Canard cycle transition at a slow–fast passage through a jump point

We introduce transitory canard cycles for slow–fast vector fields in the plane. Such cycles separate “canards without head” and “canards with head”, like for example in the Van der Pol equation. We obtain optimal upper bounds on the number of periodic orbits that can appear near the cycle under whatever condition on the related slow divergence integral I  , including the challenging case I=0$I=0$.     

Blow-up and global solutions for a class of nonlinear parabolic equations with different kinds of boundary conditions

By constructing different auxiliary functions and using Hopf’s maximum principle, the sufficient conditions for the blow-up and global solutions are presented for nonlinear parabolic equation u_t = ∇(a(u)b(x)c(t)∇u) + f(x, u, q, t) with different kinds of boundary conditions. The upper bounds of the “blow-up time” and the “upper estimates” of global solutions are provided. Finally, some examples are presented as the application of the obtained results.     

Minmax regret location–allocation problem on a network under uncertainty

We consider a robust location–allocation problem with uncertainty in demand coefficients. Specifically, for each demand point, only an interval estimate of its demand is known and we consider the problem of determining where to locate a new service when a given fraction of these demand points must be served by the utility. The optimal solution of this problem is determined by the “minimax regret” location, i.e., the point that minimizes the worst-case loss in the objective function that may occur because a decision is made without knowing which state of nature will take place. For the case where the demand points are vertices of a network we show that the robust location–allocation problem can be solved in O(min{p, n − p}n³m) time, where n is the number of demand points, p (p < n) is the fixed number of demand points that must be served by the new service and m is the number of edges of the network.     

Large eddy simulation of a sheet/cloud cavitation on a NACA0015 hydrofoil

A single fluid model of sheet/cloud cavitation is developed and applied to a NACA0015 hydrofoil. First, a cavity formation model is set up, based on a three-dimensional (3D) non-cavitation model of Navier–Stokes equations with a large eddy simulation (LES) scheme for weakly compressible flows. A fifth-order polynomial curve is adopted to describe the relationship between density coefficient ratio and pressure coefficient when cavitation occurs. The Navier–Stokes equations including cavitation bubble clusters are solved using the finite-volume approach with time-marching scheme, and MacCormack’s explicit-corrector scheme is adopted. Simulations are carried out in a 3D field acting on a hydrofoil NACA0015 at angles of attack 4°, 8° and 20°, with cavitation numbers σ = 1.0, 1.5 and 2.0, Re = 10⁶, and a 360 × 63 × 29 meshing system. We study time-dependent sheet/cloud cavitation structures, caused by the interaction of viscous objects, such as vortices, and cavitation bubbles. At small angles of attack (4°), the sheet cavity is relatively stable just by oscillating in size at the accumulation stage; at 8° it has a tendency to break away from the upper foil section, with the cloud cavitation structure becoming apparent; at 20°, the flow separates fully from the leading edge of the hydrofoil, and the vortex cavitation occurs. Comparisons with other studies, carried out mainly in the context of flow patterns on which prior experiments and simulations were done, demonstrate the power of our model. Overall, it can snapshot the collapse of cloud cavitation, and allow a study of flow patterns and their instabilities, such as “crescent-shaped regions.”     

A dynamic adaptive local search algorithm for the circular packing problem

This paper studies the circular packing problem (CPP) which consists of packing n non-identical circles C_i of known radius r_i, i ∈ N = {1, … , n}, into the smallest containing circle C. The objective is to determine the coordinates (x_i, y_i) of the center of C_i, i ∈ N, as well as the radius r and center (x, y) of C. This problem, which is a variant of the two-dimensional open dimension problem, is solved using a two-step, dynamic, adaptive, local search algorithm. At each iteration, the algorithm identifies the set of potential “best local positions” of a circle C_i, i ∈ N, given the positions of the previously packed circles, and determines for each of these positions the coordinates and radius of the smallest containing circle. The “best local position” minimizes the radius of the current containing circle. That is, every time an additional circle is packed, both the center and the radius of the containing circle are dynamically updated, and the smallest containing circle is known. The experimental results reflect the good performance of the algorithm.     

Considering the isolation set problem

Given a network, G = [N, E] the Isolation Set Problem (ISP) finds the set of arcs, D ⊆ E, that when removed will separate a predefined set of r distinguished nodes [2]. This involves eliminating connections from a specific set of nodes to the rest of a network. In our increasingly interconnected network-centric world, this might be isolating various units from Headquarters; isolating a portion of a computer network to disrupt communications or to quarantine a virus or some other form of cyber attack; or isolating a cell or sub-group in a terrorist or “dark” network, for example.     

