Polynomial Sequences in Groups

AbstractGiven a groupGwith lower central seriesG=G1⊇G2⊇G3⊇···, we say that a sequenceg: Z→Gispolynomialif for anykthere isdsuch that the sequence obtained fromgby applying the difference operatorDg(n)=g(n)−1g(n+1)dtimes takes its values inGk. We introduce the notion ofthe degree of a polynomial sequenceand we prove that polynomial sequences of degrees not exceeding a given one form a group. As an application we obtain the following extension of the Hall–Petresco theorem:THEOREM.LetG=G1⊇G2⊇G3⊇···be the lower central series of a group G.Let x∈Gk,y∈Gland letp,qbe polynomialsZ→Zof degrees k and l,respectively. Then there is a sequencez0∈G,zi∈Gifori∈N,such thatfor all n∈N

Form discrete- to continuous-time ergodic theorems

We introduce methods that allow to derive continuous-time versions of various discrete-time ergodic theorems. We then illustrate these methods by giving simple proofs and refinements of some known results as well as establishing new results of interest

Uniformity in the Wiener-Wintner theorem for nilsequences

We prove a uniform extension of the Wiener-Wintner theorem for nilsequences due to Host and Kra and a nilsequence extension of the topological Wiener-Wintner theorem due to Assani. Our argument is based on (vertical) Fourier analysis and a Sobolev embedding theorem.Comment: v3: 18 p., proof that the cube construction produces compact homogeneous spaces added, measurability issues in the proof of Theorem 1.5 addressed. We thank the anonymous referees for pointing out these gaps in v

A nilpotent IP polynomial multiple recurrence theorem

We generalize the IP-polynomial Szemer\'edi theorem due to Bergelson and McCutcheon and the nilpotent Szemer\'edi theorem due to Leibman. Important tools in our proof include a generalization of Leibman's result that polynomial mappings into a nilpotent group form a group and a multiparameter version of the nilpotent Hales-Jewett theorem due to Bergelson and Leibman.Comment: v4: switch to TeXlive 2016 and biblate

Redistribution of ions within the active layer and upper permafrost, Yamal, Russia

A landslide-affected slope was chosen to study the ionic migration in the active layer and upper portion of permafrost. The research was conducted in two stages, in 1994 and 2001. Several boreholes, in dry and wet environments of the shearing surface of a 1989-landslide, were drilled. A background borehole on an undisturbed site was sampled as well. Each sample, collected from the core, underwent a conventional chemical cation-anion analysis. The results showed desalinization of the active layer and upper permafrost, which occurred in 7 years. Different migration rates noted for various salts determine change of ionic composition from marine pattern to continental, because mobile ions are washed away by surface and subsurface runoff, while the less mobile ones are accumulating in the upper portion of the active layer due to capillary rise and at the active layer base on a geochemical barrier

Reliability, Validity, and Interpretation of the Dependence Scale in Mild to Moderately Severe Alzheimer's Disease

INTRODUCTION: The Dependence Scale (DS) was designed to measure dependence on others among patients with Alzheimer's disease (AD). The objectives of this research were primarily to strengthen the psychometric evidence for the use of the DS in AD studies. METHODS: Patients with mild to moderately severe AD were examined in 3 study databases. Within each data set, internal consistency, validity, and responsiveness were examined, and structural equation models were fit. RESULTS: The DS has strong psychometric properties. The DS scores differed significantly across known groups and demonstrated moderate to strong correlations with measures hypothesized to be related to dependence (|r| >/= .31). Structural equation modeling supported the validity of the DS concept. An anchor-based DS responder definition to interpret a treatment benefit over time was identified. DISCUSSION: The DS is a reliable, valid, and interpretable measure of dependence associated with AD and is shown to be related to--but provides information distinct from--cognition, functioning, and behavior

Geodatabase and WebGIS project for long-term permafrost monitoring at the Vaskiny Dachi Research Station, Yamal, Russia

