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A generalization of the Schwarz–Christoffel mapping to multiply connected polygonal domains is obtained by making a combined use of two preimage domains, namely, a rectilinear slit domain and a bounded circular domain. The conformal mapping from the circular domain to the polygonal region is written as an indefinite integral whose integrand consists of a product of powers of the Schottky-Klein prime functions, which is the same irrespective of the preimage slit domain, and a prefactor function that depends on the choice of the rectilinear slit domain. A detailed derivation of the mapping formula is given for the case where the preimage slit domain is the upper half-plane with radial slits. Representation formulae for other canonical slit domains are also obtained but they are more cumbersome in that the prefactor function contains arbitrary parameters in the interior of the circular domain.

This paper presents a boundary integral equation method with the adjoint generalized Neumann kernel for computing conformai mapping of unbounded multiply connected regions and its inverse onto several classes of canonical regions. For each canonical region, two integral equations are solved before one can approximate the boundary values of the mapping function. Cauchy's-type integrals are used for computing the mapping function and its inverse for interior points. This method also works for regions with piecewise smooth boundaries. Three examples are given to illustrate the effectiveness of the proposed method.

A formula for the generalized Schwarz-Christoffel mapping from a bounded multiply connected circular domain to a bounded multiply connected polygonal domain is derived. The theory of classical Schottky groups is employed. The formula for the derivative of the mapping function contains a product of powers of Schottky-Klein prime functions associated with a Schottky group relevant to the circular pre-image domain. The formula generalizes, in a natural way, the known mapping formulae for simply and doubly connected polygonal domains.

We examine the development of Cauchy's closed curve theorem, including the early contributions of Clairaut, d'Alembert, Cauchy himself, Goursat, and Pringsheim, as well as more recent approaches due to Ahlfors, Rudin, and others. A particularly simple proof was given by D. J. Newman, utilizing his original definition of a simply-connected region in the (complex) plane. We show that this definition is equivalent to the other, more familiar definitions of simple-connectedness so that Newman's approach offers an alternative and very elegant proof of the general result.

Given M a Riemannian manifold with (possibly empty) boundary, we show that its volume spectrum {ō p (M)} pεℕ satisfies a Weyl law that was conjectured by Gromov.

Neuronal dynamics display a complex spatiotemporal structure involving the precise, context-dependent coordination of activation patterns across a large number of spatially distributed regions. Functional magnetic resonance imaging (fMRI) has played a central role in demonstrating the nontrivial spatial and topological structure of these interactions, but thus far has been limited in its capacity to study their temporal evolution. Here, using high-resolution resting-state fMRI data obtained from the Human Connectome Project, we mapped time-resolved functional connectivity across the entire brain at a subsecond resolution with the aim of understanding how nonstationary fluctuations in pairwise interactions between regions relate to large-scale topological properties of the human brain. We report evidence for a consistent set of functional connections that show pronounced fluctuations in their strength over time. The most dynamic connections are intermodular, linking elements from topologically separable subsystems, and localize to known hubs of default mode and fronto-parietal systems. We found that spatially distributed regions spontaneously increased, for brief intervals, the efficiency with which they can transfer information, producing temporary, globally efficient network states. Our findings suggest that brain dynamics give rise to variations in complex network properties over time, possibly achieving a balance between efficient information-processing and metabolic expenditure.

We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call \"Morse 2-functions,\" and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is \"fiber-connected,\" and to avoid local extrema over one-dimensional submanifolds of the range, in which case the Morse 2-function is \"indefinite.\" This is foundational work for the long-range goal of defining smooth invariants from Morse 2-functions using tools analogous to classical Morse homology and Cerf theory.

Network studies of human brain structural connectivity have identified a specific set of brain regions that are both highly connected and highly central. Recent analyses have shown that these putative hub regions are mutually and densely interconnected, forming a “rich club” within the human brain. Here we show that the set of pathways linking rich club regions forms a central high-cost, high-capacity backbone for global brain communication. Diffusion tensor imaging (DTI) data of two sets of 40 healthy subjects were used to map structural brain networks. The contributions to network cost and communication capacity of global cortico-cortical connections were assessed through measures of their topology and spatial embedding. Rich club connections were found to be more costly than predicted by their density alone and accounted for 40% of the total communication cost. Furthermore, 69% of all minimally short paths between node pairs were found to travel through the rich club and a large proportion of these communication paths consisted of ordered sequences of edges (“path motifs”) that first fed into, then traversed, and finally exited the rich club, while passing through nodes of increasing and then decreasing degree. The prevalence of short paths that follow such ordered degree sequences suggests that neural communication might take advantage of strategies for dynamic routing of information between brain regions, with an important role for a highly central rich club. Taken together, our results show that rich club connections make an important contribution to interregional signal traffic, forming a central high-cost, high-capacity backbone for global brain communication.

Yoon, S.B., Kim, S.C., Baek, U., Bae, J.S., 2014. Effect of Bathymetry on Propagation of Tsunamis towards the East Coast of Korea. In: Green, A.N. and Cooper, J.A.G. (eds.), Proceedings 13th International Coastal Symposium (Durban, South Africa), Journal of Coastal Research, Special Issue No. 70, pp. 332–337, ISSN 0749-0208. In this study the effect of underwater topography of the East Sea on the propagation of tsunamis towards the Korean Peninsula is investigated using the dispersion-correction finite difference numerical model. A series of numerical simulations are conducted for three tsunami events including the 1964 Niigata Tsunami, the 1983 Central East Sea Tsunami and the 1993 Hokkaido South-West Sea Tsunami for the cases of examining the individual or combined influence of underwater topographic features. These include the Yamato Rise, a submerged ridge connecting Yamato Rise and the Shimane Peninsula of Japan, and K-shaped submerged ridges emerging from the east coast of Korea towards the East Sea. In particular, in order to evaluate quantitatively the effects of underwater topography on the propagation of tsunamis, a new concept of energy discharge per unit width is introduced. Using this concept, the quantitative analyses of energy propagation during tsunami events are performed. The analyses show that the underwater topographies including the submerged rises and ridges capture the tsunami energy and transport it to coastal areas connected to those topographies.

Sex differences in human behavior show adaptive complementarity: Males have better motor and spatial abilities, whereas females have superior memory and social cognition skills. Studies also show sex differences in human brains but do not explain this complementarity. In this work, we modeled the structural connectome using diffusion tensor imaging in a sample of 949 youths (aged 8—22 y, 428 males and 521 females) and discovered unique sex differences in brain connectivity during the course of development. Connection-wise statistical analysis, as well as analysis of regional and global network measures, presented a comprehensive description of network characteristics. In all supratentorial regions, males had greater within-hemispheric connectivity, as well as enhanced modularity and transitivity, whereas between-hemispheric connectivity and cross-module participation predominated in females. However, this effect was reversed in the cerebellar connections. Analysis of these changes developmentally demonstrated differences in trajectory between males and females mainly in adolescence and in adulthood. Overall, the results suggest that male brains are structured to facilitate connectivity between perception and coordinated action, whereas female brains are designed to facilitate communication between analytical and intuitive processing modes.