Evaluation of the degenerate scale in antiplane elasticity using null field BIE

This paper provides a numerical solution for the degenerate scale in antiplane elasticity using the null field boundary integral equation (BIE). With coordinate transformation, the BIE can be formulated in a normal scale, and a basic solution for the BIE in the normal scale is illustrated. The degenerate scale is easily obtained from the basic solution. Several numerical examples are given.     

Mathematical analysis of the guided waves in photonic crystal fibers

The propagation of guided waves in photonic crystal fibers (PCFs) is studied. The structure of a PCF can be regarded as a perfect two-dimensional photonic crystal with a line defect along the invariant direction. This problem can be treated as an eigenvalue problem for a family of noncompact self-adjoint operators. We prove that line defects do not change the essential spectrum of the associated “background” medium. This result plays a key role for studying the influence of line defects on the “background” spectrum. A modified Combes-Thomas estimate is also formulated.     

Credit,income, and causality: A contemporary co-integration analysis

In this study, we investigate the macroeconomic fundamental issue of causal relationship between credit and money income, for the US postwar economy, along the lines of the “Credit–View” theorists. In the long-run, we provide significant evidence that credit causes income. Further, we support the view that there is a rather weak causality effect from income to credit, in the long-run. Indeed, we identified that there is a dynamic causality effect in the short-run, from income changes to credit ones. A contemporary co-integration analysis is applied, using the maximum likelihood method. Additionally, considering the formulated VAR, a stability analysis of the equivalent first order dynamic system has been performed. Finally, dynamic forecasts from the corresponding ECVAR (Error Correction VAR) are obtained, in order to verify the validity of the co-integration vector used. It may be useful to mention, that we have not met in the relevant literature, the type of forecasting presented here.     

The efficient solution of the (quietly constrained) noisy, linear regulator problem

In a previous paper we gave a new, natural extension of the calculus of variations/optimal control theory to a (strong) stochastic setting. We now extend the theory of this most fundamental chapter of optimal control in several directions. Most importantly we present a new method of stochastic control, adding Brownian motion which makes the problem “noisy.” Secondly, we show how to obtain efficient solutions: direct stochastic integration for simpler problems and/or efficient and accurate numerical methods with a global a priori error of O(h3/2) for more complex problems. Finally, we include “quiet” constraints, i.e. deterministic relationships between the state and control variables. Our theory and results can be immediately restricted to the non “noisy” (deterministic) case yielding efficient, numerical solution techniques and an a priori error of O(h²). In this event we obtain the most efficient method of solving the (constrained) classical Linear Regulator Problem. Our methods are different from the standard theory of stochastic control. In some cases the solutions coincide or at least are closely related. However, our methods have many advantages including those mentioned above. In addition, our methods more directly follow the motivation and theory of classical (deterministic) optimization which is perhaps the most important area of physical and engineering science. Our results follow from related ideas in the deterministic theory. Thus, our approximation methods follow by guessing at an algorithm, but the proof of global convergence uses stochastic techniques because our trajectories are not differentiable. Along these lines, a general drift term in the trajectory equation is properly viewed as an added constraint and extends ideas given in the deterministic case by the first author.     

An extension of Glimm's method to inhomogeneous strictly hyperbolic systems of conservation laws by “weaker than weak” solutions of the Riemann problem

We construct a generalized solution of the Riemann problem for strictly hyperbolic systems of conservation laws with source terms, and we use this to show that Glimm's method can be used directly to establish the existence of solutions of the Cauchy problem. The source terms are taken to be of the form a^′G, and this enables us to extend the method introduced by Lax to construct general solutions of the Riemann problem. Our generalized solution of the Riemann problem is “weaker than weak” in the sense that it is weaker than a distributional solution. Thus, we prove that a weak solution of the Cauchy problem is the limit of a sequence of Glimm scheme approximate solutions that are based on “weaker than weak” solutions of the Riemann problem. By establishing the convergence of Glimm's method, it follows that all of the results on time asymptotics and uniqueness for Glimm's method (in the presence of a linearly degenerate field) now apply, unchanged, to inhomogeneous systems.     

