introducing a mixed type of structure on a manifold with boundary. Following this suggestion, we introduce mixed structures on a topological manifold with boundary, and describe their properties. Then we obtain connections between these structures and the classical ones, and prove that they fit in some surgery exact sequences. The relationships can be described by using certain braids of exact sequences. Finally, we discuss explicitly several geometric examples.

In this paper we present some results about $wV$ (weak property $V$ of Peł czyński) or property $wV^*$ (weak property $V^*$ of Peł czyński) in Banach spaces. We show that $E$ has property $wV$ if for any reflexive subspace $F$ of $E^*$, $^{\perp} {F}$ has property $wV$. It is shown that $G$ has property $wV$ if under some condition $K_{w^*}(E^*, F^*)$ contains the dual of $G$. Moreover, it is proved that $E^*$ contains a copy of $c_0$ if and only if $E$ contains a copy of $\ell_1$ where $E$ has property $wV^*$. Finally, the identity between $L(C(\Omega, E), F)$ and $WP(C(\Omega, E), F)$ is investigated.

It is proved that every abelian VNL-ring is an SVNL-ring, which gives a positive answer to a question of Osba et al. [7]. Some characterizations of duo VNL-rings are given and some main results of Osba et al. [7] on commutative VNL-rings are extended to right duo VNL-rings and even abelian GVNL-rings.

In this work we consider a complete spacelike submanifold $M^{n}$ immersed in the De Sitter space $S_{p}^{n+p}(1)$ with parallel mean curvature vector. We use a Simons type inequality to obtain some rigidity results characterizing umbilical submanifolds and hyperbolic cylinders in $S_p^{n+p}(1)$.

A method of choice for realizing finite groups as regular Galois groups over $\mathbb{Q}(T)$ is to find $\mathbb{Q}$-rational points on Hurwitz moduli spaces of covers. In another direction, the use of the so-called patching techniques has led to the realization of all finite groups over $\mathbb{Q}_p(T)$. Our main result shows that, under some conditions, these $p$-adic realizations lie on some special irreducible components of Hurwitz spaces (the so-called Harbater–Mumford components), thus connecting the two main branches of the area. As an application, we construct, for every projective system $(G_n)_{n\geq0}$ of finite groups, a tower of corresponding Hurwitz spaces $(\mathcal{H}_{G_n})_{n\geq0}$, geometrically irreducible and defined over some cyclotomic extension of $\mathbb{Q}$, which admits projective systems of $\mathbb{Q}_p^{\mathrm{ur}}$-rational points for all primes $p$ not dividing the orders $|G_n|$ ($n\geq0$).

Let $\Delta$ be one of the dual polar spaces $DW(5,q)$ or $DH(5,q^2)$. We consider a class of subspaces of $\Delta$, each member of which carries the structure of a near hexagon, and classify all these subspaces. Using this classification, we determine all hyperplanes of $DW(5,q)$ without ovoidal quads.

We show that the spherical subalgebra $U_{k,c}$ of the rational Cherednik algebra associated to $S_n C_{\ell}$, the wreath product of the symmetric group and the cyclic group of order $\ell$, is isomorphic to a quotient of the ring of invariant differential operators on a space of representations of the cyclic quiver of size $\ell$. This confirms a version of [5Conjecture 11.22] in the case of cyclic groups. The proof is a straightforward application of work of Oblomkov [12] on the deformed Harish–Chandra homomorphism, and of Crawley–Boevey, [3] and [4], and Gan and Ginzburg [7] on preprojective algebras.

We consider the standard action of the dihedral group $\bf{D}_n$ of order $2n$ on $\bf{C}$. This representation is absolutely irreducible and so the corresponding Hopf bifurcation occurs on $\bf{C} \oplus \bf{C}$. Golubitsky and Stewart (Hopf bifurcation with dihedral group symmetry: Coupled nonlinear oscillators. In: Multiparameter Bifurcation Series, M. Golubitsky and J. Guckenheimer, eds., Contemporary Mathematics 46, Am. Math. Soc., Providence, R.I. 1986, 131–173) and van Gils and Valkering (Hopf bifurcation and symmetry: standing and travelling waves in a circular chain. Japan J. Appl. Math.3, 207–222, 1986) prove the generic existence of three branches of periodic solutions, up to conjugacy, in systems of ordinary differential equations with $\bf{D}_n$-symmetry, depending on one real parameter, that present Hopf bifurcation. These solutions are found by using the Equivariant Hopf Theorem. We prove that generically, when $n\neq 4$ and assuming Birkhoff normal form, these are the only branches of periodic solutions that bifurcate from the trivial solution.

We provide a complete local classification of pseudo-parallel sub-manifolds with flat normal bundle of space forms, extending the classification by Dillen-Nölker for the semi-parallel case.

We combine the theory of sectorial sesquilinear forms with the theory of unbounded subnormal operators in Hilbert spaces to characterize the Friedrichs extensions of multiplication operators (with analytic symbols) in certain functional Hilbert spaces. Such characterizations lead to abstract Galerkin approximations and generalized wave equations.

In this paper, we classify 3-dimensional Lagrangian Willmore submanifolds of the nearly kaehler 6-sphere $S^6(1)$ with constant scalar curvature.

We determine for all $d$ and $p$ the maximal derived length of a soluble subgroup of the multiplicative group of a division ring of finite degree $d$ and characteristic $p\,\ge\,0$ to within one.