On the cubicity of bipartite graphs

A unit cube in k-dimension (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 on the real line of the form [a(j), a(i), + 1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. Many NP-complete graph problems can be solved efficiently or have good approximation ratios in graphs of low cubicity. In most of these cases the first step is to get a low dimensional cube representation of the given graph. It is known that for graph G, cub(G) <= left perpendicular2n/3right perpendicular. Recently it has been shown that for a graph G, cub(G) >= 4(Delta + 1) In n, where n and Delta are the number of vertices and maximum degree of G, respectively. In this paper, we show that for a bipartite graph G = (A boolean OR B, E) with |A| = n(1), |B| = n2, n(1) <= n(2), and Delta' = min {Delta(A),Delta(B)}, where Delta(A) = max(a is an element of A)d(a) and Delta(B) = max(b is an element of B) d(b), d(a) and d(b) being the degree of a and b in G, respectively , cub(G) <= 2(Delta' + 2) bar left rightln n(2)bar left arrow. We also give an efficient randomized algorithm to construct the cube representation of G in 3 (Delta' + 2) bar right arrowIn n(2)bar left arrow dimension. The reader may note that in general Delta' can be much smaller than Delta.

On the Cubicity of Interval Graphs

A k-cube (or ``a unit cube in k dimensions'') is defined as the Cartesian product R-1 x . . . x R-k where R-i (for 1 <= i <= k) is an interval of the form [a(i), a(i) + 1] on the real line. The k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that the k-cubes corresponding to two vertices in G have a non-empty intersection if and only if the vertices are adjacent. The cubicity of a graph G, denoted as cub(G), is defined as the minimum dimension k such that G has a k-cube representation. An interval graph is a graph that can be represented as the intersection of intervals on the real line - i. e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. We show that for any interval graph G with maximum degree Delta, cub(G) <= inverted right perpendicular log(2) Delta inverted left perpendicular + 4. This upper bound is shown to be tight up to an additive constant of 4 by demonstrating interval graphs for which cubicity is equal to inverted right perpendicular log(2) Delta inverted left perpendicular.

Boxicity of Halin graphs

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 is the intersection graph of a collection of k-dimensional boxes. Halin graphs are the graphs formed by taking a tree with no degree 2 vertex and then connecting its leaves to form a cycle in such a way that the graph has a planar embedding. We prove that if G is a Halin graph that is not isomorphic to K-4, then box(G) = 2. In fact, we prove the stronger result that if G is a planar graph formed by connecting the leaves of any tree in a simple cycle, then box(G) = 2 unless G is isomorphic to K4 (in which case its boxicity is 1).

Thermo-chemical interpretation of polymer degradation

A correlation has been established between the heat of depolymerization (DeltaH) of vinyl polymers for going from solid polymer state to gaseous monomer state and the activation energy (E) of degradation. On this basis it has been shown that the rate controlling step in the degradation lies in the initiation step. Attempt has been made to correlate theE and DeltaH with glass transition temperature (Tg) and melting temperature (Tm) of the polymers.[/ p]

Prediction of frequency and amplitude of foundations at resonance

This note presents the statistical analysis carried out on some of the available experimental results to predict the resonant frequency and maximum displacement amplitude of a machine foundation – soil system under vertical vibration as a function of the size and weight of the foundation and of the excitation level. A total of 442 experimental results of Fry, Novak, and Raman have been analysed using nonlinear regression analysis. The results obtained compared well with predictions obtained from the popular theoretical models, and the coefficient of correlation obtained from the analysis was satisfactory in most of the cases.

Geometrical interpretation of Inönü-Wigner contractions

The Inönü-Wigner contractions which interrelate the Lie algebras of the isometry groups of metric spaces are discussed with reference to deformations of the absolutes of the spaces. A general formula is derived for the Lie algebra commutation relations of the isometry group for anyN-dimensional metric space. These ideas are illustrated by a discussion of important particular cases, which interrelate the four-dimensional de Sitter, Poincaré, and Galilean groups.

Application of molecular orbital theory to the interpretation of x-ray absorption spectra of platinum complexes

X-ray LIII-absorption edges of platinum in nine octahedral complexes have been recorded using a bent crystal spectrograph. The edge features of the discontinuities have been interpreted with the help of qualitative molecular orbital diagrams. A correlation between the energy separation of the first two absorption maxima and the spectrochemical series of the ligands has been arrived at.

Acyclic edge coloring of subcubic graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and it is denoted by a′(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using five colors. This result is tight since there are 3-regular graphs which require five colors. In this paper we prove that any non-regular connected graph of maximum degree 3 is acyclically edge colorable using at most four colors. This result is tight since all edge maximal non-regular connected graphs of maximum degree 3 require four colors.

