On generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables

ABSTRACT

We define and discuss different enumerative methods to compute solutions of generalized Nash equilibrium problems with linear coupling constraints and mixed-integer variables. We propose both branch-and-bound methods based on merit functions for the mixed-integer game, and branch-and-prune methods that exploit the concept of dominance to make effective cuts. We show that under mild assumptions the equilibrium set of the game is finite and we define an enumerative method to compute the whole of it. We show that our branch-and-prune method can be suitably modified in order to make a general equilibrium selection over the solution set of the mixed-integer game. We define an application in economics that can be modelled as a Nash game with linear coupling constraints and mixed-integer variables, and we adapt the branch-and-prune method to efficiently solve it.     

Reducing the computational cost of NMR crystallography of organic powders at natural isotopic abundance with the help of 13C-13C dipolar couplings

Structure determination of functional organic compounds remains a formidable challenge when the sample exists as a powder. Nuclear magnetic resonance crystallography approaches based on the comparison of experimental and Density Functional Theory (DFT)-computed ¹H chemical shifts have already demonstrated great potential for structure determination of organic powders, but limitations still persist. In this study, we discuss the possibility of using ¹³C-¹³C dipolar couplings quantified on powdered theophylline at natural isotopic abundance with the help of dynamic nuclear polarization, to realize a DFT-free, rapid screening of a pool of structures predicted by ab initio random structure search. We show that although ¹³C-¹³C dipolar couplings can identify structures possessing long range structural motifs and unit cell parameters close to those of the true structure, it must be complemented with other data to recover information about the presence and the chemical nature of the supramolecular interactions.     

Local boundedness of weak solutions to the Diffusive Wave Approximation of the Shallow Water equations

In this paper we prove that weak solutions to the Diffusive Wave Approximation of the Shallow Water equations

${?}_{t}u?\mathrm{?}?({(u?z)}^{\alpha }|\mathrm{?}u{|}^{\gamma ?1}\mathrm{?}u)=f$

are locally bounded. Here, u describes the height of the water, z is a given function that represents the land elevation and f is a source term accounting for evaporation, infiltration or rainfall.     

A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as

$?F(x,u,Du,D^{2}u)=\lambda c(x)u+〈M(x)Du,Du〉+h(x)$

in a bounded domain with a Dirichlet boundary condition; here $\lambda \in \mathbb{R}$, $c,h\in L^p(\mathrm{\Omega })$, $p>n\ge 1$, $c?0$ and the matrix M satisfies $0<{\mu }_{1}I\le M\le {\mu }_{2}I$. Recently this problem was studied in the “coercive” case $\lambda c\le 0$, where uniqueness of solutions can be expected; and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, by exploiting the integral formulation of the problem. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use different techniques based on the maximum principle.We develop a new method to obtain the crucial uniform a priori bounds, which permit to us to use degree theory. This method is based on basic regularity estimates such as half-Harnack inequalities, and on a Vázquez type strong maximum principle for our kind of equations.     

Existence,nonexistence and multiplicity results for nonlocal Dirichlet problems

In this paper we establish various existence, nonexistence and multiplicity results for fully nonlinear Dirichlet problems associated to nonlocal Hamilton–Jacobi equations. This study is accomplished by a careful analysis of the principal eigenvalues of the elliptic operator. Resonance phenomena and anti maximum principles are also established.     

The regularity of the multiple higher-order poles solitons of the NLS equation

Based on the inverse scattering method, the formulae of one higher-order pole solitons and multiple higher-order poles solitons of the nonlinear Schrödinger equation (NLS) equation are obtained. Their denominators are expressed as , where is a matrix frequently constructed for solving the Riemann-Hilbert problem, and the asterisk denotes complex conjugate. We take two methods for proving is invertible. The first one shows matrix is equivalent to a self-adjoint Hankel matrix , proving . The second one considers the block-matrix form of , proving . In addition, we prove that the dimension of is equivalent to the sum of the orders of pole points of the transmission coefficient and its diagonal entries compose a set of basis.     

Remarkable Size Effect on Photophysical and Nonlinear Optical Properties of All-Carboatomic Rings,Cyclo[18]carbon and Its Analogues

Inspired by recent experimental observation of molecular morphology and theoretical predictions of multiple properties of cyclo[18]carbon, we systematically studied the photophysical and nonlinear optical properties of cyclo[2N]carbons (N=3–15) allotropes through density functional theory. This work unveils the unusual optical properties of the sp-hybridized carbon rings with different sizes. The remarkable size dependence of the optical properties of these systems and their underlying nature are profoundly explored, and the relevance between aromaticity and optical properties are highlighted. The extrapolation curves fitted for energy level of frontier molecular orbitals, maximum absorption wavelength, and (hyper)polarizability of considered carbon rings are presented, which can be used to reliably predict corresponding properties for arbitrarily large carbon rings. The findings in this study will facilitate the exploration of potential application of cyclocarbons in the field of optical materials.     

Heterobimetallic Cluster-Based Coordination Polymers: Assembly,Structures and Third-Order Nonlinear Optical Properties

Reactions of (NH₄)₂WS₄ with CuCN, CuCN/1,2-bis(4-pyridyl)propane (bppa) or [Cu(MeCN)₄]PF₆/bppa under different reaction conditions afforded a set of two- or three-dimensional W/Cu/S cluster-based coordination polymers including {[Et₄N]₂[WS₄Cu₄(μ-CN)₂(μ-I)₂]}_n ( 1 ), [WS₄Cu₄(μ-CN)₂(bppa)₂]_n ( 2 ) and {[WS₄Cu₄(bppa)₄](PF₆)₂}_n ( 3 ), respectively. Compound 2 can be readily formed from reaction of 1 with bppa under solvothermal conditions. Compounds 1 and 2 feature two-dimensional networks with a “sql” topology, while 3 possesses a two-fold interpenetrated three-dimensional net with a rare “reo” topology. Compounds 1 – 3 in DMF exhibited different third-order nonlinear optical responses, and they all showed a reverse saturable absorption while 2 held a strong self-focusing effect.