Solving the non-isospectral Ablowitz–Ladik hierarchy via the inverse scattering transform and reductions

The non-isospectral Ablowitz–Ladik hierarchy is integrated by the inverse scattering transform. In contrast with the isospectral Ablowitz–Ladik hierarchy, the eigenvalues of the non-isospectral Ablowitz–Ladik equations in the scattering data are time-dependent. The multi-soliton solution for the hierarchy is presented. The reductions to the non-isospectral discrete NLS hierarchy and the non-isospectral discrete mKdV hierarchy and their solutions are considered.     

A brief historical introduction to solitons and the inverse scattering transform–a vision of Scott Russell

The objective of this study was to evaluate biomathtutor by (i) investigating the impact of biomathtutor on the mathematics skills competencies of bioscience undergraduates, and (ii) assessing students’ and tutors’ reactions to biomathtutor, identifying whether and how tutors might integrate it into their curricula and blend it with more traditional teaching practices to enhance their students’ learning experiences. A multi-method approach was adopted in which a quasi-experiment and non-experimental evaluation of biomathtutor were used to collect both quantitative and qualitative data, using mathematics tests, questionnaires, tutor interviews and student focus groups. Eighty-nine bioscience undergraduates and eight tutors participated in the study. A comparison of student performance in the quasi-experiment, which adopted a pre-test-intervention-post-test methodology, revealed no significant difference between pre-test and post-test scores for either the ‘control’ group (no intervention) or for any of the mathematics learning support interventions used, including biomathtutor. Despite the limitations of the quasi-experiment which are discussed, tutors’ and their students’ reactions towards biomathtutor were very positive, with both groups agreeing that biomathtutor represents a very well designed and useful learning resource that has a valuable role to play in supporting mathematics learning within bioscience curricula. Students felt that using biomathtutor had helped them acquire new biological and mathematical knowledge and had increased their competence and confidence in mathematics, with many students confirming that they would use biomathtutor again. Tutors felt it would be useful to embed biomathtutor, where possible, into their curricula, perhaps linking it to assessment strategies or integrating it with their current more traditional teaching practices. Students indicated that they too would like to see biomathtutor embedded within their curricula, primarily because it would motivate them to use the resource. Modifications to biomathtutor, which may need to be considered in light of any potential further development of this resource, are discussed.     

On the moments of order statistics coming from the Topp–Leone distribution

We present recurrence relations for the moments of order statistics from the Topp–Leone distribution without any restriction for the shape parameter. Several relations are also derived when the shape parameter is an integer. An application of the results is provided.     

A first–passage time random walk distribution with five transition probabilities: a generalization of the shifted inverse trinomial

In this paper a univariate discrete distribution, denoted by GIT, is proposed as a generalization of the shifted inverse trinomial distribution, and is formulated as a first-passage time distribution of a modified random walk on the half-plane with five transition probabilities. In contrast, the inverse trinomial arises as a random walk on the real line with three transition probabilities. The probability mass function (pmf) is expressible in terms of the Gauss hypergeometric function and this offers computational advantage due to its recurrence formula. The descending factorial moment is also obtained. The GIT contains twenty-two possible distributions in total. Special cases include the binomial, negative binomial, shifted negative binomial, shifted inverse binomial or, equivalently, lost-games, and shifted inverse trinomial distributions. A subclass GIT_3,1 is a particular member of Kemp’s class of convolution of pseudo-binomial variables and its properties such as reproductivity, formulation, pmf, moments, index of dispersion, and approximations are studied in detail. Compound or generalized (stopped sum) distributions provide inflated models. The inflated GIT_3,1 extends Minkova’s inflated-parameter binomial and negative binomial. A bivariate model which has the GIT as a marginal distribution is also proposed.     

What portion of the sample makes a partial sum asymptotically stable or normal?

