Existence Of Stationary Points Research Articles

Ob ogranichennykh traektoriyakh avtonomnoy sistemy s vydelennoy polozhitel'no odnorodnoy nelineynost'yu

Bounded trajectories of an autonomous system with an isolated positively homogeneous nonlinearity that is the gradient of a smooth function are studied. We prove the existence of nonstationary bounded trajectories lying in connected components of the set of points where the positively homogeneous function is negative and nonzero stationary points in those connected components whose closure has nonzero Euler characteristic. The existence of nonstationary bounded trajectories is substantiated using the Waűewski method; and the existence of stationary points, using methods for calculating the winding number of finite-dimensional vector fields.

Evolution equation for elastic scattering of hadrons

We turn high energy elastic scattering of hadrons into an initial value problem using an evolution equation based on the Regge Field Theory, which has a form of the complex nonlinear reaction–diffusion equation, with time being played by the logarithm of energy. The initial conditions are provided by the data-driven models for the real and imaginary parts of the amplitude. Numerical calculations of pp differential cross sections and forward quantities for LHC energies agree very well with experimental data extending up to, and including the diffractive cone. Furthermore, we show that at current accessible energies the non-linear effects play an important role, as the impact parameter space profiles approach the unitarity bound. The equation also predicts some other effects discussed in the literature, like the hollowness of nuclear matter or existence of stationary points in momentum transfer t.

Combined effect of small perturbations in the Coriolis and centrifugal forces and three-body interaction on the existence of stationary points in the R3BP

In the present manuscript, the effects of small perturbations in the centrifugal and Coriolis forces with additional effect of three-body interaction on the locations as well as stability of the stationary points (SPs), and on the topology of the basins of convergence (BoCs) linked with these SPs in the restricted three-body problem (R3BP) are discussed by using the multivariate version of the Newton-Raphson iterative scheme. It is found that the positions of the SPs are highly influenced by the centrifugal force but their stability is highly influenced by combinations of the centrifugal and Coriolis forces. In addition, the correlations betwixt the basin of convergence (BoC) connected to the SPs and the associated number of iterations needed to assure pre-assumed precision are analyzed. Our analysis suggests that the mass ratio μ, three-body interaction parameter k and the effect in centrifugal and the Coriolis forces controlled by parameters ϕ and φ are very influential parameters.

Stationary points of lower semicontinuous multifunctions

We guarantee the existence of stationary points for semicompact lower semicontinuous multifunctions in metric spaces. We also show that the semicompactness condition can be replaced by some other properties. Several examples illustrating the main results are also given.

Photoisomerization dynamics and pathways of trans- and cis-azobenzene in solution from broadband femtosecond spectroscopies and calculations.

The photoisomerization of azobenzene in solution was studied experimentally and by calculations. trans-to-cis and cis-to-trans dynamics are described through broadband transient absorption, fluorescence, and stimulated Raman spectroscopy. Transient absorption was extended to cover not only the nπ* band but also the ππ* band in the ultraviolet. Isomerization yields are used for a quantitative comparison of trans and cis transient spectra under different excitation. For the trans-to-cis path upon nπ*(S(1)) excitation, the evolution develops with 0.3, 3, and 16 ps. The first two times reflect population relaxation to a local minimum S(1t )(L) and subsequent transition to a dark intermediate S(1t)(D) over an 8 kJ/mol barrier. The existence of stationary points S(1t)(L) and S(1t)(D) is confirmed by quantum-chemical calculations. The third time corresponds to S(1t) (D) → S0 relaxation to the ground state via an S1/S0 conical intersection over a 12 kJ/mol barrier. Thus, the 16 ps time constant is attributed to the isomerization process and not to vibrational cooling, contrary to the current view and in line with the previous interpretation by Lednev et al. (J. Phys. Chem. 1996, 100, 13338). The decay of the long-lived intermediate S(1t)(D) is consistent with the hula twist rather than with the inversion mechanism. For the cis-totrans reaction following nπ* excitation, signal decay is strongly nonexponential, with 0.1 and 1 ps. The latter (1 ps) is much shorter than the 16 ps decay of the trans isomer, implying different S1/S0 conical intersections and relaxation paths for the cis-totrans and trans-to-cis reaction. New results are also obtained with ππ*(Sn) excitation. Thus, for trans-azobenzene, 50% of the population relaxes to an S1 region, which is not accessible under nπ* excitation. For cis-azobenzene, up to 30% of the excited species isomerize to trans via an Sn/S1 intersection, resulting in a mixed cis/trans S1 population. The isomerization kinetics of azobenzene shows no viscosity dependence, putting into question the torsion mechanism and suggesting the hula-twist isomerization mechanism.