The research station Vaskiny Dachi (VD) in central Yamal, Western Siberia was established in 1988. Continuous monitoring of the permafrost state is conducted since 25 years, which allows collecting a large amount of data related to permafrost state and environment of this region. To store and visualise the geospatial data, containing our knowledge of the research area and research topic, we created a geodatabase (GDB) to operatively process different types of geospatial data. The produced GDB contains so far 11 vector feature datasets and raster data in the same coordinate system. The vector data represent: 1) bathymetry; 2) social-economic objects; 3) field data; 4) geomorphology; 5) hydrography; 6) landscapes; 7) permafrost; 8) snow; 9) topography; 10) vegetation; 11) long-term measurement grids and transects (Circumpolar Active Layer Monitoring (CALM) transect, CALM measurement grid). All these feature datasets contain 60 feature classes of spatial data in total. Some of the geodata layers are directly linked to data bases of field data. The raster data contain 37 layers, including a digital ele vation model with derivatives, a map of snow distribution for the key site, ba thymetric maps and other maps of different scale. Moreover, the key area is a site for international research projects and the ongoing exchange of the data is supported by the VD GDB. Geographical Information System (GIS) allows collecting, storing and processing geospatial data from different sources in a wide range of types and formats. WebGIS platforms allow displaying the geospatial data for different users, giving the impression of the general pro cesses on the certain geographic area. Also, we use the WebGIS service to publish the data and to make it available for the larger community. This paper is an overview on the permafrost studies at the VD research station, the GDB for permafrost monitoring as well as the established Yamal WebGIS project

Linear forms and quadratic uniformity for functions on ZN\mathbb{Z}_N

A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that quasirandomness can often be measured by means of certain easily described norms, known as uniformity norms. However, determining which uniformity norms work for which structures turns out to be a surprisingly hard question. In [GW09a] and [GW09b, GW09c] we gave a complete answer to this question for groups of the form $G=\mathbb{F}_p^n$, provided $p$ is not too small. In $\mathbb{Z}_N$, substantial extra difficulties arise, of which the most important is that an "inverse theorem" even for the uniformity norm $\|.\|_{U^3}$ requires a more sophisticated (local) formulation. When $N$ is prime, $\mathbb{Z}_N$ is not rich in subgroups, so one must use regular Bohr neighbourhoods instead. In this paper, we prove the first non-trivial case of the main conjecture from [GW09a].Comment: 66 page

Классификация криогенно-оползневых форм рельефа для целей картографирования и прогноза

A classification of cryogenic-landslide landforms is developed for mapping their distribution and dynamics. It is based on the previously suggested classification subdividing cryogenic landsliding into two main types: cryogenic translational landslides (or active-layer detachment slides), and cryogenic earth flows (or retrogressive thaw slumps). The increased proportion of retrogressive thaw slumps compared to active layer detachments in the North of West Siberia in the last decade creates the need for an expanded classification of cryogenic earth flows. One of the important issues is separating the process of landsliding and resulting landforms, which in English are covered by one term ‘retrogressive thaw slump’. In dealing with the landforms, we distinguish (1) open and (2) closed ones. Open cryogenic-landslide landforms are those formed by the retreating of the coast bluff due to the thaw of ice or ice-rich deposits with an additional impact from wave or stream action. Closed cryogenic-landslide landforms are those initiated on a slope landward, and thawed material is delivered to the coast or stream through an erosional channel. Morphologically we distinguish thermocirques and thermoterraces depending on the shape of the retreating headwall, crescent or linear, respectively. An important issue is the type of ground ice subjected to thaw: tabular, ice-wedge or constitutional ground ice are distinguished. Landforms can be active, stabilized or ancient. One can find both single landforms and their combination. The classification is based on a significant amount of field studies and interpretation of remote sensing data. Mapping of the cryogenic-landslide landforms is suggested using the proposed classification and indication features. The classification is based on the experience obtained mainly in the north of West Siberia. Applying it to other regions may require additional studies.Разработана классификация криогенно-оползневых форм рельефа, сформированных криогенными оползнями течения (КОТФР), для картографирования их распространения и динамики. В основе лежит значительный объем полевых исследований и интерпретации данных дистанционного зондирования Земли. Классификация включает генетические, морфологические и криолитологические особенности пород, определяющие морфологию и динамику КОТФР, их положение в рельефе, степень их активности, сочетание и комплексирование единичных КОТФР. Предложенная классификация и индикационные признаки используются для картографирования КОТФР на севере Западной Сибири