OLD AND RECENT RESULTS IN THE ANALYTIC THEORY OF DIRICHLET SERIES:A SURVEY

This survey is dedicated to the memory of Professor Jiarong Yu,who recently passed away.It is concerned by a topic of which he was fond,an interest shared by myself:the analytic theory of Dirichlet series.

Analytical Theory of the Geodesic Acoustic Mode in the Small and Large Orbit Drift Width Limits and its Application in a Study of Plasma Shaping Effect

Analytical theories of the geodesic acoustic mode （GAM） are reviewed in the small- and large-orbit drift width limits, respectively. Different physics pictures in these two limits are displayed. As an example, these two analytical methods are employed to investigate the plasma shaping effect on the frequency and collisionless damping rate of the GAM.

THE INTERIOR TRANSMISSION EIGENVALUE PROBLEM FOR AN ANISOTROPIC MEDIUM BY A PARTIALLY COATED BOUNDARY

We consider the interior transmission eigenvalue problem corresponding to the scattering for an anisotropic medium of the scalar Helmholtz equation in the case where the boundary?Ωis split into two disjoint parts and possesses different transmission conditions.Using the variational method,we obtain the well posedness of the interior transmission problem,which plays an important role in the proof of the discreteness of eigenvalues.Then we achieve the existence of an infinite discrete set of transmission eigenvalues provided that n≡1,where a fourth order differential operator is applied.In the case of n■1,we show the discreteness of the transmission eigenvalues under restrictive assumptions by the analytic Fredholm theory and the T-coercive method.

The General Theory of Law and Its Development in China

Study of the general theory of law,as an independent branch of legal research,originated in Germany in the mid to late 19th century,and thereafter became a widely propagated and well-developed subject in countries such as the UK and the US.Scholars from the Soviet Union combined the general theory of law with Marxist philosophy and adapted it from an analytical legal theory to a social legal theory.The inheritance and development of the general theory of law in China went through three stages.Specifically,from the 1950s to the early 1960s,the jurisprudence community fully adopted the legal theories promoted by the Soviet Union;from the early 1980s to the late 1990s,the studies reflected upon past theories and sought advancement;and from the late 1990s to present,research has adopted a more open and innovative approach.In addition,the philosophy of law and other interdisciplinary subjects have gradually become dominant research paradigms.A review and compilation based on the evolution of theories of legal relations showed that the study of the general theory of law in China is closely associated to the historical rate of progress of the practice of rule of law and jurisprudence in China.The overall framework of the research is deeply influenced by the model of“theories of the Soviet+civil-law prototypes”.In addition,the research methods have evolved from investigations utilizing a single approach to processes combining various approaches.The general theory of law is far from a“relic of history,”and corresponding in-depth research is recommended for future jurisprudence study in China.

Study of the Electrical Double Layer of a Spherical Micelle: Functional Theoretical Approach

By using the iterative method in functional theory, an analytic expression of the Poisson-Boltzmann equation (PB eq.), which describes the distribution of the potential of electrical double layer of a spherical micelle, has been carried out under the general potential condition for the first time. The method also can give the radius, the surface potential, and the thickness of the layer.

Global Meromorphic Solutions of Partial Differential Equations

In this paper, we survey some recent progress on analytic theory of partial differential equations.

Kinematic Solution of Spherical Stephenson-Ⅲ Six-bar Mechanism

A closed-form solution can be obtained for kinematic analysis of spatial mechanisms by using analytical method.However,extra solutions would occur when solving the constraint equations of mechanism kinematics unless the constraint equations are established with a proper method and the solving approach is appropriate.In order to obtain a kinematic solution of the spherical Stephenson-III six-bar mechanism,spherical analytical theory is employed to construct the constraint equations.Firstly,the mechanism is divided into a four-bar loop and a two-bar unit.On the basis of the decomposition,vectors of the mechanism nodes are derived according to spherical analytical theory and the principle of coordinate transformation.Secondly,the structural constraint equations are constructed by applying cosine formula of spherical triangles to the top platform of the mechanism.Thirdly,the constraint equations are solved by using Bezout’ s elimination method for forward analysis and Sylvester’ s resultant elimination method for inverse kinematics respectively.By the aid of computer symbolic systems,Mathematica and Maple,symbolic closed-form solution of forward and inverse displacement analysis of spherical Stephenson-III six-bar mechanism are obtained.Finally,numerical examples of forward and inverse analysis are presented to illustrate the proposed approach.The results indicate that the constraint equations established with the proposed method are much simpler than those reported by previous literature,and can be readily eliminated and solved.

