Abstract:
A new approach to stochastic integration is described, which is based on an a.s. pathwise approximation of the integrator by simple, symmetric random walks. Hopefully, this method is didactically more advantageous, more transparent, and technically less demanding than other existing ones. In a large part of the theory one has a.s. uniform convergence on compacts. In particular, it gives a.s. convergence for the stochastic integral of a finite variation function of the integrator, which is not c\`adl\`ag in general.

Abstract:
An active landslide in Doren, Austria, has been studied by multitemporal airborne and terrestrial laser scanning from 2003 to 2012. To evaluate the changes, we have determined the 3D motion using the range flow algorithm, an established method in computer vision, but not yet used for studying landslides. The generated digital terrain models are the input for motion estimation; the range flow algorithm has been combined with the coarse-to-fine resolution concept and robust adjustment to be able to determine the various motions over the landslide. The algorithm yields fully automatic dense 3D motion vectors for the whole time series of the available data. We present reliability measures for determining the accuracy of the estimated motion vectors, based on the standard deviation of components. The differential motion pattern is mapped by the algorithm: parts of the landslide show displacements up to 10 m, whereas some parts do not change for several years. The results have also been compared to pointwise reference data acquired by independent geodetic measurements; reference data are in good agreement in most of the cases with the results of range flow algorithm; only some special points (e.g., reflectors fixed on trees) show considerably differing motions.

Abstract:
In this paper, we consider a family of random Cantor sets on the line and consider the question of whether the condition that the sum of the Hausdorff dimensions is larger than one implies the existence of interior points in the difference set of two independent copies. We give a new and complete proof that this is the case for the random Cantor sets introduced by Per Larsson.

Abstract:
Purpose Quantifying chromosomal instability (CIN) has both prognostic and predictive clinical utility in breast cancer. In order to establish a robust and clinically applicable gene expression-based measure of CIN, we assessed the ability of four qPCR quantified genes selected from the 70-gene Chromosomal Instability (CIN70) expression signature to stratify outcome in patients with grade 2 breast cancer. Methods AURKA, FOXM1, TOP2A and TPX2 (CIN4), were selected from the CIN70 signature due to their high level of correlation with histological grade and mean CIN70 signature expression in silico. We assessed the ability of CIN4 to stratify outcome in an independent cohort of patients diagnosed between 1999 and 2002. 185 formalin-fixed, paraffin-embedded (FFPE) samples were included in the qPCR measurement of CIN4 expression. In parallel, ploidy status of tumors was assessed by flow cytometry. We investigated whether the categorical CIN4 score derived from the CIN4 signature was correlated with recurrence-free survival (RFS) and ploidy status in this cohort. Results We observed a significant association of tumor proliferation, defined by Ki67 and mitotic index (MI), with both CIN4 expression and aneuploidy. The CIN4 score stratified grade 2 carcinomas into good and poor prognostic cohorts (mean RFS: 83.8±4.9 and 69.4±8.2 months, respectively, p = 0.016) and its predictive power was confirmed by multivariate analysis outperforming MI and Ki67 expression. Conclusions The first clinically applicable qPCR derived measure of tumor aneuploidy from FFPE tissue, stratifies grade 2 tumors into good and poor prognosis groups.

Abstract:
We study an analogue of the classical moment problem in the framework where moments are indexed by graphs instead of natural numbers. We study limit objects of graph sequences where edges are labeled by elements of a topological space. Among other things we obtain strengthening and generalizations of the main results of previous papers characterizing reflection positive graph parameters, graph homomorphism numbers, and limits of simple graph sequences. We study a new class of reflection positive partition functions which generalize the node-coloring models (homomorphisms into weighted graphs).

Abstract:
Following a general program of studying limits of discrete structures, and motivated by the theory of limit objects of converge sequences of dense simple graphs, we study the limit of graph sequences such that every edge is labeled by an element of a compact second-countable Hausdorff space K. The "local structure" of these objects can be explored by a sampling process, which is shown to be equivalent to knowing homomorphism numbers from graphs whose edges are decorated by continuous functions on K. The model includes multigraphs with bounded edge multiplicities, graphs whose edges are weighted with real numbers from a finite interval, edge-colored graphs, and other models. In all these cases, a limit object can be defined in terms of 2-variable functions whose values are probability distributions on K.

Abstract:
In an earlier paper the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency conditions, and normalized, multiplicative and reflection positive graph parameters. In this paper we show that each of these structures has a related, relaxed version, which are also equivalent. Using this, we describe a further structure equivalent to graph limits, namely probability measures on countable graphs that are ergodic with respect to the group of permutations of the nodes. As an application, we prove an analogue of the Positivstellensatz for graphs: We show that every linear inequality between subgraph densities that holds asymptotically for all graphs has a formal proof in the following sense: it can be approximated arbitrarily well by another valid inequality that is a "sum of squares" in the algebra of partially labeled graphs.

Abstract:
We highlight a topological aspect of the graph limit theory. Graphons are limit objects for convergent sequences of dense graphs. We introduce the representation of a graphon on a unique metric space and we relate the dimension of this metric space to the size of regularity partitions. We prove that if a graphon has an excluded induced sub-bigraph then the underlying metric space is compact and has finite packing dimension. It implies in particular that such graphons have regularity partitions of polynomial size.

Abstract:
We study the automorphism group of graphons (graph limits). We prove that after an appropriate "standardization" of the graphon, the automorphism group is compact. Furthermore, we characterize the orbits of the automorphism group on $k$-tuples of points. Among applications we study the graph algebras defined by finite rank graphons and the space of node-transitive graphons.

Abstract:
At an Eastern Hungarian protected grassland, namely at the Lesser Mole Rat (Spalax leucodon Nordmann, 1840). Reservation of Hajdúbagos Nature Conservation Area, grazing animal husbandry formed the fa ade of the land for hundreds of years. Though, due to the unfavourable changes of the last few decades in this sector of agriculture, the primeval sand steppe meadow (Pulsatillo hungaricae-Festucetum rupicolae (Soó 1938) Borhidi 1996) plant association is endangered by the increasingly accelerating succession. To stop or at least to slow down this process the rehabilitation of the area could be necessary. The target of the restoration ecology actions is to restore the previously existing, more favourable natural status of a particular area. However, the lack of knowledge according to the conditions that refer to the original circumstances often complicates this activity. To define these reference conditions the exploration of the history of a certain landscape is very important as restoration is only successful if the restored ecosystem is similar to the original. We examined the land use changes of the research area during the last 250 years according to historical and present geographical databases using GIS technology and completed this work with the study of archival data. By our results we stated that these examinations are crucial in the proper - agriculture related - management of protected areas.