General Explicit Formula Research Articles

Diffusion and targeting centrality

This paper studies the dynamics of information diffusion within networks, encompassing both general and targeted dissemination. We first characterize the theoretical foundations of diffusion centrality. Next, we introduce two extensions of diffusion centrality: targeting centrality and reachability, that we believe to better capture situations involving targeted requests. We derive general explicit formulas for the computation of these novel centrality measures.

Observations of Flocs in an Estuary and Implications for Computation of Settling Velocity

AbstractThe settling velocity (ws) in estuarine environments can impact whether a region is eroding or accreting sediment on the bed, yet determining this rate can be an indirect process requiring a number of assumptions. Accurate determination of ws is especially needed for numerical models to reproduce observed sediment concentrations at the appropriate timescale. We collected information on suspended sediment flocculation at a channel site (13 m deep) and a shallows site (4 m deep) within South San Francisco Estuary, alongside timeseries of flow, wave statistics, turbulent shear, and bottle samples analyzed for both ws and particle size. Using the measurements of floc size and settling velocity, we performed a sensitivity analysis on the unknown parameters in the general explicit formula for settling velocity. The collected particle size distribution data show that multiple classes of flocs are present; these are characterized as flocculi, microflocs, and macroflocs. We show that ws of flocculi is closest to ws for the full distribution. The determined parameter values lead to near‐bed mass‐weighted settling velocities (standard deviation) of 1.18 (0.55) and 0.22 (0.15) mm/s at the channel and shallows sites, respectively. Modeling efforts can use this work to help select an appropriate sediment model and parameter values.

Approximate Solution for Some Continuous Optimal Control Problems Using Veita-Pell Polynomials

In this paper, Veita-Pell polynomials (VPPs) are first presented with some interesting properties. These properties are utilized to construct a general explicit formula of their operational matrix of derivative. Then an appropriate direct technique is suggested for solving quadratic optimal control problems based on VPPs and the idea of state parameterization algorithm. The resulting performance index optimal value shows the proposed method is able to provide a good treatment with fast convergence. The effectiveness of the presented method is illustrated by solving three numerical examples. The obtained results show that as the number of basis functions VPPs increase the error in the solution by the present method will be decreased and it may exactly close with the analytical one. This is the main modification of the algorithm and this contribution in the using special basis functions in obtaining an approximate solution with minimum number of VPPs and satisfactory accuracy.

Piecewise-Fitted Formula for Cable Force Identification Considering Bending Stiffness, Sag, and Inclination

The identification of cable forces plays an important role in the construction, vibration control, and health monitoring of cable structures. In practice, cable vibrations are mainly affected by the coupling of bending stiffness, sag, inclination, and boundary conditions, so it is difficult to establish an explicit functional relationship between frequency and cable force through the vibration method for facilitating application to engineering. To overcome this difficulty, the explicit expression of the frequency of the cable considering bending stiffness, sag, and inclination is derived for the first time and a novel cable force identification method for determining the piecewise-fitted formula (PFF) according to the cable parameters is proposed. The paper first reviews the cable force identification formulas based on cable frequencies. Then, the theoretical analysis model and differential equation of cable vibration considering the aforementioned factors are established. Based on these derivations, the explicit general formula of cable force identification based on fundamental frequency is derived. Next, according to the cable parameters, the value ranges of two nondimensional parameters characterizing the bending stiffness and sag of cables are divided to fit the corresponding coefficients of the PFF. Finally, numerical simulation and experimental data show that the PFF is more accurate and applicable than the existing formulas. This method can be conveniently applied to the field detection and online monitoring of cable forces.

Photoexcitation of planar Wannier excitons by twisted photons

Photoexcitation of planar Wannier excitons by twisted photons in thin semiconductor films is investigated. The excitons are assumed to be already created. The explicit general formulas for transition probabilities between exciton states are derived. The selection rules for the projection of the total angular momentum are obtained. As examples, the Coulomb and Rytova–Keldysh electron–hole interaction potentials are considered. The use of planar excitons as a pure on-chip source of twisted photons is discussed. The formulas obtained also describe photoexcitation of states of electrons and holes bound to charged impurities in planar semiconductors.

