A Full Predictor-Corrector Finite Element Method for the One-Dimensional Heat Equation with Time-Dependent Singularities

The energy norm convergence rate of the finite element solution of the heat equation is reduced by the time-regularity of the exact solution. This paper presents an adaptive finite element treatment of time-dependent singularities on the one-dimensional heat equation. The method is based on a Fourier decomposition of the solution and an extraction formula of the coefficients of the singularities coupled with a predictor-corrector algorithm. The method recovers the optimal convergence rate of the finite element method on a quasi-uniform mesh refinement. Numerical results are carried out to show the efficiency of the method.

Evolution of polarization singularities accompanied by avoided crossing in plasmonic system

The evolution of polarization singularities supported in a one-dimensional periodic plasmonic system is studied.The lateral inversion symmetry of the system,which breaks the in-plane inversion symmetry and up-down mirror symmetry simultaneously,yields abundant polarization states.A complete evolution process with geometry for the polarization states is traced.In the evolution,circularly polarized points(C points)can stem from 3 different processes.In addition to the previously reported processes occurring in an isolated band,a new type of C point appearing in two bands simultaneously due to the avoided band crossing,is observed.Unlike the dielectric system with a similar structure which only supports at-Γbound states in the continuum(BICs),accidental BICs off theΓpoint are realized in this plasmonic system.This work provides a new scheme of polarization manipulation for the plasmonic systems.

Quantum Unruh Effect on Singularities of Black Holes

It is generally believed that matter inside or once entering a black hole will gravitationally fall into the center and form a size-less singularity, where the density goes to infinity and the spacetime breaks down with infinite curvature or gravitation. In accordance to the Unruh effect, one of the most surprizing predictions of quantum field theory, however, it is found from this study that such singularity cannot be actually formed because it violates the law of energy conservation. The total Unruh radiation energy of the size-less singularity is shown to be infinite, much greater than that the collapsing matter can generate. All the energies of the collapsing matter including the gravitational potential energy, deducted, are far below the Unruh radiation energy, increased, for the collapsing matter to form the singularity. The collapsing matter actually formed is shown to be not a size-less singular point but a small sphere with a finite radius, which is found to be dependent of the mass of the singularity sphere, approximately proportional to the square root of the mass. The radius of the singularity sphere cannot be zero, unless the mass also approaches to zero. The result obtained from this study not only provides us a quantum solution to the problem of black hole singularity, but also leads to profound implications to the spacetime and cosmology. The Unruh effect excludes a black hole to form a size-less singularity, which has a finite mass but infinite density, curvature, and Unruh radiation energy. A point-like or size-less singularity can only be massless and naked.

《三体》的“Singularities”或科幻全球化时代的中国逻辑

本文旨在分析刘慈欣《三体》何以能够在科幻全球化时代以一己之力将中国科幻文学提升到世界水平这一"奇异性事件"。借用詹明信在《奇异性美学》中使用的"singularity"概念,本文认为,《三体》的科学想象力是从"奇点"这一科学边界开始的,并展开了"奇点奇观"的科学想象。《三体》超越本质主义的身份政治,展开了对"后人类"的内在多样性的思考,并建立了黑暗森林原则的宇宙社会学,从而为"独异性政治"分析提供了可能。《三体》的"奇异性美学"在于建构了《三体》的"多维时间"观念体系,并选择了不同的时间叙述方式。

On the pressure and stress singularities induced by steady flows of incompressible viscous fluids

Design for structural integrity requires an appreciation of where stress singularities can occur in structural configurations. While there is a rich literature devoted to the identification of such singular behavior in solid mechanics, to date there has been relatively little explicit identification of stress singularities caused by fluid flows. In this study, stress and pressure singularities induced by steady flows of viscous incompressible fluids are asymptotically identified. This is done by taking advantage of an earlier result that the Navier-Stokes equations are locally governed by Stokes flow in angular corners. Findings for power singularities are confirmed by developing and using an analogy with solid mechanics. This analogy also facilitates the identification of flow-induced log singularities. Both types of singularity are further confirmed for two global configurations by applying convergence-divergence checks to numerical results. Even though these flow-induced stress singularities are analogous to singularities in solid mechanics, they nonetheless render a number of structural configurations singular that were not previously appreciated as such from identifications within solid mechanics alone.

