A pair of solar spike emissions

Using the 2.6-3.8 GHz solar radio spectrometer of the National Astronomical Observatories of China (NAOC), a pair of microwave millisecond spike (MMS) emissions were observed, and their frequency drift rate was measured. The separatrix frequency of the MMS pair was at 2900 MHz. Its emission layer was about 2×104km above the photosphere. The polarization degree was wave-like variation with an average value of about 25% in LCP. An MMS pair differs greatly from the type III bursts pair. For the latter, in a certain frequency range, there is no emission around separatrix frequency. This phenomenon may help better understand the mechanism of MMS.     

Integrable Geodesic Flows on Surfaces

We propose a new condition à{{\aleph}} which enables us to get new results on integrable geodesic flows on closed surfaces. This paper has two parts. In the first, we strengthen Kozlov’s theorem on non-integrability on surfaces of higher genus. In the second, we study integrable geodesic flows on a 2-torus. Our main result for a 2-torus describes the phase portraits of integrable flows. We prove that they are essentially standard outside what we call separatrix chains. The complement to the union of the separatrix chains is C ⁰-foliated by invariant sections of the bundle.     

Stability islands in domains of separatrix crossings in slow-fast Hamiltonian systems

We consider a two-degrees-of-freedom Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. The ratio of typical velocities of changes of the slow and fast variables is the small parameter ɛ of the problem. At frozen values of the slow variables, there is a separatrix on the phase plane of the fast variables, and there is a region in the phase space (the domain of separatrix crossings) where the projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under a certain symmetry condition, we prove the existence of many (of order 1/ɛ) stable periodic trajectories in the domain of separatrix crossings. Each of these trajectories is surrounded by a stability island whose measure is estimated from below by a value of order ɛ. So, the total measure of the stability islands is estimated from below by a value independent of ɛ. The proof is based on an analysis of asymptotic formulas for the corresponding Poincaré map.     

Chaotic motions in the resonant separatrix band of a parametrically excited pendulum

The energy increment spectrum method is developed for the numerical prediction of the appearance and disappearance of a primary (2M: 1)-librational or (M: 1)-rotational resonant separatrix band in a parametrically excited pendulum. The analytical conditions for the presence of such separatrix bands are also obtained. Illustrations of the analytical and numerical results for the appearance and destruction of the resonant bands are given for a comparison.     

Stability islands in domains of separatrix crossings in slow-fast Hamiltonian systems

We consider a two-degrees-of-freedom Hamiltonian system with one degree of freedom corresponding to fast motion and the other corresponding to slow motion. The ratio of typical velocities of changes of the slow and fast variables is the small parameter ɛ of the problem. At frozen values of the slow variables, there is a separatrix on the phase plane of the fast variables, and there is a region in the phase space (the domain of separatrix crossings) where the projections of phase points onto the plane of the fast variables repeatedly cross the separatrix in the process of evolution of the slow variables. Under a certain symmetry condition, we prove the existence of many (of order 1/ɛ) stable periodic trajectories in the domain of separatrix crossings. Each of these trajectories is surrounded by a stability island whose measure is estimated from below by a value of order ɛ. So, the total measure of the stability islands is estimated from below by a value independent of ɛ. The proof is based on an analysis of asymptotic formulas for the corresponding Poincaré map. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 243–255.     

Gradient flows with wildly embedded closures of separatrices

We show that for any n ≥ 4 there exists an n-dimensional closed manifold M ⁿ on which one can define a Morse-Smale gradient flow f ^t with two nodes and two saddles such that the closure of the separatrix of some saddle of f ^t is a wildly embedded sphere of codimension 2. We also prove that the closures of separatrices of a flow with three equilibrium points are always embedded in a locally flat way.     

A continuation theorem with applications to periodically forced Liénard equations in the presence of a separatrix

In this paper we prove a result on the existence of periodic motions for the periodically forced Liénard differential equation x ^″+f(x)x ^′+g(x)=e(t) in a situation where the phase portrait of the associated autonomous equation is similar to that of a centre limited by an unbounded separatrix. The existence result, which is based on a degree theoretic continuation theorem, enables us to treat some interesting cases not previously considered in the literature. Received: April 27, 2000 Published online: December 19, 2001     

Discrete Zγ: Embedded Circle Patterns with the Square Grid Combinatorics and Discrete Painlevé Equations

We study a discrete analogue of the holomorphic map z . It is given by Schramm's circle pattern with the square grid combinatorics. We show that the corresponding circle patterns are embedded and described by special separatrix solutions of discrete Painlevé equations. We establish global properties of these solutions and of the discrete z .     

