Bilinear spaces over a fixed field are simple unstable

We study the model theory of vector spaces with a bilinear form over a fixed field. For finite fields this can be, and has been, done in the classical framework of full first-order logic. For infinite fields we need different logical frameworks. First we take a category-theoretic approach, which requires very little set-up. We show that linear independence forms a simple unstable independence relation. With some more work we then show that we can also work in the framework of positive logic, which is much more powerful than the category-theoretic approach and much closer to the classical framework of full first-order logic. We fully characterise the existentially closed models of the arising positive theory. Using the independence relation from before we conclude that the theory is simple unstable, in the sense that dividing has local character but there are many distinct types. We also provide positive version of what is commonly known as the Ryll-Nardzewski theorem for ω-categorical theories in full first-order logic, from which we conclude that bilinear spaces over a countable field are ω-categorical.     

Diffusion induced void nucleation in SnPb solder joints

Towards a simple and robust model for void-based fatigue prediction, we investigate the interaction of voids with its surrounding by using a multi-field method. We couple the concentration fields of tin c₁ and lead c₂ with an additional field c₃, where the latter is assigned with a void field. The interaction potential manifests three stable states. Two are obtained by experimental results of tin-lead (SnPb) and the void stable state is postulated by construction. The logarithmic form of the thermodynamically consistent configurational entropy is approximated within this study by a fourth order polynom. It has been shown that the interfacial energy coefficient is independent of void's size, but rather depends numerically on the mesh size, which is used in the model presented here. Both governing equations follows a Cahn-Hilliard-type equation to mimic the microstructural changes. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Fourth Painlevé Hierarchy

We use the inverse monodromy transform to find the fourth Painlevé hierarchy. The second and third members of this hierarchy are given. Special and rational solutions of the second and third members for the P ₄ hierarchy are discussed. We apply the Painlevé test to the second member of the fourth Painlevé hierarchy.     

A Note on the Stability of Geodesics on Diffeomorphism Groups with One-side Invariant Metric

In this note,we consider the stability of geodesics on volume-preserving diffeomorphism groups with one-side invariant metric.We showed that for non-Beltrami fields on a three-dimensional compact manifold,there does not exist Eulerian stable flow which is Lagrangian exponential unstable.We noticed that a stationary flow corresponding to the KdV equation can be Eulerian stable while the corresponding motion of the fluid is at most exponentially unstable.     

三项指数和四次均值的精确计算公式(2)

本文进一步深入研究了三项指数和四次均值的计算问题.运用指数和的相关性质并结合求解同余方程组的方法与技巧,利用两种不同的方法获得了两个精确的均值计算公式,揭示了三项指数和的计算与同余方程组解的个数之间的本质联系,推广了已有的结果.     

二项指数和四次混合均值的计算

本文研究了二项指数和四次均值的计算问题.利用初等数论及代数数论的方法获得了一个精确的计算公式以及一个转换公式,推广了已有的结果,揭示了均值计算与同余方程组的本质联系.     

Algebraic groups and small transcendence degree I

The purpose of this paper is to study the algebraic independence of numbers associated with one-parameter subgroups of commutative algebraic groups. Sufficient conditions are given for certain fields to have transcendence degree at least two over the field of rational numbers. As an application one may deduce generalizations of several classical results concerning the algebraic independence of values of the ordinary exponential function or the Weierstrass elliptic function.     

Simultaneous approximations of exponents by transcendental numbers of certain classes

We obtain a theorem on simultaneous approximations of values of the exponential function by elements of fields of finite transcendence degree for whose generators we have a sufficiently good estimate of the measure of algebraic independence.Translated from Matematicheskie Zametki, Vol. 20, No. 3, pp. 305–314, September, 1976.     

Lyapunov sequences for exponential dichotomies

For a linear nonautonomous dynamics with discrete time, we study the relation between nonuniform exponential dichotomies and strict Lyapunov sequences. Given such a sequence, we obtain the stable and unstable subspaces from the intersection of the images and preimages of the cones defined by each element of the sequence. The main difficulty is to extract some information about the angles between the stable and unstable subspaces (or some appropriate notion in the case of Banach spaces) from the Lyapunov sequence. In particular, for a large class of nonuniform exponential dichotomies we give a complete characterization in terms of strict quadratic Lyapunov sequences, that is, strict Lyapunov sequences defined by quadratic forms. We also construct explicitly families of strict Lyapunov sequences for each nonuniform exponential dichotomy, in terms of Lyapunov norms.     

Ax–Schanuel for linear differential equations

We generalise the exponential Ax–Schanuel theorem to arbitrary linear differential equations with constant coefficients. Using the analysis of the exponential differential equation by Kirby (The theory of exponential differential equations, 2006, Sel Math 15(3):445–486, 2009) and Crampin (Reducts of differentially closed fields to fields with a relation for exponentiation, 2006) we give a complete axiomatisation of the first order theories of linear differential equations and show that the generalised Ax–Schanuel inequalities are adequate for them.     