Upper bound for the time derivative of entropy for a stochastic dynamical system with double singularities driven by non-Gaussian noise

A stochastic dynamical system with double singularities driven by non-Gaussian noise is investigated. The Fokker Plank equation of the system is obtained through the path-integral approach and the method of transformation. Based on the definition of Shannon＇s information entropy and the Schwartz inequality principle, the upper bound for the time derivative of entropy is calculated both in the absence and in the presence of non-equilibrium constraint. The present calculations can be used to interpret the effects of the system dissipative parameter, the system singularity strength parameter, the noise correlation time and the noise deviation parameter on the upper bound.

SINGULAR INTEGRAL EQUATIONS ON THE REAL AXIS WITH SOLUTIONS HAVING SINGULARITIES OF HIGHER ORDER

We transform the singular integral equations with solutions simultaneously having singularities of higher order at infinite point and at several finite points on the real axis into ones along a closed contour with solutions having singularities of higher order, and for the former obtain the extended Neother theorem of complete equation as well as the solutions and the solvable conditions of characteristic equation from the latter. The conclusions drawn by this article contain special cases discussed before.

SINGULAR INTEGRAL EQUATIONS WITH COSECANT KERNEL IN SOLUTIONS WITH SINGULARITIES OF ORDER ONE

In this article, periodic Riemann boundary value problem with period 2aπ along closed smooth contours is discussed, and then singular integral equation with kernel csc t-t0/a along closed smooth contours restricted in the strip 0 〈 Rez 〈 aπ is discussed. Finally, the solutions with singularities of order one for the above two problems are discussed.

Time evolution of information entropy for a stochastic system with double singularities driven by quasimonochromatic noise

This paper deals with the time evolution of information entropy for a stochastic system with double singularities driven by quasimonochromatic noise. The dimension of the Fokker Planck equation is reduced by the linear transfor- mation. The exact expression of the time dependence of information entropy is obtained based on the definition of Shannon＇s information entropy. The relationships between the properties of dissipative parameters, system singularity strength parameter, quasimonochromatic noise, and their effects on information entropy are discussed.

On the pressure and stress singularities induced by steady flows of a pair of nonmiscible,incompressible,viscous fluids in contact with a wall

Design for structural integrity requires an appreciation of where stress singularities can occur in structural configurations. While there is a rich literature devoted to the identification of such singular behavior in solid mechanics, only of late has there been much in the way of corresponding identifications of flow-induced stress singularities in fluid mechanics. These recent asymptotic identifications are for a single incompressible viscous fluid： Here the asymptotic approach is extended to apply to a configuration entailing two such fluids, For this configuration, various specifications leading to power or log singularities are determined. These results demonstrate that flow-induced stress singularities can occur in a structural container at a location where no singularities are identified within solid mechanics alone.

Polarization singularities in near-field of Gaussian vortex beam diffracted by a circular aperture

Polarization singularities in the near-field of Gaussian vortex beams diffracted by a circular aperture are studied by a rigorous electromagnetic theory. It is shown that there exist C-points and L-lines, which depend on off-axis displacement parameters along the x and y directions, waist width, wavelength, and topological charge of the diffracted Gaussian vortex beam, as well as on propagation distance. The results are illustrated by numerical calculations.

Observation of Van Hove Singularities and Temperature Dependence of Electrical Characteristics in Suspended Carbon Nanotube Schottky Barrier Transistors

A Van Hove singularity(VHS) is a singularity in the phonon or electronic density of states of a crystalline solid. When the Fermi energy is close to the VHS, instabilities will occur, which can give rise to new phases of matter with desirable properties. However, the position of the VHS in the band structure cannot be changed in most materials. In this work, we demonstrate that the carrier densities required to approach the VHS are reached by gating in a suspended carbon nanotube Schottky barrier transistor. Critical saddle points were observed in regions of both positive and negative gate voltage, and the conductance flattened out when the gate voltage exceeded the critical value. These novel physical phenomena were evident when the temperature is below 100 K. Further, the temperature dependence of the electrical characteristics was also investigated in this type of Schottky barrier transistor.

