刘维尔与伽罗瓦数学手稿的发表

1846年,刘维尔在自己主办的杂志“纯粹与应用数学杂志”首次出版了伽罗瓦的数学研究,这对于伽罗瓦理论的传播与发展是具有决定意义的事件.伽罗瓦去世14年后,刘维尔发表伽罗瓦数学研究的原因是什么?采用数学历史文献分析法,得出四个重要原因:①伽罗瓦的朋友和弟弟的请求;②力图弥补科学院曾经造成的不公正;③刘维尔积极扶持年轻人的高贵品质使然;④刘维尔与利布里学术论战的促进.     

Spectral theory of Sturm–Liouville difference operators

We present several classes of explicit self-adjoint Sturm–Liouville difference operators with either a non-Hermitian leading coefficient function, or a non-Hermitian potential function, or a non-definite weight function, or a non-self-adjoint boundary condition. These examples are obtained using a general procedure for constructing difference operators realizing discrete Sturm–Liouville problems, and the minimum conditions for such difference operators to be self-adjoint with respect to a natural quadratic form. It is shown that a discrete Sturm–Liouville problem admits a difference operator realization if and only if it does not have all complex numbers as eigenvalues. Spectral properties of self-adjoint Sturm–Liouville difference operators are studied. In particular, several eigenvalue comparison results are proved.     

The minimax approach for a class of variable order fractional differential equation

This paper introduces an approximate solution for Liouville‐Caputo variable order fractional differential equations with order 0 < α(t) ≤ 1 . The solution is adapted using a family of fractional‐order Chebyshev functions with unknown coefficients. These coefficients have been obtained by using an optimization approach based on minimax technique and the least pth optimization function. Several linear and nonlinear fractional‐order differential equations are discussed using the proposed technique for fixed and variable order fractional‐order derivatives. Moreover, the response of RC charging circuit with variable order fractional capacitor is studied for different cases. Several comparisons with related published techniques have been added to illustrate the accuracy of the proposed approach.     

Discontinuous Sturm-Liouville Problems Containing Eigenparameter in the Boundary Conditions

In this paper, discontinuous Sturm-Liouville problems, which contain eigenvalue parameters both in the equation and in one of the boundary conditions, are investigated. By using an operatortheoretic interpretation we extend some classic results for regular Sturm-Liouville problems and obtain asymptotic approximate formulae for eigenvalues and normalized eigenfunctions. We modify some techniques of [Fulton, C. T., Proc. Roy. Soc. Edin. 77 （A）, 293-308 （1977）], [Walter, J., Math. Z., 133, 301-312 （1973）] and [Titchmarsh, E. C., Eigenfunctions Expansion Associated with Second Order Differential Equations I, 2nd edn., Oxford Univ. Pres, London, 1962], then by using these techniques we obtain asymptotic formulae for eigenelement norms and normalized eigenfunctions.     

Contributions to differential geometry of isotropic curves in the complex space

This work deals with classical differential geometry of isotropic curves in the complex space C⁴. First, we study spherical isotropic curves and pseudo helices. Besides, in this section we introduce some special isotropic helices (type-1, type-2 and type-3 isotropic slant helices) and express some characterizations of them in terms of É. Cartan equations. Thereafter, we prove that position vector of an isotropic curve satisfies a vector differential equation of fourth order. Finally, we investigate position vector of an arbitrary curve with respect to É. Cartan frame by a system of complex differential equations whose solution gives components of the position vector. Solutions of the mentioned system and vector differential equation have not yet been found. Therefore, in terms of special cases, we present some special characterizations.     

Maximum entropy characterizations of the multivariate Liouville distributions

A random vector X=(X₁,X₂,…,X_n) with positive components has a Liouville distribution with parameter θ=(θ₁,θ₂,…,θ_n) if its joint probability density function is proportional to , θ_i>0 [R.D. Gupta, D.S.P. Richards, Multivariate Liouville distributions, J. Multivariate Anal. 23 (1987) 233-256]. Examples include correlated gamma variables, Dirichlet and inverted Dirichlet distributions. We derive appropriate constraints which establish the maximum entropy characterization of the Liouville distributions among all multivariate distributions. Matrix analogs of the Liouville distributions are considered. Some interesting results related to I-projection from a Liouville distribution are presented.     

