Volatility of an Indian stock market : A random matrix approach

We examine volatility of an Indian stock market in terms of aspects like participation, synchronization of stocks and quantification of volatility using the random matrix approach. Volatility pattern of the market is found using the BSE index for the three-year period 2000-2002. Random matrix analysis is carried out using daily returns of 70 stocks for several time windows of 85 days in 2001 to (i) do a brief comparative analysis with statistics of eigenvalues and eigenvectors of the matrix C of correlations between price fluctuations, in time regimes of different volatilities. While a bulk of eigenvalues falls within RMT bounds in all the time periods, we see that the largest (deviating) eigenvalue correlates well with the volatility of the index, the corresponding eigenvector clearly shows a shift in the distribution of its components from volatile to less volatile periods and verifies the qualitative association between participation and volatility (ii) observe that the Inverse participation ratio for the 'last' eigenvector is sensitive to market fluctuations (the two quantities are observed to anti correlate significantly) (iii) set up a variability index, V whose temporal evolution is found to be significantly correlated with the volatility of the overall market index.Comment: 33 Pages, 19 Figure

Fractionalization of a flux quantum in a one-dimensional parallel Josephson junction array with alternating π\pi junctions

We study numerically and analytically the properties of a one-dimensional array of parallel Josephson junctions in which every {\em alternate} junction is a $\pi$ junction. In the ground state of the array, each cell contains spontaneous magnetic flux $\Phi\leq\Phi_{0}/2$ which shows {\em antiferromagnetic} ordering along the array. We find that an externally introduced $2\pi$-fluxon $\Phi_{0}$ in such an array is unstable and fractionalizes into two $\pi$ fluxons of magnitude ${1/2}\Phi_{0}$. We attribute this fractionalization to the degeneracy of the ground state of the array. The magnitude of the flux in the fractional fluxons can be controlled by changing the critical current of the $\pi$ junctions relative to the 0 junctions. In the presence of an external current, the fluxon lattice in the antiferromagnetic ground state can be depinned. We also observe a novel resonant structure in the $V$-$I$ characteristics above the depinning current due to the interaction between the fluxon lattice and the array.Comment: 6 pages, 6 figures; dvips problem correcte

On the structural properties of small-world networks with finite range of shortcut links

We explore a new variant of Small-World Networks (SWNs), in which an additional parameter ($r$) sets the length scale over which shortcuts are uniformly distributed. When $r=0$ we have an ordered network, whereas $r=1$ corresponds to the original SWN model. These short-range SWNs have a similar degree distribution and scaling properties as the original SWN model. We observe the small-world phenomenon for $r \ll 1$ indicating that global shortcuts are not necessary for the small-world effect. For short-range SWNs, the average path length changes nonmonotonically with system size, whereas for the original SWN model it increases monotonically. We propose an expression for the average path length for short-range SWNs based on numerical simulations and analytical approximations

A Shared-Constraint Approach to Multi-leader Multi-follower Games

Multi-leader multi-follower games are a class of hierarchical games in which a collection of leaders compete in a Nash game constrained by the equilibrium conditions of another Nash game amongst the followers. The resulting equilibrium problem with equilibrium constraints is complicated by nonconvex agent problems and therefore providing tractable conditions for existence of global or even local equilibria for it has proved challenging. Consequently, much of the extant research on this topic is either model specific or relies on weaker notions of equilibria. We consider a modified formulation in which every leader is cognizant of the equilibrium constraints of all leaders. Equilibria of this modified game contain the equilibria, if any, of the original game. The new formulation has a constraint structure called shared constraints, and our main result shows that if the leader objectives admit a potential function, the global minimizers of the potential function over the shared constraint are equilibria of the modified formulation. We provide another existence result using fixed point theory that does not require potentiality. Additionally, local minima, B-stationary, and strong-stationary points of this minimization are shown to be local Nash equilibria, Nash B-stationary, and Nash strong-stationary points of the corresponding multi-leader multi-follower game. We demonstrate the relationship between variational equilibria associated with this modified shared-constraint game and equilibria of the original game from the standpoint of the multiplier sets and show how equilibria of the original formulation may be recovered. We note through several examples that such potential multi-leader multi-follower games capture a breadth of application problems of interest and demonstrate our findings on a multi-leader multi-follower Cournot game.Comment: The earlier manuscript was rejected. We felt it had too many themes crowding it and decided to make a separate paper from each theme. This submission draws some parts from the earlier manuscript and adds new results. Another parts is under review with the IEEE TAC (on arxiv) and another was published in Proc IEEE CDC, 2013. This submission is under review with Set-valued and Variational Analysi

