Arbitrary Internal Structure Research Articles

A new model for overlapping communities with arbitrary internal structure

We introduce the random intersection graph with communities, a new model for networks with overlapping communities with arbitrary internal structure. We construct the model from a list of arbitrary community graphs that are the building blocks, and a separate list of individuals, each with a prescribed number of community membership tokens. Randomness is introduced by matching these tokens uniformly at random to vertices of the community graphs. We then identify the community members assigned to the same individual, thus overlaps arise due to individuals having several tokens. This gives a highly flexible model for networks with community structure.We are able to derive a wide range of analytic results on this model. We derive an asymptotic description of the local structure of the graph, which further yields the asymptotic degree distribution, local clustering coefficient, and results on the overlapping structure of the communities. For the global connectivity structure, we identify a phase transition in the size of the largest component. When the largest component constitutes a positive proportion of the graph, we can further characterize its asymptotic local structure. Finally, we study how the connectivity structure changes under a randomized attack, where we remove edges randomly, according to independent coin flips.

Stochastic methods for light propagation and recurrent scattering in saturated and nonsaturated atomic ensembles

We derive equations for the strongly coupled system of light and dense atomic ensembles. The formalism includes an arbitrary internal level structure for the atoms and is not restricted to weak excitation of atoms by light. In the low light intensity limit for atoms with a single electronic ground state, the full quantum field-theoretical representation of the model can be solved exactly by means of classical stochastic electrodynamics simulations for stationary atoms that represent cold atomic ensembles. Simulations for the optical response of atoms in a quantum degenerate regime require one to synthesize a stochastic ensemble of atomic positions that generates the corresponding quantum statistical position correlations between the atoms. In the case of multiple ground levels or at light intensities where saturation becomes important, the classical simulations require approximations that neglect quantum fluctuations between the levels. We show how the model is extended to incorporate corrections due to quantum fluctuations that result from virtual scattering processes. In the low light intensity limit we illustrate the simulations in a system of atoms in a Mott-insulator state in a 2D optical lattice, where recurrent scattering of light induces strong interatomic correlations. These correlations result in collective many-atom subradiant and superradiant states and a strong dependence of the response on the spatial confinement within the lattice sites.

Universality of Poisson Indicator and Fano Factor of Transport Event Statistics in Ion Channels and Enzyme Kinetics

We consider a generic stochastic model of ion transport through a single channel with arbitrary internal structure and kinetic rates of transitions between internal states. This model is also applicable to describe kinetics of a class of enzymes in which turnover events correspond to conversion of substrate into product by a single enzyme molecule. We show that measurement of statistics of single molecule transition time through the channel contains only restricted information about internal structure of the channel. In particular, the most accessible flux fluctuation characteristics, such as the Poisson indicator (P) and the Fano factor (F) as function of solute concentration, depend only on three parameters in addition to the parameters of the Michaelis-Menten curve that characterizes average current through the channel. Nevertheless, measurement of Poisson indicator or Fano factor for such renewal processes can discriminate reactions with multiple intermediate steps as well as provide valuable information about the internal kinetic rates.

The Motion of Point Particles in Curved Spacetime.

This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The self-force contains both conservative and dissipative terms, and the latter are responsible for the radiation reaction. The work done by the self-force matches the energy radiated away by the particle.The field’s action on the particle is difficult to calculate because of its singular nature: the field diverges at the position of the particle. But it is possible to isolate the field’s singular part and show that it exerts no force on the particle — its only effect is to contribute to the particle’s inertia. What remains after subtraction is a regular field that is fully responsible for the self-force. Because this field satisfies a homogeneous wave equation, it can be thought of as a free field that interacts with the particle; it is this interaction that gives rise to the self-force.The mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime are developed here from scratch. The review begins with a discussion of the basic theory of bitensors (Part I). It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle’s word line (Part II). It continues with a thorough discussion of Green’s functions in curved spacetime (Part III). The review presents a detailed derivation of each of the three equations of motion (Part IV). Because the notion of a point mass is problematic in general relativity, the review concludes (Part V) with an alternative derivation of the equations of motion that applies to a small body of arbitrary internal structure.

