Mechanical theorem proving in differential geometry

An automated reasoning method, based on Wu’s method and calculus of differential forms, is proposed for mechanical theorem proving in local theory of space surfaces in differential geometry. The method has been used to simplify one of Chem’s theorems: “The non-trivial families of isometric surfaces having the same principal curvatures are W-surfaces.” Some other theorems are also tested by this method. The proofs are generally simpler than those in differential geometry textbooks. Project supported partially by the National Natural Science Foundation of China.     

Ordering in mechanical geometry theorem proving

Ordering in mechanical geometry theorem proving is studied from geometric viewpoint and some new ideas are proposed. For Thebault’s theorem which is the most difficult theorem that has ever been proved by Wu’s method, a very simple proof using Wu’s method under a linear order is discovered. Project supported by the National Natural Science Foundation of China.     

A Note on the Browder＇s and Weyl＇s Theorem

Let T be a Banach space operator, E（T） be the set of all isolated eigenvalues of T and π（T） be the set of all poles of T. In this work, we show that Browder＇s theorem for T is equivalent to the localized single-valued extension property at all complex numbers λ in the complement of the Weyl spectrum of T, and we give some characterization of Weyl＇s theorem for operator satisfying E（T） = π（T）. An application is also given.     

Affine bracket algebra theory and algorithms and their applications in mechanical theorem proving

This paper discusses two problems:one is some important theories and algorithms of affine bracket algebra;the other is about their applications in mechanical theorem proving.First we give some efficient algorithms including the boundary expanding algorithm which is a key feature in application.We analyze the characteristics of the boundary operator and this is the base for the implementation of the system.We also give some new theories or methods about the exact division,the representations and structure of affine geometry and so on.In practice,we implement the mechanical auto-proving system in Maple 10 based on the above algorithms and theories.Also we test about more than 100 examples and compare the results with the methods before.     

CALCULUS ON CANTOR TRIADIC SET （I）——DERIVATIVE

For real-valued functions defined on Cantor triadic ,set. a derivative with corresponding formula of Newton-Leihniz‘s type is given In particular, for the self-simltar functions and alter-nately jumping functions defined in this paper, their derivative and exceptional sets are studied ac-curately by using ergodic theory on Е2 and Duffin-Scbaeffer‘s theorem coneerning metric diophan-tine approximation. In addition, Haar basis of L2(Е2) is constructed and Flaar expansion of stan-drd self-similar function is given.     

Ramsey’s theorem and König’s Lemma

We consider the relation between versions of Ramsey’s Theorem and König’s Infinity Lemma, in the absence of the axiom of choice.     

基于多项式组主项解耦消元法的几何定理机器证明

基于多项式组主项解耦消元法 ,将几何定理的假设条件 (多项式组 PS)化为主项只含主变元的三角型多项式组 DTS,可得到定理命题成立的不含变元的非退化条件 ,即充分必要或更接近充分必要的非退化条件 .由于多项式主系数不含变元 ,已不存在 DTS多项式之间的约化问题 ,故方法有普遍意义 .文中例为西姆松定理的机器证明 .     

A method of proving interpolation in paraconsistent extensions of the minimal logic

The interpolation property in extensions of Johansson’s minimal logic is investigated. The construction of a matched product of models is proposed, which allows us to prove the interpolation property in a number of known extensions of the minimal logic. It is shown that, unlike superintuitionistic, positive, and negative logics, a sum of J-logics with the interpolation property CIP may fail to possess _CIP, nor even the restricted interpolation property. Supported by RFBR grant No. 06-01-00358, by INTAS grant No. 04-77-7080, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-4787.2006.1. __________ Translated from Algebra i Logika, Vol. 46, No. 5, pp. 627–648, September–October, 2007.     

Newton’s method as a tool for finding the eigenvalues of certain two-parameter (multiparameter) spectral problems

An iterative algorithm is examined for finding the eigenvalues of the two-parameter (multiparameter) algebraic eigenvalue problem. This algorithm uses Newton’s method and an efficient numerical procedure for differentiating determinants. Some numerical examples are given.     