Water hammer prediction and control:the Green's function method

By Green＇s function method we show that the water hammer （WH） can be analytically predicted for both laminar and turbulent flows （for the latter, with an eddy vis- cosity depending solely on the space coordinates）, and thus its hazardous effect can be rationally controlled and mini- mized. To this end, we generalize a laminar water hammer equation of Wang et al. （J. Hydrodynamics, B2, 51, 1995） to include arbitrary initial condition and variable viscosity, and obtain its solution by Green＇s function method. The pre- dicted characteristic WH behaviors by the solutions are in excellent agreement with both direct numerical simulation of the original governing equations and, by adjusting the eddy viscosity coefficient, experimentally measured turbulent flow data. Optimal WH control principle is thereby constructed and demonstrated.

Analytical derivative techniques for TDDFT excited-state properties:Theory and application

We review our recent work on the methodology development of the excited-state properties for the molecules in vacuum and liquid solution.The general algorithms of analytical energy derivatives for the specific properties such as the first and second geometrical derivatives and IR/Raman intensities are demonstrated in the framework of the time-dependent density functional theory(TDDFT).The performance of the analytical approaches on the calculation of excited-state energy Hessian has also been shown.It is found that the analytical approaches are superior to the finite-difference method on the computational accuracy and efficiency.The computational cost for a TDDFT excited-state Hessian calculation is only 2–3 times as that for the DFT ground-state Hessian calculation.With the low computational complexity of the developed analytical approaches,it becomes feasible to realize the large-scale numerical calculations on the excited-state vibrational frequencies,vibrational spectroscopies and the electronic-structure parameters which enter the spectrum calculations of electronic absorption and emission,and resonance Raman spectroscopies for medium-to large-sized systems.

The theoretical studies on the two-photon absorption properties of symmetric l,4-bis{2-[4-(2-aryl)phenyl]vinyl}-2,5-bisdialkoxybenzenes

On the level of the time-dependent hybrid density functional theory, the one- and two-photon absorption properties of a series of symmetric 4-bis{2-[4-（2-aryl） phenyl]vinyl）-2,5-bisdialkoxybenzenes are studied respectively utilizing the analytic response theory and the few-state model methods. The calculated results show that the planarity of the geometrical structure plays a great role in enhancing the linear and nonlinear optical abilities of the molecule. However the effect of the length of the chain linked to the π-centre on the optical property is very little. For the investigated compounds, the A-π-A type charge-transfer molecules display more superior one- and two-photon absorption characteristics than the D-π-D type ones. Furthermore, the two-photon absorption results by use of few-state model are generally consistent with those by analytic response theory, demonstrating the reliability of the few-state model for evaluating the two-photon absorption cross section. The numerical simulations are in good agreement in tendency with the available experimental measurements.

The aggregation effects on the two-photon absorption properties of para-nitroaniline polymers by hydrogen-bond interactions

This paper has theoretically designed a series of aggregate polymers on the basis of several para-nitroaniline monomers by hydrogen-bond interactions. At the level of time-dependent hybrid density functional theory, it has optimized their geometrical structures and studied their two-photon absorption （TPA） properties by using analytical response theory. The calculated results exhibit that the aggregation effects not only bring out the conaiderable red shift of the excited energies but also greatly enhance the TPA intensities of the aggregate polymers in comparison with the para-nitroaniline monomer. The aggregate configurations also have an important influence on the TPA abilities of the polymers; the trimer has the largest TPA cross section. The electron transitions between the molecular orbits involving the strong TPA excitations of the trimer are depicted to illuminate the relationship between the intermolecular charge transfer and the TPA property.