Systems accounting for energy consumption and carbon emission by building

The method of systems accounting for overall energy consumption and carbon emission induced by a building is illustrated in terms of a combination of process and input–output analyses with a concrete procedure to cover various material, equipment, energy and manpower inputs. A detailed case study based on raw project data in the Bill of Quantities (BOQ) is performed for the structure engineering of the landmark buildings in E-town, Beijing (Beijing Economic–Technological Development Area). Based on the embodied energy and carbon emission intensity database for the Chinese economy in 2007, the energy consumption and the carbon emission of the structure engineering of the case buildings are quantified as 4.15E+14 J and 4.83E+04 t CO₂ Eq., corresponding to intensities of 6.91E+09 J/m² and 0.81 t CO₂ Eq./m² floor area. Steel and concrete contribute respectively about 50% and 30% of the energy consumption and the carbon emission, as a result of the reinforced-concrete structure of the case buildings. Materials contribute up to about 90% of the total energy consumption and carbon emission, in contrast to manpower, energy and equipment around 8%, 1% and 0.1%, respectively.     

Imbedded Circle Patterns with the Combinatorics of the Square Grid and Discrete Painlevé Equations

   Abstract. A discrete analogue of the holomorphic maps z ^γ and log(z) is studied. These maps are given by Schramm's circle pattern with the combinatorics of the square grid. It is shown that the corresponding circle patterns are imbedded and described by special separatrix solutions of discrete Painlevé equations. Global properties of these solutions, as well as of the discrete z ^γ and log(z) , are established.     

Passage to the limitp 2→P 1

A method is given allowing one to consider the well known formal passage to the limit P₂→P₁ as the double asymptotics of solutions of the equation P₂ in a special “transition” domain characterized by the ratio α²/x ³, where α is the parameter of P₂ and x is its argument. It is shown that the sequence of iterated Bäcklund transformations of generic solutions of P₂ is described by a generic solution of P₁. The iterations of Bäcklund transformations of solutions of P₂, both rational and separatrix for x→?∞, are studied. Bibliography: 12 titles.     

A theory for n-dimensional nonlinear dynamics on continuous vector fields

This paper systematically presents a theory for n-dimensional nonlinear dynamics on continuous vector fields. In this paper, a different view to look into the fundamental theory in dynamics is presented. The ideas presented herein are less formal and rigorous in an informal and lively manner. The ideas may give some inspirations in the field of nonlinear dynamics. The concepts of local and global flows are introduced to interpret the complexity of flows in nonlinear dynamic systems. Further, the global tangency and transversality of flows to the separatrix surface in nonlinear dynamical systems are discussed, and the corresponding necessary and sufficient conditions for such global tangency and transversality are presented. The ε-domains of flows in phase space are introduced from the first integral manifold surface. The domain of chaos in nonlinear dynamic systems is also defined, and such a domain is called a chaotic layer or band. The first integral quantity increment is introduced as an important quantity. Based on different reference surfaces, all possible expressions for the first integral quantity increment are given. The stability of equilibriums and periodic flows in nonlinear dynamical systems are discussed through the first integral quantity increment. Compared to the Lyapunov stability conditions, the weak stability conditions for equilibriums and periodic flows are developed. The criteria for resonances in the stochastic and resonant, chaotic layers are developed via the first integral quantity increment. To discuss the complexity of flows in nonlinear dynamical systems, the first integral manifold surface is used as a reference surface to develop the mapping structures of periodic and chaotic flows. The invariant set fragmentation caused by the grazing bifurcation is discussed. The global grazing bifurcation is a key to determine the global transversality to the separatrix. The local grazing bifurcation on the first integral manifold surface in a single domain without separatrix is a mechanism for the transition from one resonant periodic flow to another one. Such a transition may occur through chaos. The global grazing bifurcation on the separatrix surface may imply global chaos. The complexity of the global chaos is measured by invariant sets on the separatrix surface. The invariant set fragmentation of strange attractors on the separatrix surface is central to investigate the complexity of the global chaotic flows in nonlinear dynamical systems. Finally, the theory developed herein is applied to perturbed nonlinear Hamiltonian systems as an example. The global tangency and tranversality of the perturbed Hamiltonian are presented. The first integral quantity increment (or energy increment) for 2n-dimensional perturbed nonlinear Hamiltonian systems is developed. Such an energy increment is used to develop the iterative mapping relation for chaos and periodic motions in nonlinear Hamiltonian systems. Especially, the first integral quantity increment (or energy increment) for two-dimensional perturbed nonlinear Hamiltonian systems is derived, and from the energy increment, the Melnikov function is obtained under a certain perturbation approximation. Because of applying the perturbation approximation, the Melnikov function only can be used for a rough estimate of the energy increment. Such a function cannot be used to determine the global tangency and transversality to the separatrix surface. The global tangency and transversality to the separatrix surface only can be determined by the corresponding necessary and sufficient conditions rather than the first integral quantity increment. Using the first integral quantity increment, limit cycles in two-dimensional nonlinear systems is discussed briefly. The first integral quantity of any n-dimensional nonlinear dynamical system is very crucial to investigate the corresponding nonlinear dynamics. The theory presented in this paper needs to be further developed and to be treated more rigorously in mathematics.     

