Number Of Mass Points Research Articles

A new approach to the asymptotics of Sobolev type orthogonal polynomials

This paper deals with Mehler–Heine type asymptotic formulas for the so-called discrete Sobolev orthogonal polynomials whose continuous part is given by Laguerre and generalized Hermite measures. We use a new approach which allows to solve the problem when the discrete part contains an arbitrary (finite) number of mass points.

Transition points in the capacity-achieving distribution for the peak-power limited AWGN and free-space optical intensity channels

The capacity-achieving input distribution for many channels like the additive white Gaussian noise (AWGN) channel and the free-space optical intensity (FSOI) channel under the peak-power constraint is discrete with a finite number of mass points. The number of mass points is itself a variable, and figuring it out is a part of the optimization problem. We wish to understand the behavior of the optimal input distribution at the transition points where the number of mass points changes. To this end, we give a new set of necessary and sufficient conditions at the transition points, which offer new insights into the transition and make the computation of the optimal distribution easier. For the real AWGN channel case, we show that for the zero-mean unit-variance Gaussian noise, the peak amplitude A of 1.671 and 2.786 mark the points where the binary and ternary signaling, respectively, are no longer optimal. For the FSOI channel, we give transition points where binary gives way to ternary, and in some cases where ternary gives way to quaternary, in the presence of the peak-power constraint and with or without the average-power constraint.

The future of low-mass condensations in the core of a molecular cloud

Two scenarios have been proposed for evolution of star forming cores: gravitational fragmentation of larger structures and coalescence of smaller entities which are formed from some instabilities. Here, we turn our attention to the latter idea to investigate the evolution of observed low-mass condensations (LMCs) in the cores of molecular clouds. For this purpose, we implement the evolution of the observed LMCs of Taurus molecular cloud~1 (TMC-1). The core is modeled as a contracting cylinder with randomly spawned condensations in the middle region around its axis. For advancing bodies in their trajectories, we represent the acceleration of a particular LMC in terms of a fourth-order polynomial using the predictor-corrector scheme. Whenever two LMCs collide, they are assumed to be merged in one large condensation containing all the masses of the two progenitors. Implementations of many computer experiments with a wide variety of the free parameters show that the LMCs merge to form star-forming regions in the core. The results show that the total mechanical energy of the core increases by time, and its rate of increasing decreases by facilitating the merger. Finally, the mass spectrum index and goodness-of-fit are determined with 50% error in the number of mass points. The results show that the goodness-of-fit will be refined at the end of simulations, and the mass spectrum index inclines to the observed values for the moderate mass objects. The simulations show that the TMC-1 turns about 40% of its mass into cluster of dynamically unstable protostellar cores. In general, we suggest that the future of LMCs in a core of molecular cloud is merger to convert about half of its initial masses into a cluster of gravitationally unstable protostellar cores.

On the Capacity and Energy Efficiency of Training-Based Transmissions Over Fading Channels

In this paper, the capacity and energy efficiency of training-based communication schemes employed for transmission over a priori unknown Rayleigh block-fading channels are studied. Initially, the case in which the product of the estimate error and transmitted signal is assumed to be Gaussian noise is considered. In this case, it is shown that bit energy requirements grow without bound as the signal-to-noise ratio (SNR) goes to zero, and the minimum bit energy is achieved at a nonzero SNR value below which one should not operate. The effect of the block length on both the minimum bit energy and the SNR value at which the minimum is achieved is investigated. Flash training and transmission schemes are analyzed and shown to improve the energy efficiency in the low-SNR regime. In the second part of this paper, the capacity and energy efficiency of training-based schemes are investigated when the channel input vector in each coherence block is subject to peak power constraints. The capacity-achieving input structure is characterized and the magnitude distribution of the optimal input is shown to be discrete with a finite number of mass points. The capacity, bit energy requirements, and optimal resource allocation strategies are obtained through numerical analysis. The improvements in energy efficiency when on-off keying (OOK) with fixed peak power and vanishing duty cycle is employed are studied.

Model of a single-row flow of cars: Derivation and integration of equations

A chain of cars moving one after another in a single traffic lane is considered. It is assumed that the cars are point particles and these particles interact with adjacent ones according to the safe distance law. The differential functional equations of the particle flow are derived, and their solution is obtained. Boundary conditions at sections of particle input and output from a piece of a traffic lane are added to the equations. It is shown that models of a single-row flow of particles can be constructed as a chain of a finite number of mass points in a friction-free channel with pulse control as well as a finite-state automation model.