Applicability of Statistical Learning Algorithms for Spatial Variability of Rock Depth

Two algorithms are outlined, each of which has interesting features for modeling of spatial variability of rock depth. In this paper, reduced level of rock at Bangalore, India, is arrived from the 652 boreholes data in the area covering 220 sqa <.km. Support vector machine (SVM) and relevance vector machine (RVM) have been utilized to predict the reduced level of rock in the subsurface of Bangalore and to study the spatial variability of the rock depth. The support vector machine (SVM) that is firmly based on the theory of statistical learning theory uses regression technique by introducing epsilon-insensitive loss function has been adopted. RVM is a probabilistic model similar to the widespread SVM, but where the training takes place in a Bayesian framework. Prediction results show the ability of learning machine to build accurate models for spatial variability of rock depth with strong predictive capabilities. The paper also highlights the capability ofRVM over the SVM model.

Analysis of short interproton distances in proline peptides as a guide in the interpretation of nuclear overhauser effects

The conformational dependence of interproton distances in model proline peptides has been investigated in order to facilitate interpretation of the results of Nuclear Overhauser Effect (NOE) studies on such peptides. For this purpose two model systems, namely, Ac-Pro-NHMe and Ac-Pro-X-NHMe have been chosen and used. In the former, short interproton distances detectable in NOE experiments permit a clear distinction between conformations with Pro ψ = -300 (helical region) and those in which ψ is around 1200 (polyproline region). For the latter, the variation of distances between the protons of methyl amide and the Pro ring have been studied by superimposing on the Ramachandran map in the (φ3, ψ3) plane. The results show that β-turns and non-β-turn conformations can be readily distinguished from NOE data and such long range NOEs should be detectable for specific non-β-turn conformations. NOEs involving Cβ and Cγ protons are particularly sensitive to the state of pyrrolidine ring puckering.

Voltage and Temperature Aware Statistical Leakage Analysis Framework Using Artificial Neural Networks

Artificial neural networks (ANNs) have shown great promise in modeling circuit parameters for computer aided design applications. Leakage currents, which depend on process parameters, supply voltage and temperature can be modeled accurately with ANNs. However, the complex nature of the ANN model, with the standard sigmoidal activation functions, does not allow analytical expressions for its mean and variance. We propose the use of a new activation function that allows us to derive an analytical expression for the mean and a semi-analytical expression for the variance of the ANN-based leakage model. To the best of our knowledge this is the first result in this direction. Our neural network model also includes the voltage and temperature as input parameters, thereby enabling voltage and temperature aware statistical leakage analysis (SLA). All existing SLA frameworks are closely tied to the exponential polynomial leakage model and hence fail to work with sophisticated ANN models. In this paper, we also set up an SLA framework that can efficiently work with these ANN models. Results show that the cumulative distribution function of leakage current of ISCAS'85 circuits can be predicted accurately with the error in mean and standard deviation, compared to Monte Carlo-based simulations, being less than 1% and 2% respectively across a range of voltage and temperature values.

The Mechanics and Statistics of Active Matter

Active particles contain internal degrees of freedom with the ability to take in and dissipate energy and, in the process, execute systematic movement. Examples include all living organisms and their motile constituents such as molecular motors. This article reviews recent progress in applying the principles of nonequilibrium statistical mechanics and hydrodynamics to form a systematic theory of the behavior of collections of active particles-active matter-with only minimal regard to microscopic details. A unified view of the many kinds of active matter is presented, encompassing not only living systems but inanimate analogs. Theory and experiment are discussed side by side.

Nucleotide sequence of the 3terminal region of belladonna mottle virus (renamed Physalis mottle virus) RNA and analysis of the evolutionary relationships among tymoviral coat proteins

The 3' terminal 1255 nt sequence of Physalis mottle virus (PhMV) genomic RNA has been determined from a set of overlapping cDNA clones. The open reading frame (ORF) at the 3' terminus corresponds to the amino acid sequence of the coat protein (CP) determined earlier except for the absence of the dipeptide, Lys-Leu, at position 110-111. In addiition, the sequence upstream of the CP gene contains the message coding for 178 amino acid residues of the C-terminus of the putative replicase protein (RP). The sequence downstream of the CP gene contains an untranslated region whose terminal 80 nucleotides can be folded into a characteristic tRNA-like structure. A phylogenetic tree constructed after aligning separately the sequence of the CP, the replicase protein (RP) and the tRNA-like structure determined in this study with the corresponding sequences of other tymoviruses shows that PhMV wrongly named belladonna mottle virus [BDMV(I)] is a separate tymovirus and not another strain of BDMV(E) as originally envisaged. The phylogenetic tree in all the three cases is identical showing that any subset of genomic sequence of sufficient length can be used for establishing evolutionary relationships among tymoviruses.

An Algebraic Implicitization and Specialization of Minimum KL-Divergence Models

In this paper we study representation of KL-divergence minimization, in the cases where integer sufficient statistics exists, using tools from polynomial algebra. We show that the estimation of parametric statistical models in this case can be transformed to solving a system of polynomial equations. In particular, we also study the case of Kullback-Csiszar iteration scheme. We present implicit descriptions of these models and show that implicitization preserves specialization of prior distribution. This result leads us to a Grobner bases method to compute an implicit representation of minimum KL-divergence models.