Influence of model management systems on decision making: empirical evidence and implications

Model management (MM) regards decision models as an important organisational resource deserving prudent management. Despite the remarkable volume of model management literature compiled over the past twenty-odd years, very little is known about how decision makers actually benefit from employing model management systems (MMS). In this paper, we report findings from an experiment designed to verify the idea that the adequacy of modeling support provided by a MMS influences the decision maker's problem solving performance and behaviour. We show that the decision makers who receive adequate modelling support from MMS outperform those without such support. Also, we provide empirical evidence that the MMS help turn the decision makers' perception of problem solving from a number crunching task into development of solution strategies, consequently changing their decision making behaviour.     

The near ultraviolet emission spectrum of tetralin (tetrahydronaphthalene)

An emission spectrum of tetralin (tetrahydronaphthalene) in the near ultraviolet excited by a transformer discharge through flowing vapour has been studied and about twenty-nine bands in emission have been photographed and measured on suitable instruments. Vibrational analysis of the bands in terms of the fundamental frequencies of the molecule is proposed and is compared with those obtained in the corresponding absorption spectrum of tetrahydronaphthalene studied by the author.² The results afford further confirmation of the correctness of the analysis and particularly the assignment of the (0,0) band proposed in the absorption spectrum of the molecule.     

A duality principle for groups

The duality principle for Gabor frames states that a Gabor sequence obtained by a time-frequency lattice is a frame for L²(Rd) if and only if the associated adjoint Gabor sequence is a Riesz sequence. We prove that this duality principle extends to any dual pairs of projective unitary representations of countable groups. We examine the existence problem of dual pairs and establish some connection with classification problems for II₁ factors. While in general such a pair may not exist for some groups, we show that such a dual pair always exists for every subrepresentation of the left regular unitary representation when G is an abelian infinite countable group or an amenable ICC group. For free groups with finitely many generators, the existence problem of such a dual pair is equivalent to the well-known problem about the classification of free group von Neumann algebras.     

Electron spin resonance of Mn2+ in NaCl single crystal and lattice defects

The electron spin resonance spectra of Mn²⁺ in NaCl single crystals are investigated in detail. Seven different spectra (I, II, III₁, III₂, IV, V and VI) are observed. The spectra I–IV are the same as those observed by earlier workers, while the spectra V and VI are observed here for the first time. The local symmetry at the paramagnetic ion is orthorhombic for the spectra IV and V and tetragonal for the spectrum VI about the crystallographic [001] direction. The properties of the spectrum IV are explained in terms of an associated pair Mn²⁺: O₂ ^2? with O₂ ^2? molecular ion at the nearest neighbour anion site in the [001] direction with its internuclear axis in the (001) plane. The spectrum V is assigned to the associated pair Mn²⁺: O₂ ^2? coupled with a nearby cation vacancy in the [001] direction and the spectrum VI to Mn²⁺ ion associated with OH^? ion at the anion site in the [001] direction with a probable second neighbour cation vacancy. All the observed spectra are analysed in terms of the parameters of the usual spin-Hamiltonian.     

