Tverberg Plus Minus

We prove a Tverberg type theorem: Given a set A Rd in general position with | A| = (r- 1) (d+ 1) + 1 and k∈ { 0 , 1 , … , r- 1 } , there is a partition of A into r sets A1, … , Ar (where | Aj| ≤ d+ 1 for each j) with the following property. There is a unique zj=1raffAj and it can be written as an affine combination of the element in Aj: z=∑x∈Ajα(x)x for every j and exactly k of the coefficients α(x) are negative. The case k= 0 is Tverberg’s classical theorem. © 2018, Springer Science+Business Media, LLC, part of Springer Nature

Tverberg's theorem, a new proof

We give a new proof Tverberg's famous theorem: For every set $X \subset \R^d$ with $|X|=(r-1)(d+1)+1$, there is a partition of $X$ into $r$ sets $X_1,\ldots,X_r$ such that \bigcap_{p=1}^r \conv X_p\ne \emptyset. The new proof uses linear algebra, specially structured matrices, the theory of linear equations, and Tverberg's original ``moving the points" method

Warming of Atlantic Water in two west Spitsbergen fjords over the last century (1912-2009)

The recently observed warming of west Spitsbergen fjords has led to anomalous sea-ice conditions and has implications for the marine ecosystem. We investigated long-term trends of maximum temperature of Atlantic Water (AW) in two west Spitsbergen fjords. The data set is composed of more than 400 oceanographic stations for Isfjorden and Grønfjorden (78.1°N), spanning from 1876 to 2009. Trends throughout the last century (1912–2009) indicate an increase of 1.9°C and 2.1°C in the maximum temperature during autumn for Isfjorden and Grønfjorden, respectively. A recent warming event in the beginning of the 21st century is found to be more than 1°C higher than the second warmest period in the time series. Mean sea-level pressure (MSLP) data from ERA-40 and ERA-Interim data sets produced by the European Centre for Medium-Range Weather Forecasts and mean temperature in the core of the West Spitsbergen Current (WSC) at the Sørkapp Section along 76.3°N were used to explain the variability of the maximum temperature. A correlation analysis confirmed previous findings, showing that variability in the oceanography of the fjords can be explained mainly by two external factors: AW temperature variability in the WSC and regional patterns of the wind stress field. To take both processes into consideration, a multiple regression model accounting for temperature in the WSC core and MSLP over the area was developed. The predicted time series shows a reasonable agreement with observed maxima temperature in Isfjorden for the period 1977–2009 (N=24), with a statistically significant multiple correlation coefficient of 0.60 (R2=0.36) at P<0.05.publishedVersio

A Geometric Proof of the Colored Tverberg Theorem

The colored Tverberg theorem asserts that for every d and r there exists t=t(d,r) such that for every set C⊂ℝ d of cardinality (d+1)t, partitioned into t-point subsets C 1,C 2, ,C d+1 (which we think of as color classes; e.g., the points of C 1 are red, the points of C 2 blue, etc.), there exist r disjoint sets R 1,R 2, ,R r ⊆C that are rainbow, meaning that |R i ∩C j |≤1 for every i,j, and whose convex hulls all have a common point. All known proofs of this theorem are topological. We present a geometric version of a recent beautiful proof by Blagojević, Matschke, and Ziegler, avoiding a direct use of topological methods. The purpose of this de-topologization is to make the proof more concrete and intuitive, and accessible to a wider audienc