Splittings of free groups, normal forms and partitions of ends

Splittings of a free group correspond to embedded spheres in the 3-manifold M = # (k) S (2) x S (1). These can be represented in a normal form due to Hatcher. In this paper, we determine the normal form in terms of crossings of partitions of ends corresponding to normal spheres, using a graph of trees representation for normal forms. In particular, we give a constructive proof of a criterion determining when a conjugacy class in pi (2)(M) can be represented by an embedded sphere.

Successive refinement of structures with data of increasing resolution: a theoretical study for triclinic space groups

The method of least squares could be used to refine an imperfectly related trial structure by adoption of one of the following two procedures: (i) using all the observed at one time or (ii) successive refinement in stages with data of increasing resolution. While the former procedure is successful in the case of trial structures which are sufficiently accurate, only the latter has been found to be successful when the mean positional error (i.e.<|[Delta]r|>) for the atoms in the trial structure is large. This paper makes a theoretical study of the variation of the R index, mean phase-angle error, etc. as a function of <|[Delta]r|> for data corresponding to different esolutions in order to find the best refinement procedure [i.e. (i) or (ii)] which could be successfully employed for refining trial structures in which <|[Delta]r|> has large, medium and low values. It is found that a trial structure for which the mean positional error is large could be refined only by the method of successive refinement with data of increasing resolution.

Nonadiabatic charge pumping by oscillating potentials in one dimension: Results for infinite system and finite ring

We study charge pumping when a combination of static potentials and potentials oscillating with a time period T is applied in a one-dimensional system of noninteracting electrons. We consider both an infinite system using the Dirac equation in the continuum approximation and a periodic ring with a finite number of sites using the tight-binding model. The infinite system is taken to be coupled to reservoirs on the two sides which are at the same chemical potential and temperature. We consider a model in which oscillating potentials help the electrons to access a transmission resonance produced by the static potentials and show that nonadiabatic pumping violates the simple sin phi rule which is obeyed by adiabatic two-site pumping. For the ring, we do not introduce any reservoirs, and we present a method for calculating the current averaged over an infinite time using the time evolution operator U(T) assuming a purely Hamiltonian evolution. We analytically show that the averaged current is zero if the Hamiltonian is real and time-reversal invariant. Numerical studies indicate another interesting result, namely, that the integrated current is zero for any time dependence of the potential if it is applied to only one site. Finally we study the effects of pumping at two sites on a ring at resonant and nonresonant frequencies, and show that the pumped current has different dependences on the pumping amplitude in the two cases.

The hardness of approximating the boxicity, cubicity and threshold dimension of a graph

A k-dimensional box is the Cartesian product R-1 X R-2 X ... X R-k where each R-i is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the Cartesian product R-1 X R-2 X ... X R-k where each R-i is a closed interval oil the real line of the form a(i), a(i) + 1]. The cubicity of G, denoted as cub(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-cubes. The threshold dimension of a graph G(V, E) is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. In this paper we will show that there exists no polynomial-time algorithm for approximating the threshold dimension of a graph on n vertices with a factor of O(n(0.5-epsilon)) for any epsilon > 0 unless NP = ZPP. From this result we will show that there exists no polynomial-time algorithm for approximating the boxicity and the cubicity of a graph on n vertices with factor O(n(0.5-epsilon)) for any epsilon > 0 unless NP = ZPP. In fact all these hardness results hold even for a highly structured class of graphs, namely the split graphs. We will also show that it is NP-complete to determine whether a given split graph has boxicity at most 3. (C) 2010 Elsevier B.V. All rights reserved.

Directing role of functional groups in selective generation of C-H center dot center dot center dot pi interactions: In situ cryo-crystallographic studies on benzyl derivatives

The in situ cryo-crystallization study of benzyl derivatives reveals that the molecular packing in these compounds is either through methylene (sp(3)) C-H center dot center dot center dot pi or aromatic (sp(2)) C-H center dot center dot center dot pi interactions depending on the level of acidity of the benzyl proton. These studies of low melting compounds bring out the subtle features of such weak interactions and point to the directional preferences depending on the nature (electron withdrawing, polarizability) of the neighbouring functional group.