Search for scalar–tensor gravity theories with a non-monotonic time evolution of the speed-up factor

We present a method to detect, in the framework of scalar–tensor gravity theories, the existence of stationary points in the time evolution of the speed-up factor. An attractive aspect of this method is that, once the particular scalar–tensor theory has been specified, the stationary points are found through a simple algebraic equation which does not contain any integration. By applying this method to the three classes of scalar–tensor theories defined by Barrow and Parsons, we have found several new cosmological models with a non-monotonic evolution of the speed-up factor. The physical interest of these models is that, as previously shown by Serna and Alimi, they predict the observed primordial abundance of light elements for a very wide range of baryon density. These models are then consistent with recent CMB and Lyman-α estimates of the baryon content of the universe.

Augmented Lagrange-SQP Methods with Lipschitz-Continuous Lagrange Multiplier Updates

Augmented Lagrangian-SQP methods using Lipschitz-continuous Lagrange multiplier updates are analyzed. Kantorovich-style convergence results are proved and applied to the discretization of optimal control problems. The existence of stationary points for the discretized problems is also discussed.

Procreation, Fishing, and Hunting: Renewable Resources and Dynamic Planar Systems

AbstractMany problems in the economics of renewable resources can be put in the form of a dynamic planar system. This paper provides analytic and graphical methods for the study of the global dynamics, existence of stationary points, and local stability properties of such systems, and applies these methods to problems of (1) population and environmental quality; (2) exploitation of a stock of fish; and (3) the economics of the pleistocene extinctions and the origins of agriculture.

Variable Dimension Complexes Part I: Basic Theory

Over the past several years, much attention in the field of simplicial pivoting algorithms has been focused on the new class of variable-dimension algorithms, wherein the sequence of simplices generated varies in dimension. This study is aimed at the combinatorial nature of variable-dimension algorithms. In Part I, we introduce the notion of a V-complex, short for variable-dimension complex. We show that when a labelling function is introduced on a V-complex, certain path-following properties arise. The main result of Part I is a characterization of paths on a labelled V-complex. Part II of this study uses the path-following theory of labelled V-complexes developed in Part I to provide constructive algorithmic proofs of a variety of combinatorial lemmas in topology. We demonstrate two new dual lemmas on the n-dimensional cube, and use a generalized Sperner lemma to prove a generalization of the Knaster–Kuratowski–Mazurkiewicz covering lemma on the simplex. We also show that Tucker's lemma can be derived directly from the Borsuk–Ulam theorem. We report the interrelationships between these results, Brouwer's fixed point theorem, and the existence of stationary points on the simplex.

Variable Dimension Complexes Part II: A Unified Approach to Some Combinatorial Lemmas in Topology

Part II of this study uses the path-following theory of labelled V-complexes developed in Part I to provide constructive algorithmic proofs of a variety of combinatorial lemmas in topology. We demonstrate two new dual lemmas on the n-dimensional cube, and use a generalized Sperner lemma to prove a generalization of the Knaster–Kuratowski–Mazurkiewicz covering lemma on the simplex. We also show that Tucker's lemma can be derived directly from the Borsuk–Ulam theorem. We report the interrelationships between these results, Brouwer's fixed point theorem, and the existence of stationary points on the simplex.

On certain singular solutions of the partial differential equation u x 2 uxx+2uxuyuxy+u y 2 uyy=0

The partial differential equation treated here is the formal limit of the p-harmonic equation in R2, for p→∞. Questions related to the smoothness of solutions and the possible existence of stationary points are central for the theory. The “singular” solutions constructed here bring new light on these questions.

Magnetic mirror system with minimum-B field

The question of the existence of stationary points of the square of the magnetic field intensity is discussed and the properties of the magnetic trap suggested by Andreoletti(1) are investigated.