On arithmetic and asymptotic properties of up-down numbers

Let $\sigma=(\sigma_1,..., \sigma_N)$, where $\sigma_i =\pm 1$, and let $C(\sigma)$ denote the number of permutations $\pi$ of $1,2,..., N+1,$ whose up-down signature $\mathrm{sign}(\pi(i+1)-\pi(i))=\sigma_i$, for $i=1,...,N$. We prove that the set of all up-down numbers $C(\sigma)$ can be expressed by a single universal polynomial $\Phi$, whose coefficients are products of numbers from the Taylor series of the hyperbolic tangent function. We prove that $\Phi$ is a modified exponential, and deduce some remarkable congruence properties for the set of all numbers $C(\sigma)$, for fixed $N$. We prove a concise upper-bound for $C(\sigma)$, which describes the asymptotic behaviour of the up-down function $C(\sigma)$ in the limit $C(\sigma) \ll (N+1)!$.Comment: Recommended for publication in Discrete Mathematics subject to revision

The XFEM with an Explicit-Implicit Crack Description for Hydraulic Fracture Problems

The Extended Finite Element Method (XFEM) approach is applied to the coupled problem of fluid flow, solid deformation, and fracture propagation. The XFEM model description of hydraulic fracture propagation is part of a joint project in which the developed numerical model will be verified against large-scale laboratory experiments. XFEM forms an important basis towards future combination with heat and mass transport simulators and extension to more complex fracture systems. The crack is described implicitly using three level-sets to evaluate enrichment functions. Additionally, an explicit crack representation is used to update the crack during propagation. The level-set functions are computed exactly from the explicit representation. This explicit/implicit representation is applied to a fluid-filled crack in an impermeable, elastic solid and compared to the early-time solution of a plane-strain hydraulic fracture problem with a fluid lag

On Growth, Disorder, and Field Theory

This article reviews recent developments in statistical field theory far from equilibrium. It focuses on the Kardar-Parisi-Zhang equation of stochastic surface growth and its mathematical relatives, namely the stochastic Burgers equation in fluid mechanics and directed polymers in a medium with quenched disorder. At strong stochastic driving -- or at strong disorder, respectively -- these systems develop nonperturbative scale-invariance. Presumably exact values of the scaling exponents follow from a self-consistent asymptotic theory. This theory is based on the concept of an operator product expansion formed by the local scaling fields. The key difference to standard Lagrangian field theory is the appearance of a dangerous irrelevant coupling constant generating dynamical anomalies in the continuum limit.Comment: review article, 50 pages (latex), 10 figures (eps), minor modification of original versio

Predicting Cell Cycle Regulated Genes by Causal Interactions

The fundamental difference between classic and modern biology is that technological innovations allow to generate high-throughput data to get insights into molecular interactions on a genomic scale. These high-throughput data can be used to infer gene networks, e.g., the transcriptional regulatory or signaling network, representing a blue print of the current dynamical state of the cellular system. However, gene networks do not provide direct answers to biological questions, instead, they need to be analyzed to reveal functional information of molecular working mechanisms. In this paper we propose a new approach to analyze the transcriptional regulatory network of yeast to predict cell cycle regulated genes. The novelty of our approach is that, in contrast to all other approaches aiming to predict cell cycle regulated genes, we do not use time series data but base our analysis on the prior information of causal interactions among genes. The major purpose of the present paper is to predict cell cycle regulated genes in S. cerevisiae. Our analysis is based on the transcriptional regulatory network, representing causal interactions between genes, and a list of known periodic genes. No further data are used. Our approach utilizes the causal membership of genes and the hierarchical organization of the transcriptional regulatory network leading to two groups of periodic genes with a well defined direction of information flow. We predict genes as periodic if they appear on unique shortest paths connecting two periodic genes from different hierarchy levels. Our results demonstrate that a classical problem as the prediction of cell cycle regulated genes can be seen in a new light if the concept of a causal membership of a gene is applied consequently. This also shows that there is a wealth of information buried in the transcriptional regulatory network whose unraveling may require more elaborate concepts than it might seem at first

