Abstract:
Signaling pathways belong to a complex system of communication that governs cellular processes. They represent signal transduction from an extracellular stimulus via a receptor to intracellular mediators, as well as intracellular interactions. Perturbations in signaling cascade often lead to detrimental changes in cell function and cause many diseases, including cancer. Identification of deregulated pathways may advance the understanding of complex diseases and lead to improvement of therapeutic strategies. We propose Analysis of Consistent Signal Transduction (ACST), a novel method for analysis of signaling pathways. Our method incorporates information regarding pathway topology, as well as data on the position of every gene in each pathway. To preserve gene-gene interactions we use a subject-sampling permutation model to assess the significance of pathway perturbations. We applied our approach to nine independent datasets of global gene expression profiling. The results of ACST, as well as three other methods used to analyze signaling pathways, are presented in the context of biological significance and repeatability among similar, yet independent, datasets. We demonstrate the usefulness of using information of pathway structure as well as genes’ functions in the analysis of signaling pathways. We also show that ACST leads to biologically meaningful results and high repeatability.

Abstract:
It is shown that for non-hyperbolic real quadratic polynomials topological and quasisymmetric conjugacy classes are the same. By quasiconformal rigidity, each class has only one representative in the quadratic family, which proves that hyperbolic maps are dense.

Abstract:
Invertible compositions of one-dimensional maps are studied which are assumed to include maps with non-positive Schwarzian derivative and others whose sum of distortions is bounded. If the assumptions of the Koebe principle hold, we show that the joint distortion of the composition is bounded. On the other hand, if all maps with possibly non-negative Schwarzian derivative are almost linear-fractional and their nonlinearities tend to cancel leaving only a small total, then they can all be replaced with affine maps with the same domains and images and the resulting composition is a very good approximation of the original one. These technical tools are then applied to prove a theorem about critical circle maps.

Abstract:
The integration of ASG-EUPOS network with the national first order networks was prepared in two measurement campaigns carried out in 2008 and 2010/2011. The measurements included a total of 340 points of POLREF - the first order national GNSS network. That set of data gives the possibility for comparison of solutions prepared for comprehensive analysis of the points' position and coordinates changes in the long term. Because the results of POLREF were not published the paper includes a brief description of campaigns from 1994 and 1995 and prepared solutions as well as the results of coordinates comparison with campaigns 2008 and 2010/2011. For points of significant deviations in coordinates the analysis of causes was presented.

Abstract:
We consider fixed points of the Feigenbaum (periodic-doubling) operator whose orders tend to infinity. It is known that the hyperbolic dimension of their Julia sets go to 2. We prove that the Lebesgue measure of these Julia sets tend to zero. An important part of the proof consists in applying martingale theory to a stochastic process with non-integrable increments.

Abstract:
For exponential mappings such that the orbit of the only singular value 0 is bounded, it is shown that no integrable density invariant under the dynamics exists on the complex plane.

Abstract:
We consider infinitely renormalizable unimodal mappings with topological type which is periodic under renormalization. We study the limiting behavior of fixed points of the renormalization operator as the order of the critical point increases to infinity. It is shown that a limiting dynamics exists, with a critical point that is flat, but still having a well-behaved analytic continuation to a neighborhood of the real interval pinched at the critical point. We study the dynamics of limiting maps and prove their rigidity. In particular, the sequence of fixed points of renormalization for finite criticalities converges, uniformly on the real domain, to a mapping of the limiting type.

Abstract:
We estimate harmonic scalings in the parameter space of a one-parameter family of critical circle maps. These estimates lead to the conclusion that the Hausdorff dimension of the complement of the frequency-locking set is less than $1$ but not less than $1/3$. Moreover, the rotation number is a H\"{o}lder continuous function of the parameter.