Existence and uniqueness of solutions for acoustic scattering over infinite obstacles

The paper considers the solution of the boundary value problem (BVP) consisting of the Helmholtz equation in the region D with a rigid boundary condition on ∂D and its reformulation as a boundary integral equation (BIE), over an infinite cylindrical surface of arbitrary smooth cross-section. A boundary integral equation, which models three-dimensional acoustic scattering from an infinite rigid cylinder, illustrates the application of the above results to prove existence of solution of the integral equation and the corresponding boundary value problem.     

The inequality-satisfiability problem

We define a generalization of the satisfiability problem (SAT) where each “clause” is an or-list of inequalities in n variables. This problem is NP-complete even when containing only two inequalities per “clause”, but solvable in polynomial time when either the number of variables or the number of “clauses” is fixed.     

On the lifetime of a random binary sequence

Consider a system with m elements which is used to fulfill tasks. Each task is sent to one element which fulfills a task and the outcome is either fulfillment of the task (“1”) or the failure of the element (“0”). Initially, m tasks are sent to the system. At the second step, a complex of length m₁ is formed and sent to the system, where m₁ is the number of tasks fulfilled at the first step, and so on. The process continues until all elements fail and the corresponding waiting time defines the lifetime of the binary sequence which consists of “1” or “0”. We obtain a recursive equation for the expected value of this waiting time random variable.     

Modified algorithms for the minimum volume enclosing axis-aligned ellipsoid problem

Consider the problem of computing a (1+?)-approximation to the minimum volume axis-aligned ellipsoid (MVAE) enclosing a set of m points in Rⁿ. We first provide an extension and improvement to algorithm proposed in Kumar and Y?ld?r?m (2008) [5] (the KY algorithm) for the MVAE problem. The main challenge of the MVAE problem is that there is no closed form solution in the line search step (beta). Therefore, the KY algorithm proposed a certain choice of beta that leads to their complexity and core set results in solving the MVAE problem. We further analyze the line search step to derive a new beta, relying on an analysis of up to the fourth order derivative. This choice of beta leads to the improved complexity and core set results. The second modification is given by incorporating “away steps” into the first one at each iteration, which obtains the same complexity and core set results as the first one. In addition, since the second modification uses the idea of “dropping points”, it has the potential to compute smaller core sets in practice. Some numerical results are given to show the efficiency of the modified algorithms.     

Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations

In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate or singular when “the gradient is small”. Typical examples are either equations involving the m-Laplace operator or Bellman-Isaacs equations from stochastic control problems. We establish an Alexandroff-Bakelman-Pucci estimate and we prove a Harnack inequality for viscosity solutions of such non-linear elliptic equations.     

Universal cycles of classes of restricted words

It is well known that Universal cycles (U-cycles) of k-letter words on an n-letter alphabet exist for all k and n. In this paper, we prove that Universal cycles exist for several restricted classes of words, including non-bijections, “equitable” words (under suitable restrictions), ranked permutations, and “passwords”. In each case, proving the connectedness of the underlying de Bruijn digraph is a non-trivial step.     

Construction of a model towards EOQ and pricing strategy for gradually obsolescent products

Due to the rapid advance of technology, manufacturers’ marketing models and quickly changing consumer tastes, products such as apparel, clothing accessories and consumer electronics are likely to face the problem of short product life cycles. This study deals with the problem of determining order quantity and multi-discount selling prices for these types of gradually obsolescent products, in which two novel proposals are presented as follows: (1) without considering any exogenous factors that could affect demand, we develop a time-dependent demand model that appropriately portrays an integrated demand behavior associated with the characteristic of obsolescence; (2) we then treat the selling price as an exogenous factor influencing demand and, referring to the linear demand D = α − βp, the effect that “the increase of demand due to price change is linearly correlated with the difference between two consecutive selling prices” is incorporated, so as to make the demand model be a function of both time and selling price. Afterwards, optimization models are hereby formulated to predetermine pricing strategy with a limited number of price changes by maximizing retailer’s profit. As a result, numerical examples illustrate that the multi-discount model indeed provides higher total profit than a single discount one. This is presented along with the result analysis conducted to gain some managerial insights.     