One- and two-photon absorption properties of two metalloporphyrin complexes

The linear and nonlinear optical properties of two metalloporphyrin complexes formed by the complementary coordination of central zinc or magnesium ions to the ligand 5, 10, 15-tri-（p-tolyl）-20-phenylethynylporphyrin are theo- retically investigated by using the analytic response theory at the density functional theory level. The results indicate that the studied complexes present more symmetric geometry structures than the ligand. The charge-transfer states of the two complexes in the lower energy region are all almost degenerate but those of the ligand are well separated. The ratio of the two-photon absorption cross sections of the ligand, zinc-porphyrin and magnesium-porphyrin com- plexes is 1.0：1.5：1.8, demonstrating that the two-photon absorption capability can be greatly increased when the ligand is coordinated with a metal ion. Moreover, several physical micro-mechanisms including electron transitions and in- tramolecular charge-transfer processes are discussed to explore the differences in optical property between the ligand and two complexes.

New applications of a generalized Hooke's law for second gradient materials

We provide analytical solutions to the problems of a circular bending of a beam in plane strain and the torsion of a non-circular cross-section beam, the beams obeying a second-gradient elasticity law proposed by the author, following a previous suggestion of delrlsola et al. （2009）. The motivation was to find benchmark analytical solutions that can serve to grasp the physical foundations of second gradient elasticity laws for heterogeneous materials. The analytical solution of the circular beam problem presents the additional advantage to establish some nice properties on the unknown second gradient elastic moduli introduced by Enakoutsa （2014） model and the classical elasticity constants for both incompressible and compressible heterogeneous elastic materials. A framework to find the elastic moduli of the new model is also proposed.

A class of transient acceleration models consistent with Big Bang cosmology

Is it possible that the current cosmic accelerating expansion will turn into a decelerating one？ Can this transition be realized by some viable theoretical model that is consistent with the standard Big Bang cosmology？ We study a class of phenomeno- logical models with a transient acceleration, based on a dynamical dark energy with a very general form of equation of state Pde = αPde -- βpdem. It mimics the cosmolog- ical constant αde → const for a small scale factor a, and behaves as a barotropic gas with pde → α-3（α＋1） with α 〉 0 for large a. The cosmic evolution of four models in the class has been examined in detail, and all yield a smooth transient acceleration. Depending on the specific model, the future universe may be dominated by either dark energy or by matter. In two models, the dynamical dark energy can be explicitly real- ized by a scalar field with an analytical potential V（O）. Moreover, a statistical analysis shows that the models can be as robust as ACDM in confronting the observational data of Type Ia supernovae, cosmic microwave background （CMB） and baryon acoustic os- cillation. As improvements over previous studies, our models overcome the problem of over-abundance of dark energy during early eras, and satisfy the constraints on dark energy from WMAP observations of CMB.

The Prime Sequence: Demonstrably Highly Organized While Also Opaque and Incomputable-With Remarks on Riemann’s Hypothesis, Partition, Goldbach’s Conjecture, Euclid on Primes, Euclid’s Fifth Postulate, Wilson’s Theorem along with Lagrange’s Proof of It and Pascal’s Triangle, and Rational Human Intelligence

