Distributed nonlinear optimization algorithms for general network problems

There are a number of large networks which occur in many problems dealing with the flow of power, communication signals, water, gas, transportable goods, etc. Both design and planning of these networks involve optimization problems. The first part of this paper introduces the common characteristics of a nonlinear network (the network may be linear, the objective function may be non linear, or both may be nonlinear). The second part develops a mathematical model trying to put together some important constraints based on the abstraction for a general network. The third part deals with solution procedures; it converts the network to a matrix based system of equations, gives the characteristics of the matrix and suggests two solution procedures, one of them being a new one. The fourth part handles spatially distributed networks and evolves a number of decomposition techniques so that we can solve the problem with the help of a distributed computer system. Algorithms for parallel processors and spatially distributed systems have been described.There are a number of common features that pertain to networks. A network consists of a set of nodes and arcs. In addition at every node, there is a possibility of an input (like power, water, message, goods etc) or an output or none. Normally, the network equations describe the flows amoungst nodes through the arcs. These network equations couple variables associated with nodes. Invariably, variables pertaining to arcs are constants; the result required will be flows through the arcs. To solve the normal base problem, we are given input flows at nodes, output flows at nodes and certain physical constraints on other variables at nodes and we should find out the flows through the network (variables at nodes will be referred to as across variables).The optimization problem involves in selecting inputs at nodes so as to optimise an objective function; the objective may be a cost function based on the inputs to be minimised or a loss function or an efficiency function. The above mathematical model can be solved using Lagrange Multiplier technique since the equalities are strong compared to inequalities. The Lagrange multiplier technique divides the solution procedure into two stages per iteration. Stage one calculates the problem variables % and stage two the multipliers lambda. It is shown that the Jacobian matrix used in stage one (for solving a nonlinear system of necessary conditions) occurs in the stage two also.A second solution procedure has also been imbedded into the first one. This is called total residue approach. It changes the equality constraints so that we can get faster convergence of the iterations.Both solution procedures are found to coverge in 3 to 7 iterations for a sample network.The availability of distributed computer systems — both LAN and WAN — suggest the need for algorithms to solve the optimization problems. Two types of algorithms have been proposed — one based on the physics of the network and the other on the property of the Jacobian matrix. Three algorithms have been deviced, one of them for the local area case. These algorithms are called as regional distributed algorithm, hierarchical regional distributed algorithm (both using the physics properties of the network), and locally distributed algorithm (a multiprocessor based approach with a local area network configuration). The approach used was to define an algorithm that is faster and uses minimum communications. These algorithms are found to converge at the same rate as the non distributed (unitary) case.

Random matrices: Universality of ESDs and the circular law

Given an n x n complex matrix A, let mu(A)(x, y) := 1/n vertical bar{1 <= i <= n, Re lambda(i) <= x, Im lambda(i) <= y}vertical bar be the empirical spectral distribution (ESD) of its eigenvalues lambda(i) is an element of C, i = l, ... , n. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD mu(1/root n An) of a random matrix A(n) = (a(ij))(1 <= i, j <= n), where the random variables a(ij) - E(a(ij)) are i.i.d. copies of a fixed random variable x with unit variance. We prove a universality principle for such ensembles, namely, that the limit distribution in question is independent of the actual choice of x. In particular, in order to compute this distribution, one can assume that x is real or complex Gaussian. As a related result, we show how laws for this ESD follow from laws for the singular value distribution of 1/root n A(n) - zI for complex z. As a corollary, we establish the circular law conjecture (both almost surely and in probability), which asserts that mu(1/root n An) converges to the uniform measure on the unit disc when the a(ij) have zero mean.

Research note A novel BIMOS Schmitt trigger

A new Schmitt trigger circuit based on the lambda bipolar transistor is presented. This circuit which exhibits a hysteresis in its transfer characteristic seems to use a smaller chip area than many of the circuits proposed so far.