Analytical and Statistical Modelling of a Fast Ion Source Formed by Injection of a Neutral Beam into Magnetically Confined Plasma

Mathematical modelling of heating and current drive as well as yields and distributions of fusion products in a magnetically confined plasma subject to neutral beam injection requires, in turn, modelling of distributions of fast ions, which is a complex task including calculations of the source of suprathermal particles, i.e., the number of fast ions occurring in unit volume during unit time owing to the injection of fast atoms. The knowledge of the magnetohydrodynamic equilibrium, beam injection geometry and spatial distribution of the magnetic field are the necessary prerequisites. Explicit general analytical formulae for the source of fast ions have been obtained by two different methods. In addition, a method of statistical modelling is presented. Calculations of spatial and angular distributions of the fast ion source for a tokamak and verifications of the obtained results have been performed by a number of methods.

On the Unsteady Rotationally Symmetric Flow Between a Stationary and a Finite Rotating Disk With a Given Change in the Axial Velocity

Abstract We consider the scenario of an unsteady viscous flow between two coaxial finite disks, one stationary and the other rotating with an axial velocity changing impulsively from zero to a constant value. The three-dimensional (3D) incompressible Navier–Stokes equations are analytically solved by postulating the polynomial profiles for the axial and circumferential velocity components and by employing the open-end condition of zero pressure difference and an integral approach. It is shown that the time-dependent squeezing of the fluid between the disks and the edge effects of the finite open-ended disks in the flow domain are determined by the compressing Reynolds number and the rotating Reynolds number for the case of laminar flow at low Reynolds numbers and small aspect ratios. The general explicit formulae are derived for the velocity and pressure distributions as a function of the compressing and rotating Reynolds numbers in this unsteady flow process (and the steady-state solutions are then obtainable as the compressing Reynolds number vanishes). A simple theoretical relationship between the radial and axial pressure gradients is deduced to hinge the radial and circumferential velocity components together. The values of the compression and rotation Reynolds numbers suitable to this theory are also suggested for the problem of rotating disk flows at low Reynolds numbers. The validity of the theoretical predictions for the circumferential, radial, and axial velocity components is partially verified through comparison with previous steady experimental and numerical results. These analytical results have the immediate engineering applications of fluid flows with varying gap widths, including wet brakes, wet clutches, hydrostatic bearings, face seals, and rotating heat exchangers.

On the inverse problem of fractional Brownian motion and the inverse of infinite Toeplitz matrices

The inverse problem of fractional Brownian motion and other Gaussian processes with stationary increments involves inverting an infinite hermitian positively definite Toeplitz matrix (a matrix that has equal elements along its diagonals). The problem of inverting Toeplitz matrices is interesting on its own and has various applications in physics, signal processing, statistics, etc. A large body of literature has emerged to study this question since the seminal work of Szegö on Toeplitz forms in 1920's. In this paper we obtain, for the first time, an explicit general formula for the inverse of infinite hermitian positive definite Toeplitz matrices. Our formula is explicitly given in terms of the Szegö function associated to the spectral density of the matrix. These results are applied to the fractional Brownian motion and to m-diagonal Toeplitz matrices and we provide explicit examples.

Satisficing credibility for heterogeneous risks

As one of the earliest crucial applications of Bayesian statistics, credibility theory (see Bühlmann and Gisler, 2006) was first developed for net premium calibration in insurance by optimally combining individual claim history with other claim histories from the whole population; for a century, this research direction has become a major discipline in the interplay among actuarial science, operational research and statistics. Traditionally, for the ease of calibration, the credibility formula is a linear functional of historical observations, which greatly simplifies the underlying computational complexity; yet, its downside is the resulting high sensitivity towards outliers. To remedy this shortcoming, De Vylder (1976) proposed to first transform the observations collected by truncation in particular, and this semi-linear approach was further investigated in Bühlmann & Gisler (2006). Gisler (1980) suggested that the L2-optimal truncation point can be determined in an ad hoc manner, but the derivation of its general explicit formula is difficult. In our present work, to strike a balance between practical usage and mathematical tractability, we focus on heterogeneous risks all coming from possibly different maximum domain of attractions of the extreme value distributions, which well suffices in practice. By incorporating the satisficing method commonly used in operational research, we close the gap by providing the explicit formula for the aforementioned optimal truncation point up to a slowly varying function of the sample size in an asymptotic sense. A comprehensive numerical study also illuminates that with the aid of this newly obtained truncation point, the corresponding semi-linear credibility formula outperforms the classical Bühlmann model.