Stochastic gene expression conditioned on large deviations

The intrinsic stochasticity of gene expression can give rise to large fluctuations and rare events that drive phenotypic variation in a population of genetically identical cells. Characterizing the fluctuations that give rise to such rare events motivates the analysis of large deviations in stochastic models of gene expression. Recent developments in non-equilibrium statistical mechanics have led to a framework for analyzing Markovian processes conditioned on rare events and for representing such processes by conditioning-free driven Markovian processes. We use this framework, in combination with approaches based on queueing theory, to analyze a general class of stochastic models of gene expression. Modeling gene expression as a Batch Markovian Arrival Process (BMAP), we derive exact analytical results quantifying large deviations of time-integrated random variables such as promoter activity fluctuations. We find that the conditioning-free driven process can also be represented by a BMAP that has the same form as the original process, but with renormalized parameters. The results obtained can be used to quantify the likelihood of large deviations, to characterize system fluctuations conditional on rare events and to identify combinations of model parameters that can give rise to dynamical phase transitions in system dynamics.Comment: 10 pages, 2 figures, to appear in Physical Biolog

An Existence Result for Hierarchical Stackelberg v/s Stackelberg Games

In Stackelberg v/s Stackelberg games a collection of leaders compete in a Nash game constrained by the equilibrium conditions of another Nash game amongst the followers. The resulting equilibrium problems are plagued by the nonuniqueness of follower equilibria and nonconvexity of leader problems whereby the problem of providing sufficient conditions for existence of global or even local equilibria remains largely open. Indeed available existence statements are restrictive and model specific. In this paper, we present what is possibly the first general existence result for equilibria for this class of games. Importantly, we impose no single-valuedness assumption on the equilibrium of the follower-level game. Specifically, under the assumption that the objectives of the leaders admit a quasi-potential function, a concept we introduce in this paper, the global and local minimizers of a suitably defined optimization problem are shown to be the global and local equilibria of the game. In effect existence of equilibria can be guaranteed by the solvability of an optimization problem, which holds under mild and verifiable conditions. We motivate quasi- potential games through an application in communication networks.Comment: This submission contains some results drawn from an earlier manuscript (http://arxiv.org/abs/1206.2968v2) which was rejected. The earlier manuscript has been split into 3 parts. One part is published in the Proc IEEE CDC, 2013, another is under review with Set-valued and Varitional Analysis (on arxiv). This is the third part, under review with IEEE TA

Applications of Little's Law to stochastic models of gene expression

The intrinsic stochasticity of gene expression can lead to large variations in protein levels across a population of cells. To explain this variability, different sources of mRNA fluctuations ('Poisson' and 'Telegraph' processes) have been proposed in stochastic models of gene expression. Both Poisson and Telegraph scenario models explain experimental observations of noise in protein levels in terms of 'bursts' of protein expression. Correspondingly, there is considerable interest in establishing relations between burst and steady-state protein distributions for general stochastic models of gene expression. In this work, we address this issue by considering a mapping between stochastic models of gene expression and problems of interest in queueing theory. By applying a general theorem from queueing theory, Little's Law, we derive exact relations which connect burst and steady-state distribution means for models with arbitrary waiting-time distributions for arrival and degradation of mRNAs and proteins. The derived relations have implications for approaches to quantify the degree of transcriptional bursting and hence to discriminate between different sources of intrinsic noise in gene expression. To illustrate this, we consider a model for regulation of protein expression bursts by small RNAs. For a broad range of parameters, we derive analytical expressions (validated by stochastic simulations) for the mean protein levels as the levels of regulatory small RNAs are varied. The results obtained show that the degree of transcriptional bursting can, in principle, be determined from changes in mean steady-state protein levels for general stochastic models of gene expression.Comment: Accepted by Physical Review

