On the Complexity of Computing the Topology of Real Algebraic Space Curves

This paper presents an algorithm to compute the topology of an algebraic space curve.This is a modified version of the previous algorithm.Furthermore,the authors also analyse the bit complexity of the algorithm,which is O(N^(20)),where N=max{d,τ},d andτare the degree bound and the bit size bound of the coefficients of the defining polynomials of the algebraic space curve.To our knowledge,this is the best bound among the existing work.It gains the existing results at least N^(2).Meanwhile,the paper contains some contents of the conference papers(CASC 2014 and SNC 2014).

THE BESOV AND TRIEBEL-LIZORKIN SPACES WITH HIGH ORDER ON THE LIPSCHITZ CURVES

The Besov spaces B_p^(α,4)(Γ)and Triebel-Lizorkin spaces F_p^(α,4)(Γ)with high order x∈R on a Lipschitz curve Γ are defind,when 1≤p≤∞,1≤q≤∞.To compare to the classical case.a difference characterization of such spaces in the case|x|<1 is given also.

Conservation laws for energy and momentum in curved spaces

In arbitrary Riemannian 4-spaces, continuity equations are constructed which could be interpreted as conservation laws for the energy and momentum of the gravitational field. Special attention is given to general relativity to obtain, of natural manner, the pseudotensors of Einstein, Landau-Lifshitz, Moeller, Goldberg and Stachel, and also the conservation equations of Komar, Trautman, DuPlessis and Moss.

Equivalence of String Classical and Quantum Energy beside Equivalence of Wave Packet and Relativistic Velocity in Eucleadian and Curved Space

Plank quantum and classical string energy relations seem to be uncorrelated. This work correlated them. The relativistic energy-momentum relation has been used together with plank and de Brogglie hypothesis to prove that the wave group velocity is equal to the particle velocity in both ordinary and curved space. The plank energy relation is shown also to be related to the classical energy relation of an oscillating string. Starting from plank energy relation for n photons and performing integration, the expression of classical string energy was obtained. This means that one can treat electromagnetic waves as a collection of continuous photons having frequencies ranging from zero to w. Conversely, starting from classical string energy relation by differentiating it with respect to angular frequency, the plank quantum energy for n photons has been found. This means that the quanta results from separation of electromagnetic waves to single isolated waves. Each wave consists of n photons or quanta.

Tailoring chaotic motion of microcavity photons in ray and wave dynamics by tuning the curvature of space

Microcavity photon dynamics in curved space is an emerging interesting area at the crossing point of nanophotonics,chaotic science,and non-Euclidean geometry.We report the sharp difference between the regular and chaotic motions of cavity photons subjected to the varying space curvature.While the island modes of regular motion rise in the phase diagram in the curved space,the chaotic modes show special mechanisms to adapt to the space curvature,including the fast diffusion of ray dynamics,and the localization and hybridization of the Husimi wavepackets among different periodic orbits.These observations are unique effects enabled by the combination of the chaotic trajectory,the wave nature of light,and the non-Euclidean orbital motion,and therefore make the system a versatile optical simulator for chaotic science under quantum mechanics in curved space-time.

Modified Heisenberg Ferromagnet Model and Integrable Equation

We investigate some integrable modified Heisenberg ferromagnet models by using the prolongation structure theory. Through associating them with the motion of curve in Minkowski space, the corresponding coupled integrable equations are presented.

Forecasting the underground waters regime in the pitching aquifers in Huainan mining area

Huainan area is an important coal base of the east of China. In the early part of the 1980s, the study of the underground waters dynamic state in the area was gradually paid close attention to. This paper introduces the observation system of the groundwater dynamic state in the multilayered pitching aquifer, and expounds the hydrogeologic feature and the waterpower relations among aquifers. Furthermore, based on the analysis of the relations of the groundwater dynamic state to surface water, meteoric water and mining shaft outflow rate, this paper establishes main water filled aquifers of mining shaft (C 3-1 ,C 3-2 ,C 3-3 and O 2).In the light of the actual situation of the greatly changing aquifer occurrence and steep dip angle, the “two layer space curved surface seepage model" and the calculating step are all suggested. Since 1991,the groundwater dynamic state of the next year has been predicted (numerical simulation) every year. Contracting with the measured data, we gain a relatively ideal effect.

A RESEARCH ON URBAN RECREATIONISTS' TRAVELING BEHAVIORS IN SHANGHAI

Urban recreationists consist of those who have used urban recreational facilities, no matter they are local residents or nonlocals. Traveling models will exert impacts not only on the use of urban recreational facilities but also on the development and management of recreational resources. Taking Shanghai as a sample of destinaton and origin of urban recreationists, the author did his study based on data from comprehensive travel measurement and users' survey, including field questionnaire survey, mailing questionnaire survey, home survey, and cordon monitonng. To analyze traveing models of urban recreationists in Shanghai, the author used techniques of Recreational Activity Space (RAS) and the Use Curve (UC) Analysis. Rogarding Shanghai as an origin, the author first discussed types and characteristics of the RASs which are presented by isotourist lines in the proper and outskirts of Shanghai, the Yangtze Delta and the China's Mainland respectively. These RASs reflect the trave directional bias of Shanghainese. Then the author regarded Shanghai as a deshnahon and analyzed the RASs of tounsts from Otier provinces, which demonstrated the attractiveness of and distance decay of Shanghai. He found that a U-shaped UC model existed in the distance decay under the influence of attractiveness and distance. Finally, the author suggested, that the availability, periodicity and attractiveness, besides distance, are the basic mechanisms that affect the urban recreationists' traveling models in a city such as Shanghai.