Summary Let a sequence of independent and identically distributed random variables with the common distribution function in the domain of attraction of a stable law of index 0<2 be given. We show that if at each stage n a number k _n depending on n of the lower and upper order statistics are removed from the n-th partial sum of the given random variables then under appropriate conditions on k _n the remaining sum can be normalized to converge in distribution to a standard normal random variable. A further analysis is given to show which ranges of the order statistics contribute to asymptotic stable law behaviour and which to normal behaviour. Our main tool is a new Brownian bridge approximation to the uniform empirical process in weighted supremum norms.Work done while visiting the Bolyai Institute, Szeged University, partially supported by a University of Delaware Research Foundation Grant     

On the Su–Schrieffer–Heeger model of electron transport: Low-temperature optical conductivity by the Mellin transform

We describe the low-temperature optical conductivity as a function of frequency for a quantum-mechanical system of electrons that hop along a polymer chain. To this end, we invoke the Su–Schrieffer–Heeger tight-binding Hamiltonian for noninteracting spinless electrons on a one-dimensional (1D) lattice. Our goal is to show via asymptotics how the interband conductivity of this system behaves as the smallest energy bandgap tends to close. Our analytical approach includes: (i) the Kubo-type formulation for the optical conductivity with a nonzero damping due to microscopic collisions, (ii) reduction of this formulation to a 1D momentum integral over the Brillouin zone, and (iii) evaluation of this integral in terms of elementary functions via the three-dimensional Mellin transform with respect to key physical parameters and subsequent inversion in a region of the respective complex space. Our approach reveals an intimate connection of the behavior of the conductivity to particular singularities of its Mellin transform. The analytical results are found in good agreement with direct numerical computations.     

Optimal consumption–investment strategy under the Vasicek model: HARA utility and Legendre transform

This paper studies the optimal consumption–investment strategy with multiple risky assets and stochastic interest rates, in which interest rate is supposed to be driven by the Vasicek model. The objective of the individuals is to seek an optimal consumption–investment strategy to maximize the expected discount utility of intermediate consumption and terminal wealth in the finite horizon. In the utility theory, Hyperbolic Absolute Risk Aversion (HARA) utility consists of CRRA utility, CARA utility and Logarithmic utility as special cases. In addition, HARA utility is seldom studied in continuous-time portfolio selection theory due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the individuals. Due to the complexity of the structure of the solution to the original Hamilton–Jacobi–Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solution to the optimal consumption–investment strategy in a complete market. Moreover, some special cases are also discussed in detail. Finally, a numerical example is given to illustrate our results.     

Pauli isotonic oscillatorwith an anomalous magnetic moment in the presence of the Aharonov–Bohm effect: Laplace transform approach

A strong magnetic field significantly affects the intrinsic magnetic moment of fermions. In quantum electrodynamics, it was shown that the anomalous magnetic moment of an electron arises kinematically, while it results from a dynamical interaction with an external magnetic field for hadrons (proton). Taking the anomalous magnetic moment of a fermion into account, we find an exact expression for the boundstate energy and the corresponding eigenfunctions of a two-dimensional nonrelativistic spin-1/2 harmonic oscillator with a centripetal barrier (known as the isotonic oscillator) including an Aharonov–Bohm term in the presence of a strong magnetic field. We use the Laplace transform method in the calculations. We find that the singular solution contributes to the phase of the wave function at the origin and the phase depends on the spin and magnetic flux.     

An inventory model: What is the influence of the shape of the lead time demand distribution?

In this paper the influence of the shape of the lead time demand distribution is studied for a specific inventory model which is described in a preceding paper by Heuts and van Lieshout [4]. This continuous review inventory model uses as lead time demand distribution a Schmeiser-Deutsch distribution (S-D distribution) [9]. In a previous paper [4] an algorithm was given to solve the decision problem.In the literature attention is given to the following problem: what information on the demand during the lead time is necessary and sufficient to obtain good decisions. Using a (s, S) policy; Naddor [8] concluded that thespecific form of the lead time demand distribution is negligible, and that only its first two moments are essential. For a simple (s, q) control system Fortuin [3] comes to the same conclusion. Both authors analysed the case with known lead times and with given demand distributions from the class of two parameter distributions. So in fact their results are obvious, as the lead time demand distributions resulting from their suppositions are all nearly symmetric. We shall demonstrate that the skewness of the lead time demand distribution in our inventory model is also an important measure, which should be taken into account, as the cost differences with regard to the case where this skewness measure is not used, can be considerable.     

