Chiellini integrability and quadratically damped oscillators

In this paper a new approach to study an equation of the Liénard type with a strong quadratic damping is proposed based on Jacobi's last multiplier and Chiellini's integrability condition. We obtain a closed form solution of the transcedental characteristic equation of the Liénard type equation using the Lambert W-function.     

Tightly focusing of spirally polarized Quadratic Bessel Gaussian beam through a dielectric interface

Based on vector diffraction theory, the tight focusing properties of spirally polarized axisymmetric Bessel-modulated Gaussian beam through a dielectric interface on high numerical aperture (NA) are investigated theoretically. The optical intensity distribution in the focal region of high NA objective lens is investigated in detail by numerical calculations. The results show that the focal shift induced in the focal region is by mismatch of refractive indices across the dielectric interface. It is also found that the optical intensity in focal region of spirally polarized Quadratic Bessel Gaussian (QBG) beam can be altered considerably by changing spiral parameter C that indicates the polarization spiral degree of the incident beam.     

Formulation,analysis and numerical study of an optimization-based conservative interpolation (remap) of scalar fields for arbitrary Lagrangian–Eulerian methods

We develop and study the high-order conservative and monotone optimization-based remap (OBR) of a scalar conserved quantity (mass) between two close meshes with the same connectivity. The key idea is to phrase remap as a global inequality-constrained optimization problem for mass fluxes between neighboring cells. The objective is to minimize the discrepancy between these fluxes and the given high-order target mass fluxes, subject to constraints that enforce physically motivated bounds on the associated primitive variable (density). In so doing, we separate accuracy considerations, handled by the objective functional, from the enforcement of physical bounds, handled by the constraints. The resulting OBR formulation is applicable to general, unstructured, heterogeneous grids. Under some weak requirements on grid proximity, but not on the cell types, we prove that the OBR algorithm is linearity preserving in one, two and three dimensions. The paper also examines connections between the OBR and the recently proposed flux-corrected remap (FCR), Liska et al. [1]. We show that the FCR solution coincides with the solution of a modified version of OBR (M-OBR), which has the same objective but a simpler set of box constraints derived by using a “worst-case” scenario. Because M-OBR (FCR) has a smaller feasible set, preservation of linearity may be lost and accuracy may suffer for some grid configurations. Our numerical studies confirm this, and show that OBR delivers significant increases in robustness and accuracy. Preliminary efficiency studies of OBR reveal that it is only a factor of 2.1 slower than FCR, but admits 1.5 times larger time steps.     

Electric-field-controlled two-dimensional Raman-Nath diffraction from paraelectric potassium lithium tantalate niobate

Electric-field-controlled two-dimensional Raman-Nath diffraction has been realized using a photorefractive diffraction grating. The grating was produced by two-wave coupling (at a wavelength of 632.8 nm) at small incidence angles using a potassium lithium tantalate niobate single crystal. Results for the Raman-Nath diffraction from the g₄₄ grating are presented, in which the externally applied field is perpendicular to both the grating vector and the wave vector of the incident beam. Two pairs of coherent beams were used to record the grating for two-dimensional Raman-Nath diffraction. The wave vector and the polarization of one pair lay in the (x, z) plane, and those of the other pair lay in the (y, z) plane. The influence of the applied electric field was studied, and the results show that the intensity of the Raman-Nath diffraction could be controlled by the direction and intensity of the applied field.     

Dimer statistics on the Klein bottle

The problem of enumerating close-packed dimers, or perfect matchings, on a quadratic lattice embedded on the Klein bottle is considered. Thomassen [C. Thomassen, Tilings of the torus and the Klein Bottle and vertex-transitive graphs on a fixed surface, Trans. Amer. Math. Soc. 323(2) (1991) 605-635] characterized that there are six quadrilateral lattices embedded on the Klein bottle. Lu and Wu [W. T. Lu and F. Y. Wu, Close-packed dimers on nonorientable surfaces, Phys. Lett. A 293 (2002) 235-246] had obtained a expression for the number of close-packed dimers on one of them. In this paper we investigate four other embeddings and obtain explicit expressions of the numbers of close-packed dimers and free energy per dimer by enumerating Pfaffians.     

SUBGROUPS OF CLASS GROUPS OF ALGEBRAIC QUADRATIC FUNCTION FIELDS

Ideal class groups H(K) of algebraic quadratic function fields K are studied. Necessary and sufficient condition is given for the class group H(K) to contain a cyclic subgroup of any order n, which holds true for both real and imaginary fields K. Then several series of function fields K, including real, inertia imaginary, and ramified imaginary quadratic function fields, are given, for which the class groups H(K) are proved to contain cyclic subgroups of order n.     

Mixed boundary value problems of semilinear elliptic PDEs and BSDEs with singular coefficients

In this paper, we prove that there exists a unique weak (Sobolev) solution to the mixed boundary value problem for a general class of semilinear second order elliptic partial differential equations with singular coefficients. Our approach is probabilistic. The theory of Dirichlet forms and backward stochastic differential equations with singular coefficients and infinite horizon plays a crucial role.     

A new multicriteria approach for the analysis of efficiency in the Spanish olive oil sector by modelling decision maker preferences

The efficiency in production is often analysed as technical efficiency using the production frontier function. Efficiency scores are usually based on distance computations to the frontier in an m + s-dimensional space, where m inputs produce s outputs. In addition, efficiency improvements consider the total consumption of each input. However, in many cases, the “consumption” of each input can be divided into input-consumption sections (ICSs), and trade-off among the ICSs is possible. This share framework can be used for computing efficiency. This analysis provides information about both the total optimal consumption of each input, as does data envelopment analysis, and the most efficient allocation of the “consumption” among the ICSs. This paper studies technical efficiency using this approach and applies it to the olive oil sector in Andalusia (Spain). A non-parametrical methodology is presented, and an input-oriented Multi-Criteria Linear Programming model (MLP) is proposed. The analysis is developed at global, input and ICSs levels, defining the extent of satisfaction achieved at all these levels for each company, in accordance with their own preferences. The companies’ preferences are modelled with their utility function and their set of weights. MLP offers more detailed information to assist decision makers than other models previously proposed in the literature. In addition to this application, it is concluded that there is room for improvement in the olive oil sector, particularly in the management of the skilled labour. Additionally, the solutions with two opposite scenarios indicate that the model is suitable for the intended decision making process.     

Symmetry in RLT-type relaxations for the quadratic assignment and standard quadratic optimization problems

The reformulation–linearization technique (RLT), introduced in [Sherali, H. D., Adams. W. P. (1990). A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM Journal on Discrete Mathematics 3(3), 411–430], provides a way to compute a hierarchy of linear programming bounds on the optimal values of NP-hard combinatorial optimization problems. In this paper we show that, in the presence of suitable algebraic symmetry in the original problem data, it is sometimes possible to compute level two RLT bounds with additional linear matrix inequality constraints. As an illustration of our methodology, we compute the best-known bounds for certain graph partitioning problems on strongly regular graphs.