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Increasing digit subsystems of infinite iterated function systems
Thomas Jordan,Michal Rams
Mathematics , 2010,
Abstract: We consider infinite iterated function systems $\{f_i\}_{i=1}^{\infty}$ on $[0,1]$ with a polynomially increasing contraction rate. We look at subsets of such systems where we only allow iterates $f_{i_1}\circ f_{i_2}\circ f_{i_3}\circ...$ if $i_n>\Phi(i_{n-1})$ for certain increasing functions $\Phi:\mathbb{N}\rightarrow\mathbb{N}$. We compute both the Hausdorff and packing dimensions of such sets. Our results generalize work of Ramharter which shows that the set of continued fractions with strictly increasing digits has Hausdorff dimension 1/2.
Packing spectra for Bernoulli measures supported on Bedford-McMullen carpets
Thomas Jordan,Micha？ Rams
Mathematics , 2014,
Abstract: In this paper we consider the packing spectra for local dimension of Bernoulli measures supported on Bedford-McMullen carpets. We show that typically the packing dimension of the regular set is smaller than the packing dimension of the attractor. We also consider a specific class of measures for which we are able to calculate the packing spectrum exactly and we show that the packing spectrum is discontinuous as a function on the space of Bernoulli measures.
Multifractal analysis of weak Gibbs measures for non-uniformly expanding C^1 maps
Thomas Jordan,Michal Rams
Mathematics , 2008,
Abstract: We consider the local dimension spectrum of a weak Gibbs measure on a C^1 non-uniformly hyperbolic system of Manneville- Pomeau type. We present the spectrum in three ways: using invariant measures, uniformly hyperbolic ergodic measures and equilibrium states. We are also proving analyticity of the spectrum under additional assumptions. All three presentations are well known for smooth uniformly hyperbolic systems.
Multifractal analysis for Bedford-McMullen carpets
Thomas Jordan,Michal Rams
Mathematics , 2009,
Abstract: In this paper we compute the multifractal analysis for local dimensions of Bernoulli measures supported on the self-affine carpets introduced by Bedford-McMullen. This extends the work of King where the multifractal analysis is computed with strong additional separation assumptions.
Plasma NT-proBNP and white matter hyperintensities in type 2 diabetic patients
Henrik Reinhard, Ellen Garde, Arnold Skimminge, Per ？keson, Thomas Rams？y, Kaj Winther, Hans-Henrik Parving, Peter Rossing, Peter K Jacobsen
Cardiovascular Diabetology , 2012, DOI: 10.1186/1475-2840-11-119
Abstract: We measured P-NT-proBNP(ng/l) in 20 diabetic patients without prior stroke but with(n？=？10) or without(n？=？10) asymptomatic coronary artery disease(CAD) in order to include patients with a wide-ranging CV risk profile. All patients and 26 controls had a 3D MRI and brain volumes(ml) with WMH and brain parenchymal fraction(BPF), an indicator of brain atrophy, were determined.P-NT-proBNP was associated with WMH in linear regression analysis adjusted for CV risk factors(r？=？0.94, p？=？0.001) and with BPF in univariate analysis(r？=？0.57, p？=？0.009). Patients divided into groups of increased P-NT-proBNP levels were paralleled with increased WMH volumes(geometric mean[SD];(2.86[5.11] ml and 0.76[2.49] ml compared to patients with low P-NT-proBNP 0.20[2.28] ml, p？=？0.003)) and also when adjusted for age, sex and presence of CAD(p？=？0.017). The association was strengthened by CV risk factors and we did not find a common heart or brain specific driver of both P-NT-proBNP and WMH. Patients and particular patients with CAD had higher WMH, however no longer after adjustment for age and sex.P-NT-proBNP was associated with WMH in type 2 diabetic patients, suggesting a linkage between heart and brain disease.Manifest atherosclerosis that includes strokes is the most important determinant of the excessive morbidity and mortality in patients with type 2 diabetes, especially in patients with microalbuminuria [1]. Although medical treatment aimed at reduction of established conventional risk factors is effective in reducing the increased cardiovascular morbidity and mortality in patients with diabetes, there is still excess morbidity and mortality compared to the background population [2]. Detection of subclinical atherosclerotic manifestations in patients with diabetes in order to be able to identify patients at risk and start appropriate intervention at an early stage therefore seems evident.Elevated plasma N-terminal-proBNP (P-NT-proBNP) levels released from the heart as well as whit
Dimension of self-affine sets with holes
Andrew Ferguson,Thomas Jordan,Micha？ Rams
Mathematics , 2012,
Abstract: In this paper we compute the dimension of a class of dynamically defined non-conformal sets. Let $X\subseteq\mathbb{T}^2$ denote a Bedford-McMullen set and $T:X\to X$ the natural expanding toral endomorphism which leaves $X$ invariant. For an open set $U\subset X$ we let X_U={x\in X : T^k(x)\not\in U \text{for all}k}. We investigate the box and Hausdorff dimensions of $X_U$ for both a fixed Markov hole and also when $U$ is a shrinking metric ball. We show that the box dimension is controlled by the escape rate of the measure of maximal entropy through $U$, while the Hausdorff dimension depends on the escape rate of the measure of maximal dimension.
Multifractal analysis for expanding interval maps with infinitely many branches
Ai-Hua Fan,Thomas Jordan,Lingmin Liao,Michal Rams
Mathematics , 2011,
Abstract: In this paper we investigate multifractal decompositions based on values of Birkhoff averages of functions from a class of symbolically continuous functions. This will be done for an expanding interval map with infinitely many branches and is a generalisation of previous work for expanding maps with finitely many branches. We show that there are substantial differences between this case and the setting where the expanding map has only finitely many branches.
A reduction map for nef line bundles
Thomas Bauer,Frederic Campana,Thomas Eckl,Stefan Kebekus,Thomas Peternell,Slawomir Rams,Tomasz Szemberg,Lorenz Wotzlaw
Mathematics , 2001,
Abstract: In a recent preprint, H. Tsuji gave a number of interesting assertions on the structure of pseudo-effective line bundles on projective manifolds. In particular, he postulated the existence of an almost-holomorphic "reduction map", whose fibers are maximal subvarieties on which the line bundle is numerically trivial. The purpose of this note is to give a simple, algebraic proof of the existence of a reduction map in the case where the line bundle is nef.
On some non-conformal fractals
Michal Rams
Mathematics , 2010, DOI: 10.1088/0951-7715/23/10/004
Abstract: This paper presents a simple method of calculating the Hausdorff dimension for a class of non-conformal fractals.
On quartics with three-divisible sets of cusps
Slawomir Rams
Mathematics , 2002,
Abstract: We study the geometry and codes of quartic surfaces with many cusps. We apply Gr\"obner bases to find examples of various configurations of cusps on quartics.
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