Hierarchical coordination of periodic genes in the cell cycle of Saccharomyces cerevisiae

<p>Abstract</p> <p>Background</p> <p>Gene networks are a representation of molecular interactions among genes or products thereof and, hence, are forming causal networks. Despite intense studies during the last years most investigations focus so far on inferential methods to reconstruct gene networks from experimental data or on their structural properties, e.g., degree distributions. Their structural analysis to gain functional insights into organizational principles of, e.g., pathways remains so far under appreciated.</p> <p>Results</p> <p>In the present paper we analyze cell cycle regulated genes in <it>S. cerevisiae</it>. Our analysis is based on the transcriptional regulatory network, representing causal interactions and not just associations or correlations between genes, and a list of known periodic genes. No further data are used. Partitioning the transcriptional regulatory network according to a graph theoretical property leads to a hierarchy in the network and, hence, in the information flow allowing to identify two groups of periodic genes. This reveals a novel conceptual interpretation of the working mechanism of the cell cycle and the genes regulated by this pathway.</p> <p>Conclusion</p> <p>Aside from the obtained results for the cell cycle of yeast our approach could be exemplary for the analysis of general pathways by exploiting the rich causal structure of inferred and/or curated gene networks including protein or signaling networks.</p

1-D random landscapes and non-random data series

We study the simplest random landscape, the curve formed by joining consecutive data points $f_{1},\ldots,f_{N+1}$ with line segments, where the fi are i.i.d. random numbers and $f_{i}\ne f_{j}$. We label each segment increasing (+) or decreasing (-) and call this string of +'s and -'s the up-down signature $\sigma$. We calculate the probability $P(\sigma (f))$ for a random curve and use it to bound the algorithmic information content of f. We show that f can be compressed by $k = \log_2 {1/P(\sigma)} - N$ bits, where k is a universal currency for comparing the amount of pattern in different curves. By applying our results to microarray time series data, we blindly identify regulatory genes

Presence or absence of a prefrontal sulcus is linked to reasoning performance during child development.

The relationship between structural variability in late-developing association cortices like the lateral prefrontal cortex (LPFC) and the development of higher-order cognitive skills is not well understood. Recent findings show that the morphology of LPFC sulci predicts reasoning performance; this work led to the observation of substantial individual variability in the morphology of one of these sulci, the para-intermediate frontal sulcus (pimfs). Here, we sought to characterize this variability and assess its behavioral significance. To this end, we identified the pimfs in a developmental cohort of 72 participants, ages 6-18. Subsequent analyses revealed that the presence or absence of the ventral component of the pimfs was associated with reasoning, even when controlling for age. This finding shows that the cortex lining the banks of sulci can support the development of complex cognitive abilities and highlights the importance of considering individual differences in local morphology when exploring the neurodevelopmental basis of cognition

Hominoid-specific sulcal variability is related to face perception ability

The relationship among brain structure, brain function, and behavior is of major interest in neuroscience, evolutionary biology, and psychology. This relationship is especially intriguing when considering hominoid-specific brain structures because they cannot be studied in widely examined models in neuroscience such as mice, marmosets, and macaques. The fusiform gyrus (FG) is a hominoid-specific structure critical for face processing that is abnormal in individuals with developmental prosopagnosia (DPs)—individuals who have severe deficits recognizing the faces of familiar people in the absence of brain damage. While previous studies have found anatomical and functional differences in the FG between DPs and NTs, no study has examined the shallow tertiary sulcus (mid-fusiform sulcus, MFS) within the FG that is a microanatomical, macroanatomical, and functional landmark in humans, as well as was recently shown to be present in non-human hominoids. Here, we implemented pre-registered analyses of neuroanatomy and face perception in NTs and DPs. Results show that the MFS was shorter in DPs than NTs. Furthermore, individual differences in MFS length in the right, but not left, hemisphere predicted individual differences in face perception. These results support theories linking brain structure and function to perception, as well as indicate that individual differences in MFS length can predict individual differences in face processing. Finally, these findings add to growing evidence supporting a relationship between morphological variability of late developing, tertiary sulci and individual differences in cognition