Multiple separatrix crossing in multi-degree-of-freedom Hamiltonian flows

Summary We study separatrix crossing in near-integrablek-degree-of-freedom Hamiltonian flows, 2 <k < , whose unperturbed phase portraits contain separatrices inn degrees of freedom, 1 <n <k. Each of the unperturbed separatrices can be recast as a codimension-one separatrix in the 2k-dimensional phase space, and the collection of these separatrices takes on a variety of geometrical possibilities in the reduced representation of a Poincaré section on the energy surface. In general 0 l n of the separatrices will be available to the Poincaré section, and each separatrix may be completely isolated from all other separatrices or intersect transversely with one or more of the other available separatrices. For completely isolated separatrices, transitions across broken separatrices are described for each separatrix by the single-separatrix crossing theory of Wiggins, as modified by Beigie. For intersecting separatrices, a possible violation of a normal hyperbolicity condition complicates the analysis by preventing the use of a persistence and smoothness theory for compact normally hyperbolic invariant manifolds and their local stable and unstable manifolds. For certain classes of multi-degree-of-freedom flows, however, a local persistence and smoothness result is straightforward, and we study the global implications of such a local result. In particular, we find codimension-one partial barriers and turnstile boundaries associated with each partially destroyed separatrix. From the collection of partial barriers and turnstiles follows a rich phase space partitioning and transport formalism to describe the dynamics amongst the various degrees of freedom. A generalization of Wiggins' higher-dimensional Melnikov theory to codimension-one surfaces in the multi-separatrix case allows one to uncover invariant manifold geometry. In the context of this perturbative analysis and detailed numerical computations, we study invariant manifold geometry, phase space partitioning, and phase space transport, with particular attention payed to the role of a vanishing frequency in the limit approaching the intersection of the partially destroyed separatrices. The class of flows under consideration includes flows of basic physical relevance, such as those describing scattering phenomena. The analysis is illustrated in the context of a detailed study of a 3-degree-of-freedom scattering problem.     

Boundary layers for cellular flows at high Péclet numbers

We analyze the behavior of solutions of steady advection‐diffusion problems in bounded domains with prescribed Dirichlet data when the Péclet number Pe ? 1 is large. We show that the solution converges to a constant in each flow cell outside a boundary layer of width O(?^1/2), ? = Pe^?1, around the flow separatrices. We construct an ?‐dependent approximate “water pipe problem” purely inside the boundary layer that provides a good approximation of the solution of the full problem but has ?‐independent computational cost. We also define an asymptotic problem on the graph of streamline separatrices and show that solution of the water pipe problem itself may be approximated by an asymptotic, ?‐independent problem on this graph. Finally, we show that the Dirichlet‐to‐Neumann map of the water pipe problem approximates the Dirichlet‐to‐Neumann map of the separatrix problem with an error independent of the flow outside the boundary layers. © 2004 Wiley Periodicals, Inc.     

A new full-Newton step <Emphasis Type="Italic">O</Emphasis>(<Emphasis Type="Italic">n</Emphasis>) infeasible interior-point algorithm for semidefinite optimization

Interior-point methods for semidefinite optimization have been studied intensively, due to their polynomial complexity and practical efficiency. Recently, the second author designed a primal-dual infeasible interior-point algorithm with the currently best iteration bound for linear optimization problems. Since the algorithm uses only full Newton steps, it has the advantage that no line-searches are needed. In this paper we extend the algorithm to semidefinite optimization. The algorithm constructs strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem, close to their central paths. Two types of full-Newton steps are used, feasibility steps and (ordinary) centering steps, respectively. The algorithm starts from strictly feasible iterates of a perturbed pair, on its central path, and feasibility steps find strictly feasible iterates for the next perturbed pair. By using centering steps for the new perturbed pair, we obtain strictly feasible iterates close enough to the central path of the new perturbed pair. The starting point depends on a positive number ζ. The algorithm terminates either by finding an ε-solution or by detecting that the primal-dual problem pair has no optimal solution (X ^*,y ^*,S ^*) with vanishing duality gap such that the eigenvalues of X ^* and S ^* do not exceed ζ. The iteration bound coincides with the currently best iteration bound for semidefinite optimization problems.