Adsorption of Large Organic Molecules on Clean and Hydroxylated Rutile TiO2(110) Surfaces

Behavior of large organic molecules equipped with spacer groups (Violet Landers, VL) on the TiO₂(110)‐(1×1) surfaces is investigated by means of high‐resolution scanning tunneling microscopy (STM). Two distinct adsorption geometries are observed. We demonstrate that the molecule adsorption morphology can be alternated by well‐controlled STM tip‐induced manipulation. It is used to probe the mobility of molecules and reveals locking in one of the analyzed adsorption sites, thus allow to enhance or reduce the mobility along the [001] direction. Field induced hydrogen desorption is used to perform lateral STM manipulation on a hydroxyl‐free surface, which provides insight into the influence of surface hydroxyl groups on the molecule behavior. The ability to image with submolecular resolution both the central board and the spacer groups of the VL molecule is demonstrated.     

On nonmeasurable images

Let (X, ⫿) be a Polish ideal space and let T be any set. We show that under some conditions on a relation R ⊆ T ² × X it is possible to find a set A ⊆ T such that R(A ²) is completely ⫿-nonmeasurable, i.e, it is ⫿-nonmeasurable in every positive Borel set. We also obtain such a set A ⊆ T simultaneously for continuum many relations (R_a )_{a < 2^w} {({R_\alpha })_{\alpha < {2^\omega }}}. Our results generalize those from the papers of K. Ciesielski, H. Fejzić, C. Freiling and M. Kysiak.     

Completely nonmeasurable unions

Assume that no cardinal κ < 2 ^ω is quasi-measurable (κ is quasi-measurable if there exists a κ-additive ideal $ \mathbb{I} $ \mathbb{I} of X contains uncountably many pairwise disjoint subfamilies $ \mathbb{I} $ \mathbb{I} -Bernstein unions ∪ $ \mathbb{I} $ \mathbb{I} -Bernstein if A and X \ A meet each Borel $ \mathbb{I} $ \mathbb{I} -positive subset B ⊆ X). This result is a generalization of the Four Poles Theorem (see [1]) and results from [2] and [4].     

Descriptive properties of density preserving autohomeomorphisms of the unit interval

We prove that density preserving homeomorphisms form a Π ₁ ¹ -complete subset in the Polish space ? of all increasing autohomeomorphisms of unit interval.     

Reducing the polynomial-like iterative equations order and a generalized Zoltán Boros’ problem

We present a technique for reducing the order of polynomial-like iterative equations; in particular, we answer a question asked by Wenmeng Zhang and Weinian Zhang. Our method involves the asymptotic behaviour of the sequence of consecutive iterates of the unknown function at a given point. As an application we solve a generalized problem of Zoltán Boros posed during the 50th ISFE.     

On a theorem of E. Helly

E. Helly's theorem asserts that any bounded sequence of monotone real functions contains a pointwise convergent subsequence. We reprove this theorem in a generalized version in terms of monotone functions on linearly ordered sets. We show that the cardinal number responsible for this generalization is exactly the splitting number. We also show that a positive answer to a problem of S. Saks is obtained under the assumption of the splitting number being strictly greater than the first uncountable cardinal.

     

Homo-metathesis of vinylsilanes catalysed by ruthenium carbene complexes

Effective homo-metathesis of a series of dichloro-substituted vinylsilanes H₂C = C(H)SiCl₂R (where R = Me, OSiMe₃, C₆H₅, C₆H₄–Me-4, C₆H₄–CF₃-4) in the presence of second generation Grubbs catalyst [Cl₂(PCy₃)(IMesH₂)Ru(=CHPh)] (I) and Hoveyda–Grubbs catalyst (II) leads to selective formation of E-1,2-bis(silyl)ethenes and ethene. On the basis of the results of experiments with deuterium-labelled reagents, a metallacarbene mechanism has been suggested for these reactions.     

Density of states of a terminated superlattice with periodically arrangedδ-defects

Surface electronic structure and the local density of states (LDOS) of a periodicallyδ-doped GaAs/Al _x Ga_1−x As superlattice (SL), terminated by an Al _y Ga_1−y As substrate, are investigated using the wave-function multi-matching procedure within an effectivemass approximation and a Green-function formalism based on factorization and direct techniques. The considered system is described in terms of a semi-infinite Kronig-Penneytype of model interleaved with a periodic array ofδ-function-like potentials, representing deep-center defect sheets. It is shown that the spectrum of surface-localized states and the corresponding space-charge distributions within particular SL minibands can be tailored by varying the terminating conditions (i.e., the substrate parameters) as well as the properties of inserted defects (i.e., the weight and position ofδ-potentials). In this respect, importance of a symmetric/nonsymmetric arrangement ofδ-dopants inside SL barriers and the related effect of miniband crossing/anticrossing are emphasized. Presented at the VIII-th Symposium on Surface Physics, Třešt’ Castle, Czech Republic, June 28 – July 2, 1999. One of us (B.B.) gratefully acknowledges the support of the University of Wrocław via the project for young researchers No. 2314/W/IFD/99, while two of us (R.K. and M.S.) have been supported within the Grant No. 2016/W/IFD/98.     

Observability for regular processes

Regular processes generalize the notion of diffusion processes. We observe them with the aid of a linear map in order to determine (or to predict) the final state. The main features of observability for such processes are the phenomenon of critical time and exceeding fragility for perturbations.     

Metric characterizations of upper semicontinuity

We characterize upper semicontinuity of multifunctions in terms of upper Hausdorff semicontinuity, measure of non compactness and active boundary. The results are applicable in optimization, theory of best approximation and in metrizability theory.