Segal-Bargmann Transform and Paley-Wiener Theorems on Motion Groups

We study the Segal-Bargmann transform on a motion group R-n v K, where K is a compact subgroup of SO(n) A characterization of the Poisson integrals associated to the Laplacian on R-n x K is given We also establish a Paley-Wiener type theorem using complexified representations

Spin-1 Kitaev model in one dimension

We study a one-dimensional version of the Kitaev model on a ring of size N, in which there is a spin S > 1/2 on each site and the Hamiltonian is J Sigma(nSnSn+1y)-S-x. The cases where S is integer and half-odd integer are qualitatively different. We show that there is a Z(2)-valued conserved quantity W-n for each bond (n, n + 1) of the system. For integer S, the Hilbert space can be decomposed into 2N sectors, of unequal sizes. The number of states in most of the sectors grows as d(N), where d depends on the sector. The largest sector contains the ground state, and for this sector, for S=1, d=(root 5+1)/2. We carry out exact diagonalization for small systems. The extrapolation of our results to large N indicates that the energy gap remains finite in this limit. In the ground-state sector, the system can be mapped to a spin-1/2 model. We develop variational wave functions to study the lowest energy states in the ground state and other sectors. The first excited state of the system is the lowest energy state of a different sector and we estimate its excitation energy. We consider a more general Hamiltonian, adding a term lambda Sigma W-n(n), and show that this has gapless excitations in the range lambda(c)(1)<=lambda <=lambda(c)(2). We use the variational wave functions to study how the ground-state energy and the defect density vary near the two critical points lambda(c)(1) and lambda(c)(2).

Signal characterization using fractal dimension

Fractal Dimensions (FD) are one of the popular measures used for characterizing signals. They have been used as complexity measures of signals in various fields including speech and biomedical applications. However, proper interpretation of such analyses has not been thoroughly addressed. In this paper, we study the effect of various signal properties on FD and interpret results in terms of classical signal processing concepts such as amplitude, frequency, number of harmonics, noise power and signal bandwidth. We have used Higuchi's method for estimating FDs. This study may help in gaining a better understanding of the FD complexity measure itself, and for interpreting changing structural complexity of signals in terms of FD. Our results indicate that FD is a useful measure in quantifying structural changes in signal properties.

Boxicity and Poset Dimension

Let G be a simple, undirected, finite graph with vertex set V(G) and edge set E(C). A k-dimensional box is a Cartesian product of closed intervals a(1), b(1)] x a(2), b(2)] x ... x a(k), b(k)]. The boxicity of G, box(G) is the minimum integer k such that G can be represented as the intersection graph of k-dimensional boxes, i.e. each vertex is mapped to a k-dimensional box and two vertices are adjacent in G if and only if their corresponding boxes intersect. Let P = (S, P) be a poset where S is the ground set and P is a reflexive, anti-symmetric and transitive binary relation on S. The dimension of P, dim(P) is the minimum integer l such that P can be expressed as the intersection of t total orders. Let G(P) be the underlying comparability graph of P. It is a well-known fact that posets with the same underlying comparability graph have the same dimension. The first result of this paper links the dimension of a poset to the boxicity of its underlying comparability graph. In particular, we show that for any poset P, box(G(P))/(chi(G(P)) - 1) <= dim(P) <= 2box(G(P)), where chi(G(P)) is the chromatic number of G(P) and chi(G(P)) not equal 1. The second result of the paper relates the boxicity of a graph G with a natural partial order associated with its extended double cover, denoted as G(c). Let P-c be the natural height-2 poset associated with G(c) by making A the set of minimal elements and B the set of maximal elements. We show that box(G)/2 <= dim(P-c) <= 2box(G) + 4. These results have some immediate and significant consequences. The upper bound dim(P) <= 2box(G(P)) allows us to derive hitherto unknown upper bounds for poset dimension. In the other direction, using the already known bounds for partial order dimension we get the following: (I) The boxicity of any graph with maximum degree Delta is O(Delta log(2) Delta) which is an improvement over the best known upper bound of Delta(2) + 2. (2) There exist graphs with boxicity Omega(Delta log Delta). This disproves a conjecture that the boxicity of a graph is O(Delta). (3) There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on n vertices with a factor of O(n(0.5-epsilon)) for any epsilon > 0, unless NP=ZPP.