Characterization of multivariate stable processes

This paper deals with a characterization of a multivariate stable process using an independence property with a positive random variable. Moreover, we establish a characterization of a multivariate Lévy process based on the notion of cut in a natural exponential family. This allows us to draw some related properties. More precisely, we give the probability density function of this process and the law of the mixture of the Lévy process governed by the convolution semigroup with respect to an exponential random variable. These results are confidentially connected with the univariate case given by [G. Letac and V. Seshadri, Exponential stopping and drifted stable processes, Stat. Probab. Lett., 72:137–143, 2005].     

A lie group structure on the space of time-dependent vector fields

In this note we use a banal relation between flows of vector fields, a sort of time-dependent Campbell—Baker—Hausdorff formula, to obtain a non-linear version of the variation of constants formula. We employ this formula to calculate the tangent to the mapping which assigns to each time-dependent vector fieldV with compact support the diffeomorphism _t (V), where _t (V) is the global flow ofV. Afterwards we use these results to give a non-commutative Lie group structure to the space of time-dependent vector fields with compact support. We also treat of the Lie algebra of this group and we calculate the adjoint action and the exponential mapping. The setting is the same of [1] and we refer the reader to this book for notations.     

Development and Application of an Exponential Method for Integrating Stiff Systems Based on the Classical Runge–Kutta Method

We study numerical methods for solving stiff systems of ordinary differential equations. We propose an exponential computational algorithm which is constructed by using an exponential change of variables based on the classical Runge–Kutta method of the fourth order. Nonlinear problems are used to prove and demonstrate the fourth order of convergence of the new method.     

Hilbert Lattice Equations

There are five known classes of lattice equations that hold in every infinite dimensional Hilbert space underlying quantum systems: generalised orthoarguesian, Mayet’s E_A{\mathcal{E}_A}, Godowski, Mayet–Godowski, and Mayet’s E equations. We obtain a result which opens a possibility that the first two classes coincide. We devise new algorithms to generate Mayet–Godowski equations that allow us to prove that the fourth class properly includes the third. An open problem related to the last class is answered. Finally, we show some new results on the Godowski lattices characterising the third class of equations.     

Local nondeterminism and local times for stable processes

Summary Our main theorem gives sufficient conditions for symmetric stable processes and fields to have a jointly continuous local time. The approach is through the L ^p representation for such processes. We develop a measure of dependence for vectors in a normed linear space and use that to analyze the probabilistic independence of the increments of a stable process. Local nondeterminism is defined for stable processes and shown to be equivalent to locally approximately independent increments. Sufficient conditions for several classes of stable processes to be local nondeterministic are given. These ideas are extended to multidimensional stable random fields and we prove existence of jointly continuous local times. The results extend most Gaussian results to their stable analogs.     

Continuity,completeness, betweenness and cone-monotonicity

A non-trivial, transitive and reflexive binary relation on the set of lotteries satisfying independence that also satisfies any two of the following three axioms satisfies the third: completeness, Archimedean and mixture continuity (Dubra, 2011). This paper generalizes Dubra’s result in two ways: First, by replacing independence with a weaker betweenness axiom. Second, by replacing independence with a weaker cone-monotonicity axiom. The latter is related to betweenness and, in the case in which outcomes correspond to real numbers, is implied by monotonicity with respect to first-order stochastic dominance.     

Total variation diminishing Runge-Kutta schemes

In this paper we further explore a class of high order TVD (total variation diminishing) Runge-Kutta time discretization initialized in a paper by Shu and Osher, suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that non-TVD but linearly stable Runge-Kutta time discretization can generate oscillations even for TVD (total variation diminishing) spatial discretization, verifying the claim that TVD Runge-Kutta methods are important for such applications. We then explore the issue of optimal TVD Runge-Kutta methods for second, third and fourth order, and for low storage Runge-Kutta methods.

     

The independence of characters on non-abelian groups

We show that there are characters of compact, connected, non-abelian groups that approximate random choices of signs. The work was motivated by Kronecker's theorem on the independence of exponential functions and has applications to thin sets.

     

On polynomial reciprocity law

The well-known law of quadratic reciprocity has over 150 proofs in print. We establish a relation between polynomial Jacobi symbols and resultants of polynomials over finite fields. Using this relation, we prove the polynomial reciprocity law and obtain a polynomial analogue of classical Burde's quartic reciprocity law. Under the use of our polynomial Poisson summation formula and the evaluation of polynomial exponential map, we get a reciprocity for the generalized polynomial quadratic Gauss sums.     

Algebraic independence of some values of the exponential function

We prove general results concerning the algebraic independence of three values of the exponential function. Forβ algebraic and of degree 7 andα algebraic and ≠ 0, 1 there exist among the numbers α_β,..., \(\alpha ^{\beta ^6 } \) three which are algebraically independent. The proof employs a method due to A. O. Gel'fond and N. I. Fel'dman.