Relating tensor structures on representations of general linear and symmetric groups

For polynomial representations of $GL_n$ of a fixed degree, H. Krause defined a new internal tensor product using the language of strict polynomial functors. We show that over an arbitrary commutative base ring $k$, the Schur functor carries this internal tensor product to the usual Kronecker tensor product of symmetric group representations. This is true even at the level of derived categories. The new tensor product is a substantial enrichment of the Kronecker tensor product. E.g. in modular representation theory it brings in homological phenomena not visible on the symmetric group side. We calculate the internal tensor product over any $k$ in several interesting cases involving classical functors and the Weyl functors. We show an application to the Kronecker problem in characteristic zero when one partition has two rows or is a hook.Comment: Completely re-written manuscript with a new title. Additions include several new results and extension of earlier results. Also develops most of the necessary background in some detai

A Mini X-Ray Survey of Sub-DLAs; Searching for AGNs Formed in Protogalaxies

A significant fraction of the sub-damped Lyman-alpha (sub-DLA) absorption systems in quasar spectra appear to be metal-rich, many with even super-solar element abundances. This raises the question whether some sub-DLAs may harbor active galactic nuclei (AGN) since supersolar metallicities are observed in AGN. Here we investigate this question based on a mini-survey of 21 quasars known to contain sub-DLAs in their spectra. The X-ray observations were performed with the Chandra X-ray Observatory. In cases of no detection we estimated upper limits of the X-ray luminosities of possible AGNs at the redshifts of the sub-DLAs. In six cases we find possible X-ray emission within ~ 1 arcsec of the background quasar consistent with the presence of a nearby X-ray source. If these nearby X-ray sources are at the redshifts of the sub-DLAs, their estimated 0.2-10 keV luminosities range between 0.8 x 10^{44}h^{-2} and 4.2 x 10^{44}h^{-2} erg s^{-1}, thus ruling out a normal late-type galaxy origin, and suggesting that the emission originates in a galactic nucleus near the center of a protogalaxy. The projected distances of these possible nearby X-ray sources from the background quasars lie in the range of 3-7 h^{-1} kpc, consistent with our hypothesis that they represent AGNs centered on the sub-DLAs. Deeper follow-up X-ray and optical observations are required to confirm the marginal detections of X-rays from these sub-DLA galaxies.Comment: 12 pages, includes 4 figures, Accepted for publication in Ap

Intrinsic upper bound on two-qubit polarization entanglement predetermined by pump polarization correlations in parametric down-conversion

We study how one-particle correlations transfer to manifest as two-particle correlations in the context of parametric down-conversion (PDC), a process in which a pump photon is annihilated to produce two entangled photons. We work in the polarization degree of freedom and show that for any two-qubit generation process that is both trace-preserving and entropy-nondecreasing, the concurrence $C(\rho)$ of the generated two-qubit state $\rho$ follows an intrinsic upper bound with $C(\rho)\leq (1+P)/2$, where $P$ is the degree of polarization of the pump photon. We also find that for the class of two qubit states that is restricted to have only two non-zero diagonal elements such that the effective dimensionality of the two-qubit state is same as the dimensionality of the pump polarization state, the upper bound on concurrence is the degree of polarization itself, that is, $C(\rho)\leq P$. Our work shows that the maximum manifestation of two-particle correlations as entanglement is dictated by one-particle correlations. The formalism developed in this work can be extended to include multi-particle systems and can thus have important implications towards deducing the upper bounds on multi-particle entanglement, for which no universally accepted measure exists.Comment: 5 pages, 2 figure