Set-polynomials and polynomial extension of the Hales-Jewett Theorem

An abstract, Hales-Jewett type extension of the polynomial van der Waerden Theorem [J. Amer. Math. Soc. 9 (1996),725-753] is established: Theorem. Let r,d,q \in \N. There exists N \in \N such that for any r-coloring of the set of subsets of V={1,...,N}^{d} x {1,...,q} there exist a set a \subset V and a nonempty set \gamma \subseteq {1,...,N} such that a \cap (\gamma^{d} x {1,...,q}) = \emptyset, and the subsets a, a \cup (\gamma^{d} x {1}), a \cup (\gamma^{d} x {2}), ..., a \cup (\gamma^{d} x {q}) are all of the same color. This ``polynomial'' Hales-Jewett theorem contains refinements of many combinatorial facts as special cases. The proof is achieved by introducing and developing the apparatus of set-polynomials (polynomials whose coefficients are finite sets) and applying the methods of topological dynamics.Comment: 43 pages, published versio

Intersective polynomials and polynomial Szemeredi theorem

Let P=\{p_{1},\ld,p_{r}\}\subset\Q[n_{1},\ld,n_{m}] be a family of polynomials such that p_{i}(\Z^{m})\sle\Z, i=1,\ld,r. We say that the family $P$ has {\it PSZ property} if for any set E\sle\Z with d^{*}(E)=\limsup_{N-M\ras\infty}\frac{|E\cap[M,N-1]|}{N-M}>0 there exist infinitely many $n\in\Z^{m}$ such that $E$ contains a polynomial progression of the form \hbox{\{a,a+p_{1}(n),\ld,a+p_{r}(n)\}}. We prove that a polynomial family P=\{p_{1},\ld,p_{r}\} has PSZ property if and only if the polynomials p_{1},\ld,p_{r} are {\it jointly intersective}, meaning that for any $k\in\N$ there exists $n\in\Z^{m}$ such that the integers p_{1}(n),\ld,p_{r}(n) are all divisible by $k$. To obtain this result we give a new ergodic proof of the polynomial Szemer\'{e}di theorem, based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds. We also obtain, as a corollary, the following generalization of the polynomial van der Waerden theorem: If p_{1},\ld,p_{r}\in\Q[n] are jointly intersective integral polynomials, then for any finite partition of $\Z$, $\Z=\bigcup_{i=1}^{k}E_{i}$, there exist i\in\{1,\ld,k\} and $a,n\in E_{i}$ such that \{a,a+p_{1}(n),\ld,a+p_{r}(n)\}\sln E_{i}

Homology of iterated semidirect products of free groups

Let $G$ be a group which admits the structure of an iterated semidirect product of finitely generated free groups. We construct a finite, free resolution of the integers over the group ring of $G$. This resolution is used to define representations of groups which act compatibly on $G$, generalizing classical constructions of Magnus, Burau, and Gassner. Our construction also yields algorithms for computing the homology of the Milnor fiber of a fiber-type hyperplane arrangement, and more generally, the homology of the complement of such an arrangement with coefficients in an arbitrary local system.Comment: 31 pages. AMSTeX v 2.1 preprint styl

Modern rates of thermal denudation and thermal abrasion on western Kolguev Island