Use of running fractal dimension for the analysis of changing patterns in electroencephalograms

Running fractal dimensions were measured on four channels of an electroencephalogram (EEG) recorded from a normal volunteer. The changes in the background activity due to eye closure were clearly differentiated by the fractal method. The compressed spectral array (CSA) and the running fractal dimensions of the EEG showed corresponding changes with respect to change in the background activity. The fractal method was also successful in detecting low amplitude spikes and the changes in the patterns in the EEG. The effects of different window lengths and shifts on the running fractal dimension have also been studied. The utility of fractal method for EEG data compression is highlighted.

Lacunary Fourier series and a qualitative uncertainty principle for compact Lie groups

We define lacunary Fourier series on a compact connected semisimple Lie group G. If f is an element of L-1 (G) has lacunary Fourier series and f vanishes on a non empty open subset of G, then we prove that f vanishes identically. This result can be viewed as a qualitative uncertainty principle.

Thiol-ene Clickable Hyperscaffolds Bearing Peripheral Allyl Groups

Radical catalyzed thiol-ene reaction has become a useful alternative to the Huisgen-type click reaction as it helps to expand the variability in reaction conditions as well as the range of clickable entities. Thus, direct generation of hyper-branched polymers bearing peripheral allyl groups that could be clicked using a variety of functional thiols would be of immense value. A specifically designed AB(2) type monomer, that carries two allyl benzyl ethers groups and one alcohol functionality, was shown to undergo self-condensation under acid-catalyzed melt-transetherification to yield a hyperbranched polyether that carries numerous allyl end-groups. Importantly, it was shown that the kinetics of polymerization is not dramatically affected by the change of the ether unit from previously studied methyl benzyl ether to an allyl benzyl ether. The peripheral allyl groups were readily clicked quantitatively, using a variety of thiols, to generate an hydrocarbon-soluble octadecyl-derivative, amphiphilic systems using 2-mercaptoethanol and chiral amino acid (N-benzoyl cystine) derivatized hyperbranched structures; thus demonstrating the versatility of this novel class of clickable hyperscaffolds. (C) 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 49:1735-1744, 2011

Social selection and the evolution of cooperative groups: The example of the cellular slime moulds

In social selection the phenotype of an individual depends on its own genotype as well as on the phenotypes, and so genotypes, of other individuals. This makes it impossible to associate an invariant phenotype with a genotype: the social context is crucial. Descriptions of metazoan development, which often is viewed as the acme of cooperative social behaviour, ignore or downplay this fact. The implicit justification for doing so is based on a group-selectionist point of view. Namely, embryos are clones, therefore all cells have the same evolutionary interest, and the visible differences between cells result from a common strategy. The reasoning is flawed, because phenotypic heterogeneity within groups can result from contingent choices made by cells from a flexible repertoire as in multicellular development. What makes that possible is phenotypic plasticity, namely the ability of a genotype to exhibit different phenotypes. However, co-operative social behaviour with division of labour requires that different phenotypes interact appropriately, not that they belong to the same genotype, or have overlapping genetic interests. We sketch a possible route to the evolution of social groups that involves many steps: (a) individuals that happen to be in spatial proximity benefit simply by virtue of their number; (b) traits that are already present act as preadaptations and improve the efficiency of the group; and (c) new adaptations evolve under selection in the social context-that is, via interactions between individuals-and further strengthen group behaviour. The Dictyostelid or cellular slime mould amoebae (CSMs) become multicellular in an unusual way, by the aggregation of free-living cells. In nature the resulting group can be genetically homogeneous (clonal) or heterogeneous (polyclonal); in either case its development, which displays strong cooperation between cells (to the extent of so-called altruism) is not affected. This makes the CSMs exemplars for the study of social behaviour.

Computing Fractal Dimension of Signals using Multiresolution Box-counting Method

In this paper, we have developed a method to compute fractal dimension (FD) of discrete time signals, in the time domain, by modifying the box-counting method. The size of the box is dependent on the sampling frequency of the signal. The number of boxes required to completely cover the signal are obtained at multiple time resolutions. The time resolutions are made coarse by decimating the signal. The loglog plot of total number of boxes required to cover the curve versus size of the box used appears to be a straight line, whose slope is taken as an estimate of FD of the signal. The results are provided to demonstrate the performance of the proposed method using parametric fractal signals. The estimation accuracy of the method is compared with that of Katz, Sevcik, and Higuchi methods. In ddition, some properties of the FD are discussed.