Multiclass multiserver queueing system in the Halfin–Whitt heavy traffic regime: asymptotics of the stationary distribution

We consider a heterogeneous queueing system consisting of one large pool of O(r) identical servers, where r→∞ is the scaling parameter. The arriving customers belong to one of several classes which determines the service times in the distributional sense. The system is heavily loaded in the Halfin–Whitt sense, namely the nominal utilization is $1-a/\sqrt{r}$ where a>0 is the spare capacity parameter. Our goal is to obtain bounds on the steady state performance metrics such as the number of customers waiting in the queue Q ^r (∞). While there is a rich literature on deriving process level (transient) scaling limits for such systems, the results for steady state are primarily limited to the single class case. This paper is the first one to address the case of heterogeneity in the steady state regime. Moreover, our results hold for any service policy which does not admit server idling when there are customers waiting in the queue. We assume that the interarrival and service times have exponential distribution, and that customers of each class may abandon while waiting in the queue at a certain rate (which may be zero). We obtain upper bounds of the form $O(\sqrt{r})$ on both Q ^r (∞) and the number of idle servers. The bounds are uniform w.r.t. parameter r and the service policy. In particular, we show that $\limsup_{r} \mathbb {E}\exp(\theta r^{-{1\over2}}Q^{r}(\infty))<\infty$ . Therefore, the sequence $r^{-{1\over2}}Q^{r}(\infty)$ is tight and has a uniform exponential tail bound. We further consider the system with strictly positive abandonment rates, and show that in this case every weak limit $\hat{Q}(\infty)$ of $r^{-{1\over2}}Q^{r}(\infty)$ has a sub-Gaussian tail. Namely, $\mathbb {E}[\exp(\theta(\hat{Q}(\infty ))^{2})]<\infty$ , for some θ>0.     

Evolving to the edge of chaos: Chance or necessity?

We show that ecological systems evolve to edges of chaos (EOC). This has been demonstrated by analyzing three diverse model ecosystems using numerical simulations in combination with analytical procedures. It has been found that all these systems reside on EOC and display short-term recurrent chaos (strc). The first two are non-linear food chains and the third one is a linear food chain. The dynamics of first two is dictated by deterministic changes in system parameters. In contrast to this, dynamics of the third model system (the linear food chain) is governed by both deterministic changes in system parameters as well as exogenous stochastic perturbations (unforeseen changes in initial conditions) of these dynamical systems.     

Inverse scattering transform for the two-component Gerdjikov–Ivanov equation with nonzero boundary conditions: Dark-dark solitons

The two-component Gerdjikov–Ivanov equation with nonzero boundary conditions is studied by the inverse scattering transform. A fundamental set of analytic eigenfunctions is obtained with the aid of the associated adjoint problem. Three symmetry conditions are discussed to curb the scattering data. The behavior of the Jost functions and the scattering matrix at the branch points is discussed. The inverse scattering problem is formulated by a matrix Riemann–Hilbert problem. The trace formula in terms of the scattering data and the so-called asymptotic phase difference for the potential are obtained. The solitons classification is described in detail. When the discrete eigenvalues lie on the circle, the dark-dark soliton is obtained for the first time in this work. And the discrete eigenvalues off the circle generate the dark-bright, bright-bright, breather-breather, M(-type)-W(-type) solitons, and their interactions.     

Is the Home Equity Conversion Mortgage in the United States sustainable? Evidence from pricing mortgage insurance premiums and non-recourse provisions using the conditional Esscher transform

The purpose of this paper is to build a modeling and pricing framework to investigate the sustainability of the Home Equity Conversion Mortgage (HECM) program in the United States under realistic economic scenarios, i.e., whether the premium payments cover the fair premiums for the inherent risks in the HECM program. We note that earlier HECM models use static mortality tables, neglecting the dynamics of mortality rates and extreme mortality jumps. The earlier models also assume housing prices follow a geometric Brownian motion, which contradicts the fact that housing prices exhibit strong autocorrelation and varying volatility over time. To solve these problems, we propose a generalized Lee-Carter model with asymmetric jump effects to fit the mortality data, and model the house price index via an ARIMA-GARCH process. We then employ the conditional Esscher transform to price the non-recourse provision of reverse mortgages and compare it with the calculated mortgage insurance premiums. The HECM program turns out to be sustainable based on our model setup and parameter settings.