Destruction mechanisms and dynamics of the Arctic coast, also in the western sector of the Russian Arctic, are studied in detail, including the use of remote sensing data. However, data on thermal abrasion and thermo denudation of Kolguev island is quite limited. Some estimates were presented in article of M.A.Velikotsky (1998). Estimation of thermos denudation rates near the Sauchiha river mouth for the period 1948-2002 years was done by the authors earlier (Kizyakov& Perednya, 2003). To obtain data about the modern (after 2002) shoreline retreat rates and growth of thermal cirque a high resolution remote sensing data were involved in our research. Part of the western coast of the Kolguev island was inspected in field work conducted on 2002 by ECI SB RAS, together with VNIIOkeangeologia. The object of research was the part of coast, including a group of three coastal thermal cirques, located 3.5 km south of the Sauchiha river mouth. In 2012, within the framework of the project ‘Geoportal of MSU’ operational satellite imaging was done on Kolguev island by satellite FORMOSAT-2. High resolution satellite imagery provides ample opportunities for visual interpretation of coastal landforms. Aerial photographs (1948 and 1968), surveying materials (2002), high-resolution satellite images (2009 and 2012) became basis to study the dynamics of the coast and thermal cirques in the key area. For key area were calculated: retreat rates of the edge of the coastal terraces and thermal cirques for the periods 1948-1968, 1968-2002, 2002-2009, 2009-2012; retreat rates of the foot of the coastal terrace for the periods 2002-2009, 2009-2012; volume of the material enters the coastal zone by the thermal abrasion for one linear km of a coast (Kizyakov et al., 2013). Average long-term rates of retreat of the coastal terrace during 1948-2012 varied from 0.7 to 2.4 m/year; 2002-2012 varied from 1.7 to 2.4 m/year. Identified rates are distinctive for the part of coast from the mouth of Krivaya river to the curve of coastline near the mouth of the Gusinaya river - a length is 60.5 km. These rates are in 1.1-1.5 times lower than average rates of retreat of thermal cirque edges which are connected with melting of massive ice deposits. Averaged growth rates of the thermal cirques in 1948-2002 was 2.4 m/year; in 2002-2012 was - 2.6 m/year. The maximum growth rate on some sections in 2009-2012 were 14.5-15.1 m/year. These rates are the largest for the previously recorded in the Western sector of the Russian Arctic. The cause of the abnormally high rates is an increase the annual amount of positive air temperatures, which in 2011-2012 was 1.4-1.5 times higher than the long-term average. The determined rates of the development of thermal cirque can be extended to the north from the key area (near the Sauchiha river mouth) to the Gusinaya river mouth with total length of 32.3 km. The next plans on studying the coastal dynamics on Kolguev Island - using additional satellite images for the purposes of: detailization of interannual dynamics through the analysis of more short time span series of satellite images, definition of variations of the coastal destruction rates on the Western and Northern coasts. References: Velikotsky M.A. Characteristics of modern coastal dynamics of the Kolguev Island // Dynamics of the Russian Arctic coasts, Moscow, MSU – 1989 – P.93- 101 (In Russian) Kizyakov A.I., Perednya D.D. Destruction of coasts on the Yugorsky Peninsula and on Kolguev Island (Russia) // Permafrost: Abstr. of the 8th Intern. Conf. (Zurich, Switzerland, 21–25 July 2003). Zurich, Switzerland – 2003 – P. 79–80. Kizyakov A.I., Zimin M.V., Leibman M.O., Pravikova N.V. Monitoring the rate of thermal denudation and thermal abrasion on the western coast of Kolguev Island using high resolution satellite images // Earth Cryosphere (Kriosfera Zemli). – 2013, XVII, No. 4 – P. 15-25 (In Russian)

Northern Hemisphere permafrost map based on TTOP modelling for 2000-2016 at 1 km²scale

Permafrost is a key element of the cryosphere and an essential climate variable in the Global Climate Observing System. There is no remote-sensing method available to reliably monitor the permafrost thermal state. To estimate permafrost distribution at a hemispheric scale, we employ an equilibrium state model for the temperature at the top of the permafrost (TTOP model) for the 2000–2016 period, driven by remotely-sensed land surface temperatures, down-scaled ERA-Interim climate reanalysis data, tundra wetness classes and landcover map from the ESA Landcover Climate Change Initiative (CCI) project. Subgrid variability of ground temperatures due to snow and landcover variability is represented in the model using subpixel statistics. The results are validated against borehole measurements and reviewed regionally. The accuracy of the modelled mean annual ground temperature (MAGT) at the top of the permafrost is ±2 °C when compared to permafrost borehole data. The modelled permafrost area (MAGT 0) is around 21 × 106 km2 (22% of exposed land area), which is approximately 2 × 106 km2 less than estimated previously. Detailed comparisons at a regional scale show that the model performs well in sparsely vegetated tundra regions and mountains, but is less accurate in densely vegetated boreal spruce and larch forests

Northern Hemisphere permafrost map based on TTOP modelling for 2000–2016 at 1 km2 scale

Permafrost is a key element of the cryosphere and an essential climate variable in the Global Climate Observing System. There is no remote-sensing method available to reliably monitor the permafrost thermal state. To estimate permafrost distribution at a hemispheric scale, we employ an equilibrium state model for the temperature at the top of the permafrost (TTOP model) for the 2000–2016 period, driven by remotely- sensed land surface temperatures, down-scaled ERA-Interim climate reanalysis data, tundra wetness classes and landcover map from the ESA Landcover Climate Change Initiative (CCI) project. Subgrid variability of ground temperatures due to snow and landcover variability is represented in the model using subpixel statistics. The results are validated against borehole measurements and reviewed regionally. The accuracy of the modelled mean annual ground temperature (MAGT) at the top of the permafrost is ±2 °C when compared to permafrost borehole data. The modelled permafrost area (MAGT 0) is around 21 × 106 km2 (22% of exposed land area), which is approximately 2 × 106 km2 less than estimated previously. Detailed comparisons at a regional scale show that the model performs well in sparsely vegetated tundra regions and mountains, but is less accurate in densely vegetated boreal spruce and larch forests