Elliptic complex numbers with dual multiplication

Investigated is a number system in which the square of a basis number: (w)², and the square of its additive inverse: (−w)², are not equal. Termed W space, a vector space over the reals, this number system will be introduced by restating defining relations for complex space C, then changing a defining conjugacy relation from conj(z) + z = 0 in the complexes to conj(z) + z = 1 for W space. This change produces a dual-represented vector space consisting of two dual, isomorphic fields, which are unified under one “context-sensitive” multiplication. Fundamental algebraic and geometric properties will be investigated. W space can be interpreted as a generalization of the complexes but is characterized by an interacting duality which seems to produce two of everything: two representations, two multiplications, two norm values, and two solutions to a linear equation. W space will be compared to a previous suggestion of a similar algebra, and then possible applications will be offered, including a W space fractal.     

Applications of bulk queues to group testing models with incomplete identification

A population of items is said to be “group-testable”, (i) if the items can be classified as “good” and “bad”, and (ii) if it is possible to carry out a simultaneous test on a batch of items with two possible outcomes: “Success” (indicating that all items in the batch are good) or “failure” (indicating a contaminated batch). In this paper, we assume that the items to be tested arrive at the group-testing centre according to a Poisson process and are served (i.e., group-tested) in batches by one server. The service time distribution is general but it depends on the batch size being tested. These assumptions give rise to the bulk queueing model M/G^(m,M)/1, where m and M(>m) are the decision variables where each batch size can be between m and M. We develop the generating function for the steady-state probabilities of the embedded Markov chain. We then consider a more realistic finite state version of the problem where the testing centre has a finite capacity and present an expected profit objective function. We compute the optimal values of the decision variables (m, M) that maximize the expected profit. For a special case of the problem, we determine the optimal decision explicitly in terms of the Lambert function.     

Mathematical model of liquid–liquid equilibrium for a ternary system using the GMDH-type neural network and genetic algorithm

A GMDH type-neural network was used to predict liquid phase equilibrium data for the (water + ethanol + trans-decalin) ternary system in the temperature range of 300.2–315.2 K. In order to accomplish modeling, the experimental data were divided into train and test sections. The data set was divided into two parts: 70% were used as data for “training” and 30% were used as a test set. The predicted values were compared with those of experimental values in order to evaluate the performance of the GMDH neural network method. The results obtained by using GMDH type neural network are in excellent agreement with the experimental results.     

Hidden lessons: How a focus on slope-like properties of quadratic functions encouraged unexpected generalizations

This article presents secondary students’ generalizations about the connections between algebraic and graphical representations of quadratic functions, focusing specifically on the roles of the parameters a, b, and c in the general form of a quadratic function, y = ax² + bx + c. Students’ generalizations about these connections led to a surprising finding: two-thirds of the students interviewed identified the parameter a as the “slope” of the parabola. Analysis of qualitative data from interviews and classroom observations led to the development of three focusing phenomena in the classroom environment that inadvertently supported a focus on slope-like properties of quadratic functions: (a) the use of linear analogies, (b) the rise over run method, and (c) viewing a as dynamic rather than static.     

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.     

Modified Whittle estimation of multilateral models on a lattice

In the estimation of parametric models for stationary spatial or spatio-temporal data on a d-dimensional lattice, for d?2, the achievement of asymptotic efficiency under Gaussianity, and asymptotic normality more generally, with standard convergence rate, faces two obstacles. One is the “edge effect”, which worsens with increasing d. The other is the possible difficulty of computing a continuous-frequency form of Whittle estimate or a time domain Gaussian maximum likelihood estimate, due mainly to the Jacobian term. This is especially a problem in “multilateral” models, which are naturally expressed in terms of lagged values in both directions for one or more of the d dimensions. An extension of the discrete-frequency Whittle estimate from the time series literature deals conveniently with the computational problem, but when subjected to a standard device for avoiding the edge effect has disastrous asymptotic performance, along with finite sample numerical drawbacks, the objective function lacking a minimum-distance interpretation and losing any global convexity properties. We overcome these problems by first optimizing a standard, guaranteed non-negative, discrete-frequency, Whittle function, without edge-effect correction, providing an estimate with a slow convergence rate, then improving this by a sequence of computationally convenient approximate Newton iterations using a modified, almost-unbiased periodogram, the desired asymptotic properties being achieved after finitely many steps. The asymptotic regime allows increase in both directions of all d dimensions, with the central limit theorem established after re-ordering as a triangular array. However our work offers something new for “unilateral” models also. When the data are non-Gaussian, asymptotic variances of all parameter estimates may be affected, and we propose consistent, non-negative definite estimates of the asymptotic variance matrix.     