h-Vectors of Gorenstein polytopes

We show that the Ehrhart h-vector of an integer Gorenstein polytope with a regular unimodular triangulation satisfies McMullen's g-theorem; in particular, it is unimodal. This result generalizes a recent theorem of Athanasiadis (conjectured by Stanley) for compressed polytopes. It is derived from a more general theorem on Gorenstein affine normal monoids M: one can factor K[M] (K a field) by a “long” regular sequence in such a way that the quotient is still a normal affine monoid algebra. This technique reduces all questions about the Ehrhart h-vector of P to the Ehrhart h-vector of a Gorenstein polytope Q with exactly one interior lattice point, provided each lattice point in a multiple cP, c∈N, can be written as the sum of c lattice points in P. (Up to a translation, the polytope Q belongs to the class of reflexive polytopes considered in connection with mirror symmetry.) If P has a regular unimodular triangulation, then it follows readily that the Ehrhart h-vector of P coincides with the combinatorial h-vector of the boundary complex of a simplicial polytope, and the g-theorem applies.     

Smooth zero-contact-angle solutions to a thin-film equation around the steady state

In the simplest case of a linearly degenerate mobility, we view the thin-film equation as a classical free boundary problem. Our focus is on the regularity of solutions and of their free boundary in the “complete wetting” regime, which prescribes zero slope at the free boundary. In order to rule out of the analysis possible changes in the topology of the positivity set, we zoom into the free boundary by looking at perturbations of the stationary solution. Our strategy is based on a priori energy-type estimates which provide “minimal” conditions on the initial datum under which a unique global solution exists. In fact, this solution turns out to be smooth for positive times and to converge to the stationary solution for large times. As a consequence, we obtain smoothness and large-time behavior of the free boundary.     

A stochastic and asymmetric-information framework for a dominant-manufacturer supply chain

Consider a dominant manufacturer wholesaling a product to a retailer, who in turn retails it to the consumers at $p/unit. The retail-market demand volume varies with p according to a given demand curve. This basic system is commonly modeled as a manufacturer-Stackelberg ([mS]) game under a “deterministic and symmetric-information” (“det-sym-i”) framework. We first explain the logical flaws of this framework, which are (i) the dominant manufacturer-leader will have a lower profit than the retailer under an iso-elastic demand curve; (ii) in some situations the system’s “correct solution” can be hyper-sensitive to minute changes in the demand curve; (iii) applying volume discounting while keeping the original [mS] profit-maximizing objective leads to an implausible degenerate solution in which the manufacturer has dictatorial power over the channel. We then present an extension of the “stochastic and asymmetric-information” (“sto-asy-i”) framework proposed in Lau and Lau [Lau, A., Lau, H.-S., 2005. Some two-echelon supply-chain games: Improving from deterministic–symmetric-information to stochastic-asymmetric-information models. European Journal of Operational Research 161 (1), 203–223], coupled with the notion that a profit-maximizing dominant manufacturer may implement not only [mS] but also “[pm]”—i.e., using a manufacturer-imposed maximum retail price. We show that this new framework resolves all the logical flaws stated above. Along the way, we also present a procedure for the dominant manufacturer to design a profit-maximizing volume-discount scheme using stochastic and asymmetric demand information.     

Nonlocal second-order geometric equations arising in tomographic reconstruction

In this paper, we study new nonlocal geometric equations which are related to tomographic reconstruction when using the level-set approach. We treat two additional difficulties which make the work original. On one hand, the level lines do not evolve along normal directions and the nonlocal term is not of “convolution type”. On the other hand, the speed is not necessarily bounded compared to the nonlocal term. We prove an existence and uniqueness result for our equation.