S. Janeczko

Formation of Highly Ordered Nanostructures by Drying Micrometer Colloidal Droplets

Nanoparticles with well-defined chemical compositions can act as building blocks for the construction of functional structures, such as highly ordered aggregates, as well as porous and hollow aggregates. In this work, a spray-drying technique is used to form a crystal-like structure with nanoparticle building blocks. When spray-drying uniform spherical particles, tightly packed aggregates with either simple or broken symmetries (quasicrystalline) were formed. Using polystyrene (PS) particles with varied zeta potentials as templates, it is also possible to form highly ordered porous and hollow aggregates from inorganic colloidal particles. Essential to the production of quasicrystalline structures is the use of monodisperse colloidal particles in spray drying, as the quasicrystalline form is not achievable when two different sizes of colloidal particles are used in the precursor suspension. With varying colloidal particles sizes, smaller colloidal particles fill the spaces formed between the larger particles, resulting in adjustment of colloidal crystallization. A geometric model that considers the tight packing of several spheres into frustrated clusters (quasicrystal form) with short-range icosahedral symmetry is compared to experimentally produced structures and found to quantitatively explain experimental observations.

Generating families for images of Lagrangian submanifolds and open swallowtails

In this paper we study the symplectic relations appearing as the generalized cotangent bundle liftings of smooth stable mappings. Using this class of symplectic relations the classification theorem for generic (pre) images of lagrangian submanifolds is proved. The normal forms for the respective classified puilbacks and pushforwards are provided and the connections between the singularity types of symplectic relation, mapped lagrangian submanifold and singular image, are established. The notion of special symplectic triplet is introduced and the generic local models of such triplets are studied. We show that the open swallowtails are canonically represented as pushforwards of the appropriate regular lagrangian submanifolds. Using the SL 2 (ℝ) invariant symplectic structure of the space of binary forms of n appropriate dimension we derive the generating families for the open swallowtails and the respective generating functions for its regular resolutions.

Bifurcations of the center of symmetry

We use the concept of measure of symmetry for ovals to generalize the notion of center of symmetry. We show that for generic plane convex curves the center symmetry set has only fold and cusp singularities.

Singularities of Lagrangian varieties and related topics

We study the classification problem for generic projections of Lagrangian submanifolds. A classification list for symmetric Lagrangian submanifolds is obtained and the generic evolutions of symmetric caustics are illustrated. We show how the singular Lagrangian varieties appear in the invariant theory of binary forms and we introduce the basic concepts of the desingularization procedure. Applications to differential geometry, geometrical optics, and mechanics are presented.

Lagrangian submanifolds in product symplectic spaces

We analyze the global structure of Lagrangian Grassmannian in the product symplectic space and investigate the local properties of generic symplectic relations. The cohomological symplectic invariant of discrete dynamical systems is generalized to the class of generalized canonical mappings. Lower bounds for the number of two-point and three-point symplectic invariants for billiard-type dynamical systems are found and several examples of symplectic correspondences encountered from physics are presented.

Generalized Luneburg canonical varieties and vector fields on quasicaustics

Some aspects of a particular class of bifurcation varieties which are provided by simple and unimodal boundary singularities are studied. Their correspondence to diffraction theory is established. The generic caustics by diffraction on apertures are derived and their generating families for the corresponding Lagrangian varieties are calculated. It is proved that the quasicaustics associated to simple singularities are smooth hypersurfaces or Whitney’s cross‐caps. The procedure for calculating the modules of logarithmic vector fields is given, and the minimal sets of the corresponding generators are explicitly calculated. The general boundary singularities are constructed and the structure of quasicaustics defined by parabolic singularities is investigated.

LINEAR AUTOMORPHISMS THAT ARE SYMPLECTOMORPHISMS

Let K be the field of real or complex numbers. Let (X ∼ K 2n ,ω ) be a symplectic vector space and take 0 <k < n, N =( 2n

On Martinet's singular symplectic structures

Introduction. Let V be a stratified subspace of R . We call it symplectic if there exists a differential 2-form ω on R such that the restriction of ω to each stratum is a symplectic form. In the Marsden-Weinstein singular reduction theory these spaces were studied by several authors [5, 4, 9, 1]. In this paper we classify the symplectic spaces modelled on the so-called symplectic flag S. First we prove the corresponding Darboux theorem and then we show that the only reasonable symplectic structures on S are those with underlying Martinet’s singular symplectic structure of type Σ2,0. Finally we find the normal form for this structure and show the similar result for an example of a stratified symplectic space with singular boundary of the maximal stratum.

Equivariant singularities of Lagrangian manifolds and uniaxial ferromagnet

A classification of typical 3-dimensional Lagrangian singularities with

Stability criteria and classification of singularities for equivariant lagrangian submanifolds

Z_2 \oplus Z_2