Lagrangian Theory of Tensor Fields over Spaces with Contravariant and Covariant Affine Connections and Metrics and Its Application to Einstein's Theory of Gravitation in V 4 Spaces

The Lagrangian formalism for tensor fields over differentiable manifolds with contravariant and covariant affine connections (whose components differ not only by sign) and a metric is considered. The functional, the Lie, the covariant, and the total variations of a Lagrangian density, depending on components of tensor fields (with finite rank) and their first and second covariant derivatives, are established. A variation operator is determined and the corollaries of its commutation relations with the covariant and the Lie differential operators are found. The canonical (common) method of Lagrangians with partial derivatives (MLPD) and the method of Lagrangians with covariant derivatives (MLCD) are outlined. They differ each other by the commutation relations the variation operator has to obey with the covariant and the Lie differential operator. The covariant Euler–Lagrange equations are found on the basis of the MLCD. The energy-momentum tensors are considered on the basis of the Lie variation and the covariant Noether identities.As an application of the investigated general scheme, (pseudo) Riemannian spaces with contravariant and covariant affine connections (whose components differ not only by sign) are considered as a special case of -spaces with Riemannian metric, symmetric covariant connection and a weaker definition of dual vector basis with conformal noncanonical contraction operator . The geodesic and autoparallel equations in -spaces are found as different equations in contrast to the case of V ₄-spaces. The Euler–Lagrange equations as Einstein's field equations in -spaces and the corresponding energy-momentum tensors (EMTs) are obtained and compared with the Einstein equations and the EMTs in V ₄-spaces. The geodesic and the auto-parallel equations are discussed.     

About the definition of mass in (Machian) classical mechanics

We want in this note to clarify some aspects of the Machian foundation of the concept of mass in classical mechanics; specifically, we show how the relations of transitivity for the mass-ratios, necessary for a well grounded definition of mass, can be derived from Machian postulates.     

Computing Eigenelements of Real Symmetric Matrices via Optimization

In certain circumstances, it is advantageous to use an optimization approach in order to solve the generalized eigenproblem, Ax = Bx, where A and B are real symmetric matrices and B is positive definite. In particular, this is the case when the matrices A and B are very large the computational cost, prohibitive, of solving, with high accuracy, systems of equations involving these matrices. Usually, the optimization approach involves optimizing the Rayleigh quotient.We first propose alternative objective functions to solve the (generalized) eigenproblem via (unconstrained) optimization, and we describe the variational properties of these functions.We then introduce some optimization algorithms (based on one of these formulations) designed to compute the largest eigenpair. According to preliminary numerical experiments, this work could lead the way to practical methods for computing the largest eigenpair of a (very) large symmetric matrix (pair).     

The Inverse Problem of the Calculus of Variations for Sixth- and Eighth-order Scalar Ordinary Differential Equations

On the equation manifold of the 2nth-order scalar ordinary differential equation, n3,

Positivity and Negativity of Solutions to a Schrödinger Equation in ℝ N

Weak L ² -solutions u of the Schrödinger equation, –u + q(x) u – u = f(x) in L ² , are represented by a Fourier series using spherical harmonics in order to prove the following strong maximum and anti-maximum principles in (N 2): Let ₁ denote the positive eigenfunction associated with the principal eigenvalue ₁ of the Schrödinger operator . Assume that the potential q(x) is radially symmetric and grows fast enough near infinity, and f is a `sufficiently smooth' perturbation of a radially symmetric function, f 0 and 0 f / C const a.e. in . Then u is ₁-positive for - < < ₁ (i.e., u c ₁ with c const > 0) and ₁-negative for ₁ < < ₁ + (i.e., u –c₁ with c const > 0), where > 0 is a number depending on f. The constant c > 0 depends on both and f.     

Porosity of ill-posed problems

We prove that in several classes of optimization problems, including the setting of smooth variational principles, the complement of the set of well-posed problems is -porous.

     

压力及杂质氢对金属锂弹性特性的影响

针对六角密堆金属锂16个原子超晶胞(supercell)、填隙一个氢原子的周期单元，采用基于密度泛函理论的平面波-赝势方法，研究了零温条件下压力及填隙氢掺杂对体系弹性性质的影响.结果表明氢掺杂导致体系的体模量增加.常压下掺杂体系的弹性常数C₁₁，C₃₃，C₆₆和C₁₂高于单质体系，剪切模量C₄₄有所下降，而C₁₃则与单质体系持平.压力作用下C₁₁，C₃₃和C₆₆一直大于单质体系，但C₁₂的值低于单质体系.在2GPa—4GPa压力区间内，弹性常数C₁₃呈反常变化，小于单质体系；在高压区掺杂体系的C₄₄和C₁₃则高于单质体系的相应量值，压力导致掺杂体系和单质体系之间剪切模的偏离加剧.掺杂体系在压力作用下依然保持压缩模的各向同性，具有和单质体系相似的特性. 关键词： 第一性原理 压力效应 弹性常数 金属锂     

On the Validity of Entropy Production Principles for Linear Electrical Circuits

We explain the (non-)validity of close-to-equilibrium entropy production principles in the context of linear electrical circuits. Both the minimum and the maximum entropy production principles are understood within dynamical fluctuation theory. The starting point are Langevin equations obtained by combining Kirchoff’s laws with a Johnson-Nyquist noise at each dissipative element in the circuit. The main observation is that the fluctuation functional for time averages, that can be read off from the path-space action, is in first order around equilibrium given by an entropy production rate. That allows to understand beyond the schemes of irreversible thermodynamics (1) the validity of the least dissipation, the minimum entropy production, and the maximum entropy production principles close to equilibrium; (2) the role of the observables’ parity under time-reversal and, in particular, the origin of Landauer’s counterexample (1975) from the fact that the fluctuating observable there is odd under time-reversal; (3) the critical remark of Jaynes (1980) concerning the apparent inappropriateness of entropy production principles in temperature-inhomogeneous circuits.