Maximum order-index of matrices over commutative inclines: an answer to an open problem

This paper proves that the maximum order-index of n × n matrices over an arbitrary commutative incline equals (n − 1)² + 1. This is an answer to an open problem “Compute the maximum order-index of a member of M_n(L)”, proposed by Cao, Kim and Roush in a monograph Incline Algebra and Applications, 1984, where M_n(L) is the set of all n × n matrices over an incline L.     

The Proof of a Conjecture of Additive Number Theory

The aim of this paper is to show that for any n∈N, n>3, there exist a, b∈N* such that n=a+b, the “lengths” of a and b having the same parity (see the text for the definition of the “length” of a natural number). Also we will show that for any n∈N, n>2, n≠5, 10, there exist a, b∈N* such that n=a+b, the “lengths” of a and b having different parities. We will prove also that for any prime p≡7(mod 8) there exist a, b∈N* such that p=a²+b, the “length” of b being an even number.     

What drives value creation in investment projects? An application of sensitivity analysis to project finance transactions

Evaluating the economic attractiveness of large projects often requires the development of large and complex financial models. Model complexity can prevent management from obtaining crucial information, with the risk of a suboptimal exploitation of the modelling efforts. We propose a methodology based on the so-called “differential importance measure (D)$(D)$” to enhance the managerial insights obtained from financial models. We illustrate our methodology by applying it to a project finance case study. We show that the additivity property of D grants analysts and managers full flexibility in combining parameters into any group and at the desired aggregation level. We analyze investment criteria related to both the investors’s and lenders’ perspectives. Results indicate that exogenous factors affect investors (sponsors and lenders) in different ways, whether exogenous variables are considered individually or by groups.     

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.     

A bivariate optimal replacement policy for a cold standby repairable system with preventive repair

In this paper, the repair-replacement problem for a deteriorating cold standby repairable system is investigated. The system consists of two dissimilar components, in which component 1 is the main component with use priority and component 2 is a supplementary component. In order to extend the working time and economize the running cost of the system, preventive repair for component 1 is performed every time interval T, and the preventive repair is “as good as new”. As a supplementary component, component 2 is only used at the time that component 1 is under preventive repair or failure repair. Assumed that the failure repair of component 1 follows geometric process repair while the repair of component 2 is “as good as new”. A bivariate repair-replacement policy (T, N) is adopted for the system, where T is the interval length between preventive repairs, and N is the number of failures of component 1. The aim is to determine an optimal bivariate policy (T, N)^∗ such that the average cost rate of the system is minimized. The explicit expression of the average cost rate is derived and the corresponding optimal bivariate policy can be determined analytically or numerically. Finally, a Gamma distributed example is given to illustrate the theoretical results for the proposed model.     

Inclusions of von Neumann algebras and quantum groupoïds III

In a former article, in collaboration with Jean-Michel Vallin, we have constructed two “quantum groupo?¨ds” dual to each other, from a depth 2 inclusion of von Neumann algebras M₀⊂M₁, in such a way that the canonical Jones’tower associated to the inclusion can be described as a tower of successive crossed-products by these two structures. We are now investigating in greater details these structures in the presence of an appropriate modular theory on the basis , and we show how these examples fit with Lesieur's “measured quantum groupo?¨ds”.     

Constraint programming for computing non-stationary (R, S) inventory policies

This paper proposes a constraint programming model for computing the finite horizon single-item inventory problem with stochastic demands in discrete time periods with service-level constraints under the non-stationary version of the “periodic review, order-up-to-level” policy (i.e., non-stationary (R, S) or, simply (R_n, S_n)). It is observed that the modeling process is more natural and the required number of variables is smaller compared to the MIP formulation of the same problem. The computational tests show that the CP approach is more tractable than the conventional MIP formulation. Two different domain reduction methods are proposed to improve the computational performance of solution algorithms. The numerical experiments confirmed the effectiveness of these methods.     

Forecasting accuracy of behavioural models for participation in the arts

This paper assesses the forecasting performance of count data models applied to arts attendance. We estimate participation models for two artistic activities that differ in their degree of popularity – museums and jazz concerts – with data derived from the 2002 release of the Survey of Public Participation in the Arts for the United States. We estimate a finite mixture model – a zero-inflated negative binomial model – that allows us to distinguish between “true” non-attendants and “goers” and their respective behaviour regarding participation in the arts. We evaluate the predictive (in-sample) and forecasting (out-of-sample) accuracy of the estimated model using bootstrapping techniques to compute the Brier score. Overall, the results indicate the model performs well in terms of forecasting. Finally, we draw certain policy implications from the model’s forecasting capacity, thereby allowing the identification of target populations.