The inertia tensor of a magic cube

Magic cubes are shown to have maximally symmetric inertia tensors if they are interpreted as rigid body mass distributions. This symmetry is due to their semi-magic property where each row, column, and pillar has the same mass sum. The moment of inertia depends only on this property and the number of point masses in each row, column, and pillar. Because magic cubes do not possess detailed cubic symmetry, other scenarios that result in maximally symmetric inertia tensors are discussed.

Sound radiation from a baffled rectangular plate under a variable line constraint

Sound radiation from plates in baf.es, and shells is a well-studied subject. More recently, researchers are trying to optimize these structures for minimizing sound radiation [1]. A few typical parameters which are optimized include material tailoring [2], location and number of point masses attached to the structure [3], dynamic compliance [4], and damping layer placements [5]. These techniques include numerical methods such as BEM and FEM.

Minimum velocity dispersion in stable stellar disks. Numerical simulations

N-body dynamical simulations are used to analyze the conditions for the gravitational stability of a three-dimensional stellar disk in the gravitational field of two rigid spherical components—a bulge and halo whose central concentrations and relative masses vary over wide ranges. The number of point masses N in the simulations varies from 40 to 500 000 and the evolution of the simulated systems is followed over 10–20 rotation periods of the outer edge of the disk. The initially unstable disks are heated and, as a rule, reach a quasi-stationary equilibrium with a steady-state radial-velocity dispersion c r over five to eight turns. The radial behavior of the Toomre stability parameter Q T(r) for the final state of the disk is estimated. Simple models are used to analyze the dependence of the gravitational stability of the disk on the relative masses of the spherical components, disk thickness, degree of differential rotation, and initial state of the disk. Formal application of existing, analytical, local criteria for marginal stability of the disk can lead to errors in cr of more than a factor of 1.5. It is suggested that the approximate constancy of Q T⋍1.2–1.5 for r⋍(1–2)×L (where L is the radial scale of disk surface density), valid for a wide range of models, can be used to estimate upper limits for the mass and density of a disk based on the observed distributions of the rotational velocity of the gaseous component and of the stellar velocity dispersion.

Equilibrium Partitioning of Flexible Macromolecules in Fibrous Membranes and Gels

A theory was developed to describe the equilibrium partitioning of a flexible polymeric solute between a dilute solution and a medium consisting of a randomly oriented array of fibers (e.g., a fibrous membrane or gel). The fibers were regarded as long, rigid cylinders, and the solute was modeled as a chain consisting of n mass points joined by n − 1 rectilinear segments. Results were obtained for freely jointed chains and for chains with fixed bond angles, using a combination of analytical and computational (e.g., Monte Carlo) techniques. The predicted partition coefficient (Φ, the solute concentration in the fibrous material divided by that in the bulk solution) proved to be sensitive to the number of fibers included; typically, 50 or more nearest-neighboring fibers were needed to obtain a convergent result. The values of Φ decreased as fiber volume fraction, molecular size, or number of mass points was increased selectively. As the bond angle was increased from 0° (a chain folding back on itself) to 180° (a rodlike chain), Φ first decreased and then increased. The model predictions agreed reasonably well with literature partitioning data for various polymeric solutes in hydrogels, especially when the radius of gyration was no more than a few times the fiber radius. With the radius of gyration specified, the model predictions were fairly insensitive to whether the bond angle was fixed or random.

Equilibrium properties of polyampholytes in electric fields

An analytically tractable model for the description of a charged polymer chain (polyampholyte) in an external electric field is presented. The chain is comprised of mass points and their connectivity is taken into account by harmonic constraints. In particular the finite extensibility of a chain is captured by our approach. Equilibrium properties for chains of various charge distributions are calculated, especially for intermediate and strong electric fields. Particularly, chains of alternating positive and negative charges are considered within a discrete chain model. For intermediate electric fields such chains exhibit stretched configurations, whereas for strong electric fields a collapse or stretching occurs depending on the number of mass points. In addition a continuous chain with random charge density is considered. Our investigations demonstrate that the equilibrium properties of polyampholytes in strong electric fields decisively depend on the charge distribution along the chain.

A Method of Calculating Space Free Resonance Modes of an Aircraft

The problem of deducing resonance modes of vibration of an aircraft in free space is a concomitant of flutter calculations if the number of degrees of freedom used is to be small. When the structure is complex in that it involves wings, fuselage and tailplane, each of which possesses infinitely many normal modes, it becomes apparent that the number of point masses which must be considered, in constructing a dynamical equivalent to give a sufficient coverage of the frequency range in which flutter is likely, is very large. For example, it may be necessary to use four rods per wing and per tailplane and four point masses on the fuselage. This would involve 8 + 8 + 4 degrees of freedom, and if the usual technique of characteristic roots is used it would be necessary to consider a characteristic root matrix of order 20 × 20.