Synthesis, Structure and Optical Studies of a Family of Three-Dimensional Rare-Earth Aminoisophthalates M(mu(2)-OH)(C8H5NO4)] (M = Y3+, La3+, Pr3+, Nd3+, Sm3+, Eu3+, Gd3+, Dy3+, and Er3+)

A hydrothermal reaction of the acetate salts of the rare-earths, 5-aminoisophthalic acid (H(2)AIP), and NaOH at 150 degrees C for 3 days gave rise to a new family of three-dimensional rare-earth aminoisophthalates, M(mu(2)-OH)(C8H5NO4)] M = Y3+ (I), La3+ (II), Pr3+ (III), Nd3+ (IV), Sm3+ (V), Eu3+ (VI), Gd3+ (VII), Dy3+ (VIII), and Er3+ (IX)]. The structures contain M-O(H)-M chains connected by AIP anions. The AIP ions are connected to five metal centers and each metal center is connected with five AIP anions giving rise to a unique (5,5) net. To the best of our knowledge, this is the first observation of a (5,5) net in metal-organic frameworks that involve rare-earth elements. The doping of Eu3+/(3+) ions in place of Y3+/ La3+ in the parent structures gave rise to characteristic metal-centered emission (red = Eu3+, green = Tb3+). Life-time studies indicated that the excited emission states in the case of Eu3+ (4 mol-% doped) are in the range 0.287-0.490 ms and for Tb3+ (4 mol-% doped) are in the range of 1.265-1.702 ms. The Nd3+-containing compound exhibits up-conversion behavior based on two-photon absorption when excited using lambda = 580 nm.

On the Fekete-Szego problem for concave univalent functions

We consider the Fekete-Szego problem with real parameter lambda for the class Co(alpha) of concave univalent functions. (C) 2010 Elsevier Inc. All rights reserved.

Criticality of the exponential rate of decay for the largest nearest-neighbor link in random geometric graphs

Let n points be placed independently in d-dimensional space according to the density f(x) = A(d)e(-lambda parallel to x parallel to alpha), lambda, alpha > 0, x is an element of R-d, d >= 2. Let d(n) be the longest edge length of the nearest-neighbor graph on these points. We show that (lambda(-1) log n)(1-1/alpha) d(n) - b(n) converges weakly to the Gumbel distribution, where b(n) similar to ((d - 1)/lambda alpha) log log n. We also prove the following strong law for the normalized nearest-neighbor distance (d) over tilde (n) = (lambda(-1) log n)(1-1/alpha) d(n)/log log n: (d - 1)/alpha lambda <= lim inf(n ->infinity) (d) over tilde (n) <= lim sup(n ->infinity) (d) over tilde (n) <= d/alpha lambda almost surely. Thus, the exponential rate of decay alpha = 1 is critical, in the sense that, for alpha > 1, d(n) -> 0, whereas, for alpha <= 1, d(n) -> infinity almost surely as n -> infinity.

Crystal structure of cadaverine dihydrochloride monohydrate

The structure of cadaverine dihydrochloride monohydrate has been determined by X-ray crystallography with the following features: NH3+(CH2)5NH3+.2Cl-.H2O, formula weight 191.1, monoclinic, P2, a = 11.814(2) angstrom, b = 4.517(2) angstrom, c = 20.370(3) angstrom, beta = 106.56-degrees(1): V = 1041.9(2) angstrom3, lambda = 1.541 angstrom; mu = 53.4 1; T = 296-degrees; Z = 4, D(x) = 1.218 g.cm-3, R = 0.101 for 1383 observed reflections. The crystal is highly pseudosymmetric with 2 molecules of cadaverine, 4 chloride ions and 2 partially disordered water molecules present in the asymmetric unit. Though both the cadaverine molecules in the asymmetric unit have an all trans conformation, the carbon backbones are slightly bent. Between the concave surfaces of two bent cadaverine molecules exists water channels all along the short b axis. The water molecules present in the channels are partially disordered

Crystal structures of propanediamine complexed with L and DL- glutamic acid: effect of chirality on propanediamine.