Barbilian spaces: The history of a geometric idea

Barbilian spaces are metric spaces with a metric induced by a special procedure of metrization that is inspired by the study of the models of non-Euclidean geometry. In this note we discuss the history of Barbilian spaces and the evolution of the theory. We point out that some of the current references to work done in Barbilian spaces refer to Barbilian's contribution from 1934, while his construction has been greatly extended in four works published in Romanian in 1959–1962.     

Spinning gas clouds: Liouville integrable cases

We consider the class of ellipsoidal gas clouds expanding into a vacuum [1, 2] which has been shown to be a Liouville integrable Hamiltonian system [3]. This system presents several interesting features, such as the Painlevé property [4, 5], the existence of Bäcklund transformations and the separability of variables, all shown to be present in at least several sub-cases. A remarkable result that emerged from the study of the cases of rotation around a fixed principal axis, was that the Liouville torus, which is the locus of trajectories of the representative point of the cloud when all the constants of motion are fixed, could be assimilated with a quartic surface presenting 16 conic point singularities. The geometry of such surfaces is entirely determined by the datum of a 6th degree polynomial in one variable, and the consideration of the corresponding natural coordinate system then led to the separation of variables for these cases [6]. Further, the equation of the surface takes the form of a 4×4 determinant, which constitutes a generalization of Stieltjes 4 × 4 determinant formulation of the addition formula for elliptic functions; and the corresponding matrix also defines the system of the equations of motion; so that it can be said that the differential system is completely determined by the surface’s geometry. Forsaking now the assumption of a fixed rotation axis, in cases where the energy constant takes its minimum value compatible with the other constants of motion, we found that the Liouville torus was still reducible to the form of a quartic surface, presenting 15 conic points only instead of 16 (16 conic points were indeed present originally, but one of them had to disappear in the process of reducing the surface to the 4th degree). The geometry of these surfaces is entirely determined by the datum of a plane unicursal quartic (which is the transformed version of the missing conic point). The system can be reduced to the form of a differential equation of second degree, the coefficients of which are polynomials of degree 7, which are determined by the surface’s geometry, except for their quadratic dependence on a single free parameter, z. Defining u the (time-like) independent variable, and Φ the integration constant (which are functions defined on the Liouville torus), it is found that Φ depends linearly on the parameter: Φ = Φ(z) and then u may be taken to coincide with Φ(z′), for any value of z′ distinct from z. Solving the system for one particular value of z therefore also solves it for all other values of the parameter. It appears that the geometry alone does not specify in this case any particular value of z, but then any two values lead to differential systems which (although their solutions differ) turn out to be equivalent. It may also be worth pointing out that changing z may be viewed as exchanging the roles of u and Φ. Finally, in degenerate cases the Liouville torus presents a double line of self-intersection, and the separation of variables can be achieved. Sections by planes through the double line are conic sections, which may be labeled by a parameter w, say. Denoting α the eccentric anomaly on the conic, the differential system in fact takes a remarkably simple form: da/dw = f(w), and involves an elliptic integral.     

The Liouville equation in an annulus

We give the solutions to the Liouville equation in an annulus A of R² that satisfy a certain Neumann condition on each component of ∂A. As a consequence, we classify all the metrics of constant curvature in A that have constant geodesic curvature on ∂A.     

Liouville theorem of axially symmetric Navier–Stokes equations with growing velocity at infinity

In the paper Koch et al. (2009), the authors make the following conjecture: any bounded ancient mild solution of the 3D axially symmetric Navier–Stokes equations is constant. And it is proved in the case that the solution is swirl free. Our purpose of this paper is to improve their result by allowing that the solution can grow with a power smaller than 1 with respect to the distance to the origin. Also, we will show that such a power is optimal to prove the Liouville type theorem since we can find counterexamples for the Navier–Stokes equations such that the Liouville theorem fails if the solution can grow linearly.     