Guaranteed passive balancing transformations for model order reduction

The major concerns in state-of-the-art model reduction algorithms are: achieving accurate models of sufficiently small size, numerically stable and efficient generation of the models, and preservation of system properties such as passivity. Algorithms, such as PRIMA, generate guaranteed-passive models for systems with special internal structure, using numerically stable and efficient Krylov-subspace iterations. Truncated balanced realization (TBR) algorithms, as used to date in the design automation community, can achieve smaller models with better error control, but do not necessarily preserve passivity. In this paper, we show how to construct TBR-like methods that generate guaranteed passive reduced models and in addition are applicable to state-space systems with arbitrary internal structure.

Relaxation of disordered polymer networks: Regular lattice made up of small-world Rouse networks

As models for inhomogeneous polymer networks, we investigate the Rouse dynamics of regular lattices built from subunits with arbitrary internal structure. We analyze as an example a two-dimensional lattice, consisting of small-world networks (SWNs). Using analytical and numerical calculations we study the stretching of such a structure under an external force. We find that the network shows interesting relaxation features and an unusual behavior in the intermediate time (frequency) domain, which lies in the region between the modes of the SWN subunits and those of the lattice. This behavior is related to the SWN-density of states, which leads to the appearance of a “pseudogap” between the highest lattice eigenvalue and the lowest SWN eigenvalue.

Multiobjective L1/H∞ Controller Design for Systems with Frequency and Time Domain Constraints

In this paper we develop an optimal mixed-norm L1 bound/H∞ and l1 bound/H∞ controller synthesis framework for continuous-time and discrete-time systems. This multiobjective problem is treated by forming a convex combination ofboth L1 (time domain worst-case peak amplitude response) and entropy (frequency domain worst-case H∞ disturbance attenuation) performance measures. For flexibility in controller synthesis, we adopt the approach of fixedstructure controller design which allows consideration of arbitrary controller structures, including order, internal structure, and decentralization. Finally, using a quasi-Newton continuation algorithm, we demonstrate the effectiveness of the proposed mixed-norm L1/ H∞ approach via two numerical design examples.

Mixed-norm H2/L1 controller synthesis via fixed-order dynamic compensation: A Riccati equation approach

One of the fundamental problems in feedback control design is the ability of the control system to reject uncertain exogenous disturbances. Since a single performance objective is seldom adequate to capture multiple and often conflicting system disturbances, in this paper we develop a Riccati equation approach for mixed H2/L1 controller synthesis via fixed-order dynamic compensation. This multiobjec2 tive problem is treated by forming a convex combination of both the H2 (quadratic) and L1 bound (worst-case peak amplitude response) performance measures. For flexibility in controller synthesis, we adopt the approach of fixed-structure controller design which allows consideration of arbitrary controller structures, including order, internal structure, and decentralization. Finally, using a quasi-Newton continuation algorithm, we demonstrate the effectiveness of the proposed mixed-norm H2/L1 Riccati equation approach via several design examples.

Complete decomposition of stochastic Petri nets representing generalized service networks

Complete decomposition is a new strategy for evaluating the performance of a network of generalized service centers, represented in the notation of Generalized Stochastic Petri Nets (GSPNs). Each service center can have arbitrary internal structure (including internal parallelism), but it must conserve tokens at the boundaries, and its inputs must be i/o-connected to its outputs. Routing between centers can depend on the state of the departure center. The new method adapts a delay equivalence decomposition technique used previously. Within this framework, it reduces the solution complexity of the auxiliary models which must be solved in the iteration. This new method is applied to a scalable model for computer system performance. >

Complete decomposition of stochastic Petri nets representing generalized service networks

Complete decomposition is a new strategy for evaluating the performance of a network of generalized service centers, represented in the notation of Generalized Stochastic Petri Nets (GSPNs). Each service center can have arbitrary internal structure (including internal parallelism), but it must conserve tokens at the boundaries, and its inputs must be i/o-connected to its outputs. Routing between centers can depend on the state of the departure center. The new method adapts a delay equivalence decomposition technique used previously. Within this framework, it reduces the solution complexity of the auxiliary models which must be solved in the iteration. This new method is applied to a scalable model for computer system performance.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Rotational distortion of stars having arbitrary structure described by fourth-order sectorial harmonics

In a series of papers, the equilibrium configurations of highly rotating fluid bodies have been derived. The deformation of these inhomogeneous self-gravitating fluid, of arbitrary internal structure are due to centrifugation potential. These level surfaces are expressed in terms of fourth-order sectorial harmonics.