The structures of complexes of 1,3-diaminopropane With L- and DL-glutamic acid have been determined. L-Glutamic acid complex: C3H12N22+.2C5H8NO4-, M(r) = 368.4, orthorhombic. P2(1)2(1)2(1), a = 5.199 (1), b = 16.832 (1). c = 20.076 (3) angstrom, V = 1756.6 (4) angstrom3, z = 4, D(x) = 1.39 g cm-3, lambda(Mo K-alpha) = 0.7107 angstrom, mu = 1.1 cm-1, F(000) = 792. T = 296 K, R = 0.044 for 1276 observed reflections. DL-Glutamic acid complex: C3H12N22+.2C5H8NO4-, M(r) = 368.4, orthorhombic, Pna2(1), a = 15.219(2), b = 5.169 (1), c 22.457 (4) angstrom, V = 1766.6 (5) angstrom3 Z = 4, D(x) = 1.38 g cm-3, lambda(Mo K-alpha) = 0.7107 angstrom, mu = 1.1 cm F(000) = 792, T = 296 K, R = 0.056 for 993 observed reflections. The conformation of diaminopropane is all-trans in the DL complex but trans-gauche in the L complex. The main packing feature in the L complex is the arrangement of diaminopropane around dimers of antiparallel L-glutamic acid molecules. The diaminopropane in the DL complex is sandwiched between two antiparallel glutamic acid molecules of the same chirality and this forms the basic packing unit. This might be the dominant form of interaction between L-glutamic acid and diaminopropane in solution. The structures reveal the adaptability of the polyamine backbone to different environments and the probable reasons for their choice as biological cations.

Structure of a monochloro bridged polymeric copper(II)-di-2-pyridylamine complex: [Cu(dipyam)Cl(NO3)].0.5H2O

Di-2-pyridylaminechloronitratocopper(II) hemihydrate, [CuCl(NO3)(C10H9N3)].0.5H2O, M(r) = 341.21, monoclinic, P2(1)/a, a = 7.382 (1), b = 21.494 (4), c = 8.032 (1) angstrom, beta = 94.26 (1)-degrees, V = 1270.9 angstrom 3, Z = 4, D(m) = 1.78, D(x) = 1.782 g cm-3, lambda(Mo K-alpha) = 0.7107 angstrom, mu(Mo K-alpha) = 19.47 cm-1, F(000) = 688. The structure was solved by the heavy-atom method and refined to a final R value of 0.034 for 2736 reflections collected at 294 K. The structure consists of polymeric [Cu(dipyam)Cl(NO3)] units bridged by a chloride ion.

Unsteady mixed convection flow in stagnation region adjacent to a vertical surface

The unsteady two-dimensional laminar mixed convection flow in the stagnation region of a vertical surface has been studied where the buoyancy forces are due to both the temperature and concentration gradients. The unsteadiness in the flow and temperature fields is caused by the time-dependent free stream velocity. Both arbitrary wall temperature and concentration, and arbitrary surface heat and mass flux variations have been considered. The Navier-Stokes equations, the energy equation and the concentration equation, which are coupled nonlinear partial differential equations with three independent variables, have been reduced to a set of nonlinear ordinary differential equations. The analysis has also been done using boundary layer approximations and the difference between the solutions has been discussed. The governing ordinary differential equations for buoyancy assisting and buoyancy opposing regions have been solved numerically using a shooting method. The skin friction, heat transfer and mass transfer coefficients increase with the buoyancy parameter. However, the skin friction coefficient increases with the parameter lambda, which represents the unsteadiness in the free stream velocity, but the heat and mass transfer coefficients decrease. In the case of buoyancy opposed flow, the solution does not exist beyond a certain critical value of the buoyancy parameter. Also, for a certain range of the buoyancy parameter dual solutions exist.

Complete Pick positivity and unitary invariance

The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foias. Just as a contraction is related to the Szego kernel k(S) (z, w) = (1 - z (w) over tilde)(-1) for |z|, |w| < 1, by means of (1/k(S))(T,T*) >= 0, we consider an arbitrary open connected domain Omega in C-n, a complete Pick kernel k on Omega and a tuple T = (T-1, ..., T-n) of commuting bounded operators on a complex separable Hilbert space H such that (1/k)(T,T*) >= 0. For a complete Pick kernel the 1/k functional calculus makes sense in a beautiful way. It turns out that the model theory works very well and a characteristic function can be associated with T. Moreover, the characteristic function is then a complete unitary invariant for a suitable class of tuples T.

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).