Stokes parameters of optical singularities of random electromagnetic beams

In this paper, the relation between the spectral degree of coherence and degree of polarization of random electromagnetic beams is derived by the Stokes parameters. And the concept of polarization singularity is extended from spatially fully coherent beams to partially coherent electromagnetic beams. Theoretical analysis shows that correlation vortices are linearly polarized singularities. The results are illustrated by numerical examples.

ON THE SINGULARITIES OF COMPOSITION OF ENTIRE FUNCTIONS

Let f be an entire function. A point Zo is called a critical point of f if f′（zo） = O, and f（zo） is called a critical value （or an algebraic singularity） of f. Next a ∈ C is said to be an asymptotic value （or a transcendental singularity） of f if there exists a curve Г ： [0, 1） → C such that limt→1 F（t） = ∞ and limt→1（f o Г）（t） = a. In this paper we find relations between the asymptotic values of f, 9 and f o 9, relations between critical points of f, 9 and f o 9 and also in the case when the two functions f and 9 are semi-conjugated with another entire function.

Singular Integral Equations with Cosecant Kernel in Solutions with Singularities of High Order

We have discussed and solved the boundary value problem with period 2aπ and the singular integral equation with kernel csc t-tv/a in solution having singularities of high order, where the smooth contour of integration is in the strip 0<Rez<aπ.

CONORMAL SINGULARITIES FOR SOLUTION OF SEMILINEAR PARTIAL DIFFERENTIAL EQUATIONS

In the studies of nonlinear partial differential equations, the influence, from the singularities of coefficients to the singularities of solution, is a field that has not been dealt with. In this paper, we discuss a simple case of semilinear equations under the frame of the space of conormal distributions. We prove the result that the solution has the same singularities on the hypersurface in which the coefficients have the conormal singularities.

Pointwise Approximation Theorems for Combinations of Bernstein Polynomials with Inner Singularities

It is well-known that Bernstein polynomials are very important in studying the characters of smoothness in theory of approximation. A new type of combinations of Bernstein operators are given in [1]. In this paper, we give the Bernstein-Markov inequalities with step-weight functions for combinations of Bernstein polynomials with inner singularities as well as direct and inverse theorems.

Application of Interpolation Inequalities to the Study of Operators with Linear Fractional Endpoint Singularities in Weighted Hölder Spaces

In this paper we consider operators with endpoint singularities generated by linear fractional Carleman shift in weighted H&#246;lder spaces. Such operators play an important role in the study of algebras generated by the operators of singular integration and multiplication by function. For the considered operators, we obtained more precise relations between norms of integral operators with local singularities in weighted Lebesgue spaces and norms in weighted H&#246;lder spaces, making use of previously obtained general results. We prove the boundedness of operators with linear fractional singularities.

Patchwise Mapping Method for Solving Elliptic Boundary Value Problems Containing Multiple Singularities

In the paper [1], the geometrical mapping techniques based on Non-Uniform Rational B-Spline (NURBS) were introduced to solve an elliptic boundary value problem containing a singularity. In the mapping techniques, the inverse function of the NURBS geometrical mapping generates singular functions as well as smooth functions by an unconventional choice of control points. It means that the push-forward of the NURBS geometrical mapping that generates singular functions, becomes a piecewise smooth function. However, the mapping method proposed is not able to catch singularities emerging at multiple locations in a domain. Thus, we design the geometrical mapping that generates singular functions for each singular zone in the physical domain. In the design of the geometrical mapping, we should consider the design of control points on the interface between/among patches so that global basis functions are in C0?space. Also, we modify the B-spline functions whose supports include the interface between/among them. We put the idea in practice by solving elliptic boundary value problems containing multiple singularities.

INTERACTION OF STRONG AND WEAK SINGULARITIES FOR HYPERBOLIC SYSTEM OF CONSERVATION LAWS IN MULTIDIMENSIONAL SPACE

In this paper we study the interaction of strong and weak singularities for hyperbolic system of conservation laws in multidimensional space. Under the assumption of transversal intersect of the shock front with the bicharacteristics bearing weak singularities we proved a theorem on regularity propagation across the shock front.