ON THE EQUIVALENCE OF SOME CONVOLUTIONAL EQUALITIES IN SPACES OF SEQUENCES

The problem of the equivalence of two systems with $n$ convolutional equalities arose in investigation of the conditions of similarity in spaces of sequences of operators which are left inverse to the $n$-th degree of the generalized integration operator. In this paper we solve this problem. Note that we first prove the equivalence of two corresponding systems with $n$ equalities in the spaces of analytic functions, and then, using this statement, the main result of paper is obtained. Let $X$ be a vector space of sequences of complex numbers with K$\ddot{\rm o}$the normal topology from a wide class of spaces, ${\mathcal I}_{\alpha}$ be a generalized integration operator on $X$, $\ast$ be a nontrivial convolution for ${\mathcal I}_{\alpha}$ in $X$, and $(P_q)_{q=0}^{n-1}$ be a system of natural projectors with $\displaystyle x = \sum\limits_{q=0}^{n-1} P_q x$ for all $x\in X$. We established that a set $(a^{(j)})_{j=0}^{n-1}$ with $$ \max\limits_{0\le j \le n-1}\left\{\mathop{\overline{\lim}}\limits_{m\to\infty} \sqrt[m]{\left|\frac{a_{m}^{(j)}}{\alpha_m}\right|}\right\}<\infty $$ and a set $(b^{(j)})_{j=0}^{n-1}$ of elements of the space $X$ satisfy the system of equalities $$ b^{(j)}=a^{(j)}+\sum\limits_{k=0}^{n-1}({\mathcal I}_{\alpha}^{n-k-1} a^{(k)}) \ast {(P_{k}b^{(j)})}, \quad j = 0, 1, ... \, , \, n-1, $$ if and only if they satisfy the system of equalities $$ b^{(j)}=a^{(j)}+\sum\limits_{k=0}^{n-1}({\mathcal I}_{\alpha}^{n-k-1} b^{(k)}) \ast {(P_{k}a^{(j)})}, \quad j = 0, 1, ... \, , \, n-1. $$ Note that the assumption on the elements $(a^{(j)})_{j=0}^{n-1}$ of the space $X$ allows us to reduce the solution of this problem to the solution of an analogous problem in the space of functions analytic in a disc.

Cubic phases of self-assembled amphiphilic aggregates

In this paper we give an overview of cubic liquid-crystalline mesophases formed by amphiphiles. In § 1 we present brief descriptions of the principal types of translationally ordered lyotropic phases, and describe the locations in the phase diagrams where the different types of cubic phase occur. In §2 we discuss the various forces that act between bilayers. These transverse interactions are relatively straightforward to quantify in the case of lamellar phases, but are more complex for cubic phases, because of the non-planar geometry. In §3 we show how an intrinsic desire for interfacial curvature can lead to a state of physical frustration. We then introduce the curvature elastic energy, and describe how this may be related to the stress profile across the bilayer. In the following sections we focus attention on the inverse (water-in-oil) versions of the non-lamellar phases, although analogous effects also operate in the normal topology (oil-in-water) structures. In §4 we briefly describe the inverse hexagonal phase, which is the simplest inverse phase with curved interfaces. This allows us to illustrate the role of hydrocarbon chain packing frustration in a rather clear way before coming on to the more subtle interplay between packing and curvature frustration, characteristic of the bicontinuous cubic phases, which is discussed in §5. In §6 we describe an entirely different class of cubic phases, with positive interfacial gaussian curvature. These cubic phases are composed of complex packings of discrete micellar or inverse micellar aggregates, which may be quasi-spherical and/or anisotropic in shape. Finally, in §7 we discuss geometric aspects of transitions between lamellar, hexagonal and cubic phases, and show how determination of the epitaxial relations between phases can shed light on the precise mechanisms of the phase transitions.

Some criteria for nuclearity

For a given locally convex space, it is always of interest to find conditions for its nuclearity. Well known results of this kind – by now already familiar – involve the use of tensor products, diametral dimension, bilinear forms, generalized sequence spaces and a host of other devices for the characterization of nuclear spaces (see [9]). However, it turns out, these nuclearity criteria are amenable to a particularly simple formulation in the setting of certain sequence spaces; an elegant example is provided by the so-called Grothendieck–Pietsch (GP, for short) criterion for nuclearity of a sequence space (in its normal topology) in terms of the summability of certain numerical sequences.

On the Dvoretzky-Rogers theorem

The classical Dvoretzky-Rogers theorem states that if E is a normed space for which l1(E)=l1{E} (or equivalently , then E is finite dimensional (see [12] p. 67). This property still holds for any lp (l<p<∞) in place of l1 (see [7]p. 104 and [2] Corollary 5.5). Recently it has been shown that this result remains true when one replaces l1 by any non nuclear perfect sequence space having the normal topology (see [14]). In this context, De Grande-De Kimpe [4] gives an extension of the Devoretzky-Rogers theorem for perfect Banach sequence spaces.