Closed Timelike Curves in Type Ⅱ Non-Vacuum Spacetime

Here we present a cyclicly symmetric non-vacuum spacetime, admitting closed timelike curves(CTCs) which appear after a certain instant of time,i.e.,a time-machine spacetime. The spacetime is asymptotically flat, freefrom curvature singularities and a four-dimensional extension of the Misner space in curved spacetime. The spacetime is of type Ⅱ in the Petrov classification scheme and the matter field pure radiation satisfy the energy condition.

Families of hyperelliptic curves with maximal slopes

For each integer g≥2, we construct a family of hyperelliptic curves of genus g whose slope reaches the upper bound obtained by Xiao.

ON THE RECOVERY OF A CURVE ISOMETRICALLY IMMERSED IN E^n

It is known from classical differential geometry that one can reconstruct a curve with (n - 1) prescribed curvature functions, if these functions can be differentiated a certain number of times in the usual sense and if the first (n - 2) functions are strictly positive. It is established here that this result still holds under the assumption that the curvature functions belong to some Sobolev spaces, by using the notion of derivative in the distributional sense. It is also shown that the mapping which associates with such prescribed curvature functions the reconstructed curve is of class C∞.

A Knowledge Graph-Based Deep Learning Framework for Efficient Content Similarity Search of Sustainable Development Goals Data

Sustainable development denotes the enhancement ofliving standards in the present without compromising future generations'resources.Sustainable Development Goals(SDGs)quantify the accomplishment of sustainable development and pave the way for a world worth living in for future generations.Scholars can contribute to the achievement of the SDGs by guiding the actions of practitioners based on the analysis of SDG data,as intended by this work.We propose a framework of algorithms based on dimensionality reduction methods with the use of Hilbert Space Filling Curves(HSFCs)in order to semantically cluster new uncategorised SDG data and novel indicators,and efficiently place them in the environment of a distributed knowledge graph store.First,a framework of algorithms for insertion of new indicators and projection on the HSFC curve based on their transformer-based similarity assessment,for retrieval of indicators and loadbalancing along with an approach for data classification of entrant-indicators is described.Then,a thorough case study in a distributed knowledge graph environment experimentally evaluates our framework.The results are presented and discussed in light of theory along with the actual impact that can have for practitioners analysing SDG data,including intergovernmental organizations,government agencies and social welfare organizations.Our approach empowers SDG knowledge graphs for causal analysis,inference,and manifold interpretations of the societal implications of SDG-related actions,as data are accessed in reduced retrieval times.It facilitates quicker measurement of influence of users and communities on specific goals and serves for faster distributed knowledge matching,as semantic cohesion of data is preserved.

RATIONAL GENERAL SOLUTIONS OF HIGHER ORDER ALGEBRAIC ODES

This paper generalizes the method of Ng6 and Winkler （2010, 2011） for finding rational general solutions of a first order non-autonomous algebraic ordinary differential equation （AODE） to the case of a higher order AODE, provided a proper parametrization of its solution hypersurface. The authors reduce the problem of finding the rational general solution of a higher order AODE to finding the rational general solution of an associated system. The rational general solutions of the original AODE and its associated system are in computable 1-1 correspondence. The authors give necessary and sufficient conditions for the associated system to have a rational solution based on proper reparametrization of invariant algebraic space curves. The authors also relate invariant space curves to first integrals and characterize rationally solvable systems by rational first integrals.

IMPLICITIZATION USING UNIVARIATE RESULTANTS

Among several implicitization methods, the method based on resultant computation is a simple and direct one, but it often brings extraneous factors which are difficult to remove. This paper studies a class of rational space curves and rational surfaces by implicitization with univaxiate resultant computations. This method is more efficient than the other algorithms in finding implicit equations for this class of rational curves and surfaces.

Modular Invariants and Singularity Indices of Hyperelliptic Fibrations

The modular invariants of a family of curves are the degrees of the pullback of the corresponding divisors by the moduli map. The singularity indices were introduced by Xiao(1991) to classify singular fibers of hyperelliptic fibrations and to compute global invariants locally. In semistable case, the author shows that the modular invariants corresponding to the boundary divisor classes are just the singularity indices. As an application,the author shows that the formula of Xiao for relative Chern numbers is the same as that of Cornalba-Harris in semistable case.

On singular Lüroth quartics

Plane quartics containing the ten vertices of a complete pentalateral and limits of them are called Lüroth quartics.The locus of singular Lüroth quartics has two irreducible components,both of codimension two in P14.We compute the degree of them and discuss the consequences of this computation on the explicit form of the Lüroth invariant.One important tool is the Cremona hexahedral equations of the cubic surface.We also compute the class in M 3 of the closure of the locus of nonsingular Lüroth quartics.

A topological way of finding solutions to the Yang–Mills equation

We propose a systematic way of finding solutions to the classical Yang–Mills equation with nontrivial topology. This approach is based on one of the Wightman axioms for quantum field theory, which is referred to as the form invariance condition in this paper. For a given gauge group and a spacetime with certain isometries, thanks to this axiom that imposes strong constraints on the general ansatz, a systematic way of solving the Yang–Mills equation can be obtained in both flat and curved spacetimes. In order to demonstrate this method, we recover various known solutions as special cases, as well as producing new solutions not previously reported in the literature.

A genus-4 topological recursion relation for Gromov-Witten invariants

In this paper,we give a new genus-4 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds via Pixton’s relations on the moduli space of curves.As an application,we prove that Pixton’s relations imply a known topological recursion relation on Mg,1 for genus g≤4.