A note on "effects of couple stresses on the blood flow through an artery with mild stenosis"

Some errors have been observed in the analytical expression for the resistance to flow (lambda R), and in the computation of shear stress distribution (tau R) in the analysis of Prawal Sinha and Chandan Singh (1). These errors have been rectified in the present analysis. Also, better values have been suggested for the couple stress parameter alpha for getting better results for lambda R and tau R.

Molecular structure of Lantadene-B&C, triterpenoids ofantana camara, red variety: Lantadene-B, 22β-angeloyloxy-3-oxoolean-12-en-28-oic acid; Lantadene-C, 22 β-(S)-2′-methyl butanoyloxy-3-oxoolean-12-en-28-oic acid

(I)Lantadene-B: C35H52O5,M r =552.80, MonoclinicC2,a=25.65(1),b=6.819(9),c=18.75(1) Å,beta=100.61(9),V=3223(5) Å3,Z=4,D x =1.14 g cm–3 CuKagr (lambda=1.5418A),mgr=5.5 cm–1,F(000)=1208,R=0.118,wR=0.132 for 1527 observed reflections withF o ge2sgr(F o ). (II)Lantadene-C: C35H54O5·CH3OH,Mr=586.85, Monoclinic,P21,a=9.822(3),b=10.909(3),c=16.120(8)Å,beta=99.82(4),V=1702(1)Å3,Z=2,D x =1.145 g cm–3, MoKagr (lambda=0.7107Å), mgr=0.708 cm–1 F(000)=644,R=0.098, wR=0.094 for 1073 observed reflections. The rings A, B, C, D, and E aretrans, trans, trans, cis fused and are in chair, chair, sofa, half-chair, chair conformations, respectively, in both the structures. In the unit cell the molecules are stabilized by O-HctdotO hydrogen bonds in both the structures, however an additional C-HctdotO interaction is observed in the case of Lantadene-C.

A Shapley Value-Based Approach to Discover Influential Nodes in Social Networks

Our study concerns an important current problem, that of diffusion of information in social networks. This problem has received significant attention from the Internet research community in the recent times, driven by many potential applications such as viral marketing and sales promotions. In this paper, we focus on the target set selection problem, which involves discovering a small subset of influential players in a given social network, to perform a certain task of information diffusion. The target set selection problem manifests in two forms: 1) top-k nodes problem and 2) lambda-coverage problem. In the top-k nodes problem, we are required to find a set of k key nodes that would maximize the number of nodes being influenced in the network. The lambda-coverage problem is concerned with finding a set of k key nodes having minimal size that can influence a given percentage lambda of the nodes in the entire network. We propose a new way of solving these problems using the concept of Shapley value which is a well known solution concept in cooperative game theory. Our approach leads to algorithms which we call the ShaPley value-based Influential Nodes (SPINs) algorithms for solving the top-k nodes problem and the lambda-coverage problem. We compare the performance of the proposed SPIN algorithms with well known algorithms in the literature. Through extensive experimentation on four synthetically generated random graphs and six real-world data sets (Celegans, Jazz, NIPS coauthorship data set, Netscience data set, High-Energy Physics data set, and Political Books data set), we show that the proposed SPIN approach is more powerful and computationally efficient. Note to Practitioners-In recent times, social networks have received a high level of attention due to their proven ability in improving the performance of web search, recommendations in collaborative filtering systems, spreading a technology in the market using viral marketing techniques, etc. It is well known that the interpersonal relationships (or ties or links) between individuals cause change or improvement in the social system because the decisions made by individuals are influenced heavily by the behavior of their neighbors. An interesting and key problem in social networks is to discover the most influential nodes in the social network which can influence other nodes in the social network in a strong and deep way. This problem is called the target set selection problem and has two variants: 1) the top-k nodes problem, where we are required to identify a set of k influential nodes that maximize the number of nodes being influenced in the network and 2) the lambda-coverage problem which involves finding a set of influential nodes having minimum size that can influence a given percentage lambda of the nodes in the entire network. There are many existing algorithms in the literature for solving these problems. In this paper, we propose a new algorithm which is based on a novel interpretation of information diffusion in a social network as a cooperative game. Using this analogy, we develop an algorithm based on the Shapley value of the underlying cooperative game. The proposed algorithm outperforms the existing algorithms in terms of generality or computational complexity or both. Our results are validated through extensive experimentation on both synthetically generated and real-world data sets.