Design of a PID-Lead Compensator for a Twin Rotor Aerodynamic System (TRAS)

This paper deals with a Twin Rotor Aerodynamic System (TRAS). It is a Multi-Input Multi-Output (MIMO) system with high crosscoupling between its two channels. It proposes a hybrid design procedure that combines frequency response and root locus approaches. The proposed controller is designated as PID-Lead Compensator (PIDLC); the PID controller was designed in previous work using frequency response design specifications, while the lead compensator is proposed in this paper and is designed using the root locus method. A general explicit formula for angle computations in any of the four quadrants is also given. The lead compensator is designed by shifting the dominant closed-loop poles slightly to the left in the s-plane. This has the effect of enhancing the relative stability of the closed-loop system by eliminating the oscillation in its transient part but at the expense of greater rise time. However, for some applications, long rise time may be an allowable price to get rid of undesired oscillation. To demonstrate the proposed hybrid controller's performance numerically, a new performance index, designated by Integral Reciprocal Time Absolute Error (IRTAE), is defined as a figure to measure the oscillation of the response in its transient part. The proposed controller enhances this performance index by 0.6771%. Although the relative enhancement of the performance index is small, it contributes to eliminating the oscillation of the response in its transient part. Simulation results are performed on the MATLAB/Simulink environment.

Design of a PID-Lead Compensator for a Twin Rotor Aerodynamic System (TRAS)

This paper deals with a Twin Rotor Aerodynamic System (TRAS). It is a Multi-Input Multi-Output (MIMO) system with high crosscoupling between its two channels. It proposes a hybrid design procedure that combines frequency response and root locus approaches. The proposed controller is designated as PID-Lead Compensator (PIDLC); the PID controller was designed in previous work using frequency response design specifications, while the lead compensator is proposed in this paper and is designed using the root locus method. A general explicit formula for angle computations in any of the four quadrants is also given. The lead compensator is designed by shifting the dominant closed-loop poles slightly to the left in the s-plane. This has the effect of enhancing the relative stability of the closed-loop system by eliminating the oscillation in its transient part but at the expense of greater rise time. However, for some applications, long rise time may be an allowable price to get rid of undesired oscillation. To demonstrate the proposed hybrid controller's performance numerically, a new performance index, designated by Integral Reciprocal Time Absolute Error (IRTAE), is defined as a figure to measure the oscillation of the response in its transient part. The proposed controller enhances this performance index by 0.6771%. Although the relative enhancement of the performance index is small, it contributes to eliminating the oscillation of the response in its transient part. Simulation results are performed on the MATLAB/Simulink environment.

The structure and free resolutions of the symbolic powers of star configurations of hypersurfaces

Star configurations of points are configurations with known (and conjectured) extremal behaviors among all configurations of points in $\mathbb P_k^n$; additional interest come from their rich structure, which allows them to be studied using tools from algebraic geometry, combinatorics, commutative algebra and representation theory. In the present paper we investigate the more general problem of determining the structure of symbolic powers of a wide generalization of star configurations of points (introduced by Geramita, Harbourne, Migliore and Nagel) called star configurations of hypersurfaces in $\mathbb P_k^n$. Here (1) we provide explicit minimal generating sets of the symbolic powers $I^{(m)}$ of these ideals $I$, (2) we introduce a notion of $\delta$-c.i. quotients, which generalize ideals with linear quotients, and show that $I^{(m)}$ have $\delta$-c.i. quotients, (3) we show that the shape of the Betti tables of these symbolic powers is determined by certain Koszul strands and we prove that a little bit more than the bottom half of the Betti table has a regular, almost hypnotic, pattern, and (4) we provide a closed formula for all the graded Betti numbers in these strands. As a special case of (2) we deduce that symbolic powers of ideals of star configurations of points have linear quotients. We also improve and extend results by Galetto, Geramita, Shin and Van Tuyl, and provide explicit new general formulas for the minimal number of generators and the symbolic defects of star configurations. Finally, inspired by Young tableaux, we introduce a technical tool which may be of independent interest: it is a canonical way of writing any monomial in any given set of polynomials. Our methods are characteristic--free.