A complete analogue of Hardy’s theorem on SL2(ℝ) and characterization of the heat kernel

A theorem of Hardy characterizes the Gauss kernel (heat kernel of the Laplacian) on ℝ from estimates on the function and its Fourier transform. In this article we establisha full group version of the theorem for SL₂(ℝ) which can accommodate functions with arbitraryK-types. We also consider the ‘heat equation’ of the Casimir operator, which plays the role of the Laplacian for the group. We show that despite the structural difference of the Casimir with the Laplacian on ℝⁿ or the Laplace—Beltrami operator on the Riemannian symmetric spaces, it is possible to have a heat kernel. This heat kernel for the full group can also be characterized by Hardy-like estimates.     

A method of summation of divergent series to any accuracy

We propose a new method of summation to any accuracy for a wide class of divergent series, using only a finite number of terms of the series. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 24–35, July, 2000.     

On the convergence of the secant method under the gamma condition

We provide sufficient convergence conditions for the Secant method of approximating a locally unique solution of an operator equation in a Banach space. The main hypothesis is the gamma condition first introduced in [10] for the study of Newton’s method. Our sufficient convergence condition reduces to the one obtained in [10] for Newton’s method. A numerical example is also provided.      

On a quadratically convergent method using divided differences of order one under the gamma condition

We re-examine a quadratically convergent method using divided differences of order one in order to approximate a locally unique solution of an equation in a Banach space setting [4, 5, 7]. Recently in [4, 5, 7], using Lipschitz conditions, and a Newton-Kantorovich type approach, we provided a local as well as a semilocal convergence analysis for this method which compares favorably to other methods using two function evaluations such as the Steffensen’s method [1, 3, 13]. Here, we provide an analysis of this method under the gamma condition [6, 7, 19, 20]. In particular, we also show the quadratic convergence of this method. Numerical examples further validating the theoretical results are also provided.      

On the difference between the number of prime divisors at consecutive integers

Suppose thatg(n) is equal to the number of divisors ofn, counting multiplicity, or the number of divisors ofn, a≠0 is an integer, andN(x,b)=|{n∶n≤x, g(n+a)−g(n)=b orb+1}|. In the paper we prove that sup _b N(x,b)≤C(a)x)(log log 10 _x )^−1/2 and that there exists a constantC(a,μ)>0 such that, given an integerb |b|≤μ(log logx)^1/2,x≥x _o, the inequalityN(x,b)≥C(a,μ)x(log logx(^−1/2) is valid. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 579–595, October, 1999.     

Arnold’s web and diffusion in the Stark-Quadratic-Zeeman problem

The Arnold web and the Arnold diffusion arise when an integrable Hamiltonian system is slightly perturbed: the first concerns the peculiar topology characterizing the set of the resonance lines in phase space, the latter the extremaly slow motion (if any) along these lines. While Arnold has proved the possibility of diffusion, it is still unknown if the phenomenon is generic in realistic physical systems. The system we consider is the Hydrogen atom (or Kepler problem) subject to the combined action of a constant electric and magnetic field, which is known as Stark-Zeeman problem. We describe the results of numerical experiments: the Arnold web is clearly highlighted and, looking at the behaviour of the KAM frequencies on orbits of 10⁸ revolutions, evidence for the diffusion existence is reached.      

Prime witnesses in the Shor algorithm and the Miller-Rabin algorithm

We prove that prime witnesses in the Miller-Rabin algorithm coincide with those in the Shor algorithm which satisfy the condition of Fermat’s little theorem. We describe the set of natural numbers, whose prime witnesses in the Miller-Rabin algorithm coincide with those in the Shor algorithm. We find all such numbers less than 100,000,000 and experimentally study the rate of increase of the ratio of the quantity of such numbers to the quantity of Carmichael numbers.     

Small set in a large box

Let K ? ?^d be a compact convex set which is an intersection of half-spaces defined by at most two coordinates. Let Q be the smallest axes-parallel box containing K. We show that, as the dimension d grows, the ratio diam Q/ diam K can be arbitrarily large. We also give examples of compact sets in Banach spaces which are not contained in any compact contractive set.