The Mittag-Leffler Theorem: The origin,evolution, and reception of a mathematical result, 1876–1884

The Swedish mathematician Gösta Mittag-Leffler (1846–1927) is well-known for founding Acta Mathematica, often touted as the first international journal of mathematics. A “post-doctoral” student in Paris and Berlin between 1873 and 1876, Mittag-Leffler built on Karl Weierstrass? work by proving the Mittag-Leffler Theorem, which states that a function of rational character (i.e. a meromorphic function) is specified by its poles, their multiplicities, and the coefficients in the principal part of its Laurent expansion.     

Spectral Dimension of Liouville Quantum Gravity

This paper is concerned with computing the spectral dimension of (critical) 2d-Liouville quantum gravity. As a warm-up, we first treat the simple case of boundary Liouville quantum gravity. We prove that the spectral dimension is 1 via an exact expression for the boundary Liouville Brownian motion and heat kernel. Then we treat the 2d-case via a decomposition of time integral transforms of the Liouville heat kernel into Gaussian multiplicative chaos of Brownian bridges. We show that the spectral dimension is 2 in this case, as derived by physicists (see Ambjørn et al. in JHEP 9802:010, 1998) 15 years ago.     

Explicit construction of a time superoperator for quantum unstable systems

A time superoperator T conjugate to the Liouville superoperator L_H=[H,] is constructed for a quantum system with one excited state or unstable particle. While there is no time operator conjugate to the Hamiltonian in the wave function space due to the positivity of energy, T may exist in the density matrix space as the spectrum of L_H covers all the real axis. This is the first example of an observable that can only be formulated in the Liouville–von Neumann space of density matrices. In our example the expectation value of T gives the lifetime of the unstable particle. Once the time superoperator is obtained it is easy to define an entropy superoperator.     

The Finite Spectrum of Sturm–Liouville Problems with Transmission Conditions and Eigenparameter-Dependent Boundary Conditions

We study the finite spectrum of Sturm–Liouville problems with transmission conditions and eigenparameter-dependent boundary conditions. For any positive integers m and n, we construct a class of regular Sturm–Liouville problems with transmission conditions and eigenparameter-dependent boundary conditions, which have at most m + n + 4 eigenvalues.     

Insights into Dedekind and Weber’s edition of Riemann’s <Emphasis Type="Italic">Gesammelte Werke</Emphasis>

In this paper, I present and analyse Dedekind’s and Weber’s editorial work which led to the edition of Riemann’s Gesammelte Werke in 1876. With several examples, I suggest that this editorial work is to be understood as a mathematical activity in and of itself and provide evidence for it.     

From Archimedean to Liouville copulas

We use a recent characterization of the d-dimensional Archimedean copulas as the survival copulas of d-dimensional simplex distributions (McNeil and Nešlehová (2009) [1]) to construct new Archimedean copula families, and to examine the relationship between their dependence properties and the radial parts of the corresponding simplex distributions. In particular, a new formula for Kendall’s tau is derived and a new dependence ordering for non-negative random variables is introduced which generalises the Laplace transform order. We then generalise the Archimedean copulas to obtain Liouville copulas, which are the survival copulas of Liouville distributions and which are non-exchangeable in general. We derive a formula for Kendall’s tau of Liouville copulas in terms of the radial parts of the corresponding Liouville distributions.     

Symmetry properties for solutions of nonlocal equations involving nonlinear operators

We pursue the study of one-dimensional symmetry of solutions to nonlinear equations involving nonlocal operators. We consider a vast class of nonlinear operators and in a particular case it covers the fractional p-Laplacian operator. Just like the classical De Giorgi's conjecture, we establish a Poincaré inequality and a linear Liouville theorem to provide two different proofs of the one-dimensional symmetry results in two dimensions. Both approaches are of independent interests. In addition, we provide certain energy estimates for layer solutions and Liouville theorems for stable solutions. Most of the methods and ideas applied in the current article are applicable to nonlocal operators with general kernels where the famous extension problem, given by Caffarelli and Silvestre, is not necessarily known.