The main design of this paper is to determine once and for all the true nature and status of the sequence of the prime numbers, or primes—that is, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and so on. The main conclusion revolves entirely around two points. First, on the one hand, it is shown that the prime sequence exhibits an extremely high level of organization. But second, on the other hand, it is also shown that the clearly detectable organization of the primes is ultimately beyond human comprehension. This conclusion runs radically counter and opposite—in regard to both points—to what may well be the default view held widely, if not universally, in current theoretical mathematics about the prime sequence, namely the following. First, on the one hand, the prime sequence is deemed by all appearance to be entirely random, not organized at all. Second, on the other hand, all hope has not been abandoned that the sequence may perhaps at some point be grasped by human cognition, even if no progress at all has been made in this regard. Current mathematical research seems to be entirely predicated on keeping this hope alive. In the present paper, it is proposed that there is no reason to hope, as it were. According to this point of view, theoretical mathematics needs to take a drastic 180-degree turn. The manner of demonstration that will be used is direct and empirical. Two key observations are adduced showing, 1), how the prime sequence is highly organized and, 2), how this organization transcends human intelligence because it plays out in the dimension of infinity and in relation to π. The present paper is part of a larger project whose design it is to present a complete and final mathematical and physical theory of rational human intelligence. Nothing seems more self-evident than that rational human intelligence is subject to absolute limitations. The brain is a material and physically finite tool. Everyone will therefore readily agree that, as far as reasoning is concerned, there are things that the brain can do and things that it cannot do. The search is therefore for the line that separates the two, or the limits beyond which rational human intelligence cannot go. It is proposed that the structure of the prime sequence lies beyond those limits. The contemplation of the prime sequence teaches us something deeply fundamental about the human condition. It is part of the quest to Know Thyself.

One-loop contributions for h→ℓℓγ and e−e+→hγ in the U(1)_(B−L) extension of the standard model

We present one-loop contributions for h→ℓℓγ with ℓ=νe,μ,τ,e,μ and e−e+→hγ in the U(1)_(B−L) extension of the standard model. In the phenomenological results, the signal strengths for h→ℓℓγ at the Large Hadron Collider and for e−e+→hγ at future lepton colliders are analyzed in the physical parameter space for both the vector and chiral B−L models. We found that the contributions from the neutral gauge boson Z′ to the signal strengths are rather small. Consequently, the effects will be difficult to probe at future colliders. However, the impacts of charged Higgs and CP-odd Higgs in the chiral B−L model on the signal strengths are significant and can be measured with the help of the initial polarization beams at future lepton colliders.

Assumed stress quasi-conforming triangular element for couple stress theory

In this paper, a 3-node triangular element for couple stress theory is proposed based on the assumed stress quasi-conforming method. The formulation starts from polynomial approx- imation of stresses. Then the stress-function matrix is treated as the weighted function to weaken the strain-displacement equations. Finally, the string-net functions are introduced to calculate strain integration and the stress smooth technique is adopted to improve the stress accuracy. Numerical results show that the proposed new model can pass the Co- 1 patch test with excellent precision, does not exhibit extra zero energy modes and can cap- ture the scale effects of microstructure.

One-loopformulas for H→Zν_(ι)ν_(ι),forιe,μ,τin't Hooft-Veltman gauge

In this paper,we present analytical results for one-loop contributions to the decay processes H-Zvivi(for I=e,μ,T).The calculations are performed within the Standard Model framework in the't Hooft-Veltman gauge.One-loop form factors are then written in terms of scalar one-loop functions in the standard notations of LoopTools.As a result,one-loop decay rates for the decay channels can be evaluated numerically by using the package.Furthermore,we analyze the signals of H→Zvivi via the production processes e-e+→ZH*-Z(H*→Zν_(ι)ν_(ι)),including the initial beam polarizations at future lepton colliders.The Standard Model backgrounds,such as the processes e-e+→ν_(ι)ν_(ι)ZZ,are also examined in this study.Numerical results indicate that one-loop corrections make contributions of approximately 10%to the decay rates.These are sizeable contributions and should be taken into account at future colliders.We show that the signals H-Zν_(ι)ν_(ι)are clearly visible at the center-of-mass energy√s=250 GeV and are difficult to probe in higher-energy regions owing to the dominant backgrounds.

Polynomial solutions of quasi-homogeneous partial differential equations

By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation $\left\{ {\delta _\tau } \right\}{\text{ }}_{\tau< 0} $ given by ( a1, …, an). Assume that either a1, …, an are positive rational numbers or $m{\text{ = }}\sum\limits_{j = 1}^n {\alpha _j \alpha _j } $ for some $\alpha {\text{ = }}\left( {\alpha _1 ,{\text{ }} \ldots {\text{ }},\alpha _n } \right) \in l _ + ^n $ Then the dimension of the space of polynomial solutions of the equationp[u] = 0 on ?n must be infinite