Revising and Extending the Linear Response Theory for Statistical Mechanical Systems: Evaluating Observables as Predictors and Predictands

Linear response theory, originally formulated for studying how near-equilibrium statistical mechanical systems respond to small perturbations, has developed into a formidable set of tools for investigating the forced behaviour of a large variety of systems, including non-equilibrium ones. Mathematically rigorous derivations of linear response theory have been provided for systems obeying stochastic dynamics as well as for deterministic chaotic systems. In this paper we provide a new angle on the problem. We study under which conditions it is possible to perform predictions of the response of a given observable of a forced system, using, as predictors, the response of one or more different observables of the same system. This allows us to bypass the need to know all the details of the acting perturbation. Thus, we break the rigid separation between forcing and response, which is key in linear response theory, and revisit the concept of causality. We find that that not all observables are equally good as predictors when a given forcing is applied. In fact, the surrogate Green function one constructs for predicting the response of an observable of interest using a “bad” observable as predictor has support that is not limited to the nonnegative time axis. We explain the mathematical reasons behind the fact that an observable is an inefficient predictor. We derive general explicit formulas that, in absence of such pathologies, allow one to reconstruct the response of an observable of interest to N independent external forcings by using as predictors N other observables, with Ngeq 1 . We provide a thorough test of the theory and of the possible pathologies by using numerical simulations of the paradigmatic Lorenz’96 model. Our results are potentially relevant for problems like the reconstruction of data from proxy signals, like in the case of paleoclimate, and, in general, the analysis of signals and feedbacks in complex systems where our knowledge on the system is limited, as in neurosciences. Our technique might also be useful for reconstructing the response to forcings of a spatially extended system in a given location by looking at the response in a separate location.

A multi-parameter generalization of the symmetric algorithm

Abstract The symmetric algorithm is a variant of the well-known Euler-Seidel method which has proven useful in the study of linearly recurrent sequences. In this paper, we introduce a multivariate generalization of the symmetric algorithm which reduces to it when all parameters are unity. We derive a general explicit formula via a combinatorial argument and also an expression for the row generating function. Several applications of our algorithm to the q-Fibonacci and q-hyper-Fibonacci numbers are discussed. Among our results is an apparently new recursive formula for the Carlitz Fibonacci polynomials. Finally, a (p, q)-analogue of the algorithm is introduced and an explicit formula for it in terms of the (p, q)-binomial coefficient is found.

Unique Normal Form and the Associated Coefficients for a Class of Three-Dimensional Nilpotent Vector Fields

In this paper, we mainly focus on the unique normal form for a class of three-dimensional vector fields via the method of transformation with parameters. A general explicit recursive formula is derived to compute the higher order normal form and the associated coefficients, which can be achieved easily by symbolic calculations. To illustrate the efficiency of the approach, a comparison of our result with others is also presented.

Multiple scales scheme for bifurcation in a delayed extended van der Pol oscillator

The van der Pol(VDP) oscillator is an essential nonlinear oscillator, which has been extensively studied. However, the results of delayed generalized VDP oscillator are relatively less. The issue of bifurcation control for a delayed extended VDP oscillator is fully considered in this paper. Firstly, a nonlinear time-delayed feedback controller is carefully designed to control the dynamics for the proposed oscillator, and the results of stability of such oscillator are obtained by means of stability switches theory. Secondly, a general explicit formula is derived to establish the properties of Hopf bifurcation by utilizing the method of multiple scales. Moreover, the impact of feedback gain on the bifurcation point for the controlled oscillator is demonstrated numerically. Finally, the effectiveness of theoretical results is verified via a numerical example.