Debye–Hückel theory for particles of arbitrary electrical structure

Classical linearized Debye–Hückel theory is formulated for a finite fluid system, of arbitrary shape, composed of rigid particles with arbitrary internal electrical structure. The multipole description is eschewed in favor of the more basic description of a particle in terms of its charge density function. This function is left arbitrary, so the particles may be charged or neutral, polar or nonpolar, etc. The theory implies that the direct correlation function c(12)=−v(12)/kT, where v(12) is the Coulomb interaction energy between the charge densities of particles 1 and 2. In the case of uncharged polar molecules, the dielectric constant may be evaluated in closed form from c(12); the result is the Langevin–Debye equation. This development removes the nonuniqueness in the original formulation of dipolar Debye–Hückel theory [J. Chem. Phys. 64, 3666 (1976)], and demonstrates that this nonuniqueness was an artifact of the multipole description rather than the mean-field approximation. Specialization to the case of simple finite dipoles shows that the nonuniqueness is associated with premature passage to the point dipole limit.

Clairaut coordinates and the vibrational stability of distorted stars

The aim of the present paper will be to generalize the concept of the ‘Roche coordinates’, introduced previously by the author (see Kopal, 1969, 1970, 1971) for a treatment of dynamical phenomena in close binary systems, to ‘Clairaut's coordinates’ in which the Roche potential of a rotating dipole is replaced by the actual potential of configurations of finite density concentration and arbitrary structure. By virtue of an identification of the potential with the radial coordinate of our three-dimensional system, the Roche and Clairaut coordinates are both bound to be curvilinear if the star in question departs from spherical form. However, unlike Roche coordinates, the Clairaut coordinates introduced in this paper will not be required to constitute an orthogonal system; and, as a result of the freedom so preserved, their angular variables will be identified with the angles θ and ϕ of spherical polars. Such an adoption entails advantages and disadvantages. In the orthodox Roche system, the radial coordinate (i.e., the potential ξ) is given to us in a closed form; but their angular variables η and ζ must, in general, be obtained by an integration of partial differential equations constituting the orthogonality conditions. On the other hand for the Clairaut (non-orthogonal) system of coordinates no such integration is necessary — and, in fact, the angular variables can be adopted at will. However, their radial coordinate (i.e., the potential of a star of arbitrary structure and distortion) is no longer available in a closed form and must be constructed by a sequence of successive approximations — a process initiated in the 18th century by Clairaut (1743), which can be developed to any desired accuracy. As is well known, investigations of the stability of self-gravitating configurations of arbitrary internal structure must be conducted on the basis of fundamental equations of stellar hydrodynamics, which for small oscillations can be reduced to linear forms. In Section 2 the explicit form of these fundamental equations will be set up in Clairaut's coordinates and linearized in Section 3 to the case of small oscillations, while in Section 4 a critical comparison of the Clairaut and Roche coordinates will be made. However their application to rotating stars will be the subject of subsequent papers.

Relativistic effects in many-body systems of finite size, internal structure, and internal motions: I. ?Self-acceleration? of astrophysical systems

It is proved that the post-Newtonian general relativistic center of rest mass of a bounded physical system composed of a number of bodies characterized by finite dimensions, arbitrary internal structure, and arbitrary internal motions cannot in general move uniformly, contrary to what was conventionally accepted up to now. Mathematical expressions are derived and discussed describing, in terms of the above characteristics of the bodies, the position and velocity vectors of the center of rest mass of the system and the force, which is responsible for its nonuniform motion, with respect to the uniformly moving post-Newtonian general relativistic center of inertial mass of the physical system. In the special case of a binary star it is shown that the center of rest mass should describe, around the uniformly moving center of inertial mass, an ellipse of the same period and eccentricity as those of the Newtonian elliptical orbit of the relative motion. The length of the axes of this ellipse depends on the internal characteristics of the members of the binary star, and the motion of the center of rest mass becomes important when these characteristics are strong enough. Finally, it is proved that in every effort for describing theoretically a binary system, the internal characteristics of the members of which are ignored, and for accurate measurements of their positions and velocities, the above fictitious motion of the center of rest mass has to be taken into account; otherwise, the results of the measurements will not be consistent with the theoretical predictions.

Newtonian dynamics of systems of extended bodies

A consistent method for the description of the Newtonian dynamics of a bounded system composed of a number of perfect-fluid bodies of finite dimensions, with arbitrary internal structure and internal motions is presented. It is explored under what conditions the dynamical equations for such a system become formally the same as the corresponding ones valid for a system of point-masses. For this purpose a consistent and complete dynamical description of each one of the bodies of the system is simultaneously developed. The relative accuracy with which the dynamical equations describing bounded systems of point masses and bounded perfect-fluid bodies apply to systems of extended bodies, and the time-scale over which the application of these equations becomes doubtful are explicitly derived. Generally these two quantities depend on three ratios. From these ratios the first one depending on the internal structure and internal motions of each body represents its own contribution. The second, which is intimately related to the space density of the system, represents the contribution of the system as a whole. Finally, the third one being always a power of the ratio of the linear dimensions of the bodies over their mutual distances represents the interaction between each body and the system. In most of the cases of astrophysical interest the relative accuracies are small numbers, while the time scales are always much larger than the typical Keplerian orbital period of the corresponding system.

Relativistic equations of motion of extended bodies

We derive the most general equations of motion in the post-Newtonian approximation of general relativity for a bounded system of extended bodies with arbitrary internal structure and internal motions, and we explore in detail the conditions under which the motion of the bodies deviates from the geodesic motion.

On the energy in relativistic stellar dynamics

Using the results of the preceding paper we evaluate in the post-Newtonian approximation the energy integral for a system of extended bodies with arbitrary internal structure and internal motions.

On secular stability of tidally distorted stars of arbitrary structure

The aim of the present paper will be to develop a theory which should make it possible to investigate secular stability of close binary systems, consisting of tidally-distorted components of arbitrary internal structure, by a minimization of the potential energy of the system as a whole. In the second section which follows brief introductory remarks, appropriate expressions for the total potential energy of a close binary will be formulated. Section 3 will be concerned mainly with the nature of the tide-generating potential, and its effects on the shape of each star. In Section 4, the amplitudes of partial tides raised by this potential will be specified, for stars of arbitrary structure, correctly to terms of second order in superficial distortion; and in Section 5 we shall investigate the effects of interaction between rotation and tides to the same degree of approximation. The concluding Section 6 will then contain an explicit formulation of different constituents adding up to the total potential energy of the system, which can be used as a basis for its secular stability by the methods outlined already in our previous investigation (Kopal, 1973).

On secular stability of rapidly rotating stars of arbitrary structure

The aim of the present paper will be to utilize Poincare's criteria to investigate secular stability of self-gravitating configurations, of arbitrary structure, in the state of rapid rotation. The potential energy, a knowledge of which is necessary for application of these criteria, will be determined by an extension of Clairaut's method; and its evaluation in terms of suitably chosen generalized coordinates carried out explicitly to quantities of fourth order in superficial oblateness, for configurations of arbitrary internal structure. The method employed can, moreover, clearly be extended to attain accuracy of any order — at the expense of mere manipulative work which lends itself to machine automation; and the angular velocity of axial rotation can be an arbitrary function of position as well as of the time. An application of our results to homogeneous configurations in rigig-body rotation will be undertaken to demonstrate that our method, when applied to a case for which a closed solution exists, leads to results which are consistent with it.

New Method for Generating Density Expansions

The calculus of finite differences is used to develop a new method for expressing the thermodynamic limit of a reasonably arbitrary statistical-mechanical average as a power series in the number density ρ. The method is simple, straightforward, and purely analytic: it involves no intermediate expansion in powers of the activity and it avoids the use of graph theory. Moreover, the method is developed independently of the prescription for computing the statistical average, a fact which lends to the results an especially wide range of applicability. In particular, these results may be used in classical or quantum statistical mechanics, for intermolecular potentials which are not spherically symmetric or pairwise additive, for molecules of arbitrary internal structure and complexity, and for polar molecules. A general formula is obtained for the coefficient of ρk in the series; as usual, the most difficult problem one need solve in order to compute this coefficient is the evaluation of a k-molecule average. It is shown that if all the coefficients exist and if the density is less than a certain well-defined critical density, then the series converges to the thermodynamic limit of the average in question. The practical use of the method is clarified by examples.