Exact likelihood ratio calculations for pairwise cases

Some practical and theoretical aspects of evaluation of evidence based on the likelihood ratio (LR) in kinship cases are discussed. If relationships are complex or if complicating factors like mutation, correction for population structure or silent alleles need to be accounted for, available software may fail. We present an explicit general formula for non-inbred pairwise cases. Equipped with this formula it is possible to evaluate, say, how strongly a shared rare allele, points towards a specific relationship. Moreover, a general expression as the one presented, adds to the understanding of models and the underlying biological mechanisms. It is also useful for checking software and defining the limitations of programs. Some ideas for improving software may also be generated by the derivation of exact expressions.We argue that a proportional mutation model is well suited from a pragmatic point of view and derive some theoretical properties of this model. Several examples based on the general pairwise formula and its implementation in the freely available R package mut are presented.

A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities

In this paper, we consider the energy decay of a damped hyperbolic system of wave-wave type which is coupled through the velocities. We are interested in the asymptotic properties of the solutions of this system in the case of indirect nonlinear damping, i.e. when only one equation is directly damped by a nonlinear damping. We prove that the total energy of the whole system decays as fast as the damped single equation. Moreover, we give a one-step general explicit decay formula for arbitrary nonlinearity. Our results shows that the damping properties are fully transferred from the damped equation to the undamped one by the coupling in velocities, different from the case of couplings through displacements as shown in [F. Alabau, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999) 1015–1020; F. Alabau, P. Cannarsa and V. Komornik, J. Evol. Equ. 2 (2002) 127–150; F. Alabau, SIAM J. Control Optim. 41 (2002) 511–541; F. Alabau-Boussouira and M. Léautaud, ESAIM: COCV 18 (2012) 548–582] for the linear damping case, and in [F. Alabau-Boussouira, NoDEA 14 (2007) 643–669] for the nonlinear damping case. The proofs of our results are based on multiplier techniques, weighted nonlinear integral inequalities and the optimal-weight convexity method of [F. Alabau-Boussouira, Appl. Math. Optim. 51 (2005) 61–105; F. Alabau-Boussouira, J. Differ. Equ. 248 (2010) 1473–1517].

Dynamics and control in an $$({\varvec{n}}+{\varvec{2}})$$ ( n + 2 ) -neuron BAM network with multiple delays

The issues of the stability and bifurcation for a delayed BAM network involving two neurons in the I-layer and arbitrary neurons in the J-layer are concerned in the present paper. By adopting the sum of the delays as the bifurcation parameter, we discuss the distribution of the roots of the characteristic equation for high-dimension system in terms of stability switches theory and further present some sufficient conditions for the occurrence of Hopf bifurcations. Analysis reveals that Hopf bifurcation will emerge after the given system loses its stability. Moreover, we derive explicit general formulae to determine the properties of bifurcation via the normal theory and the center manifold theorem. It is demonstrated that the sum of the delays can effectively affect the dynamics of the proposed system. Finally, an illustrative example is employed to verify the validity of the theoretical results obtained.

The star-shaped Λ-coalescent and Fleming–Viot process

ABSTRACTThe star-shaped Λ-coalescent and corresponding Λ-Fleming–Viot process, where the Λ measure has a single atom at unity, are studied in this article. The transition functions and stationary distribution of the Λ-Fleming–Viot process are derived in a two-type model with mutation. The distribution of the number of non-mutant lines back in time in the star-shaped Λ-coalescent is found. Extensions are made to a model with d types, either with parent-independent mutation or general Markov mutation, and an infinitely-many-types model, when d → ∞. An eigenfunction expansion for the transition functions is found, which has polynomial right eigenfunctions and left eigenfunctions described by hyperfunctions. A further star-shaped model with general frequency-dependent change is considered and the stationary distribution in the Fleming–Viot process derived. This model includes a star-shaped Λ-Fleming–Viot process with mutation and selection. In a general Λ-coalescent explicit formulae for the transition functions and stationary distribution, when there is mutation, are unknown. However, in this article, explicit formulae are derived in the star-shaped coalescent.