中心—焦点区分问题的I l'yashenko算法

对二维平面系统的二维系统的中心焦点区分问题 ,I l'yashenko曾建议一个算法 ,本文给出此法的详细证明 .据此 ,我们讨论了区分问题在 Arnold意义下的代数可解性与不可解性     

Degrees of logics with Henkin quantifiers in poor vocabularies

We investigate some logics with Henkin quantifiers. For a given logic L, we consider questions of the form: what is the degree of the set of L–tautologies in a poor vocabulary (monadic or empty)? We prove that the set of tautologies of the logic with all Henkin quantifiers in empty vocabulary L* is of degree 0. We show that the same holds also for some weaker logics like L(H) and L(E). We show that each logic of the form L^(k)(Q), with the number of variables restricted to k, is decidable. Nevertheless – following the argument of M. Mostowski from [Mos89] – for each reasonable set theory no concrete algorithm can provably decide L^(k)(Q), for some (Q). We improve also some results related to undecidability and expressibility for logics L(H₄) and L(F₂) of Krynicki and M. Mostowski from [KM92].Mathematics Subject Classification (2000): 03C80, 03D35, 03B25Revised version: 28 August 2003     

On effectively computable realizations of choice functions: Dedicated to Professors Kenneth J. Arrow and Anil Nerode

Let X be a compact, convex subset of R_n, and let $\text{〈R(X),}\text{F}{}_{\mathrm{R}}\text{〉}$ be a recursive space of alternatives, where R(X) is the image of X in a recursive metric space, and $\text{F}{}_{\mathrm{R}}$ is the family of all recursive subsets of R(X). If $\text{C:}\text{F}{}_{\mathrm{R}}\text{→}\text{F}{}_{\mathrm{R}}$ is a non-trivial recursively representable choice function that is rational in the sense of Richter, we prove that C has no recursive realization within Church's Thesis. Our proof is not a diagonalization argument and uses no paradoxical statements from formal systems. Instead, the proof is a Kleene-Post reduction style argument and uses the Turing equivalence between mechanical devices of computation and the recursive functions of Gödel and Kleene.     

Blow-up for Joseph–Egri equation: Theoretical approach and numerical analysis

This work develops the theory of the blow-up phenomena for Joseph–Egri equation. The existence of the nonextendable solution of two initial-boundary value problems (on a segment and a half-line) is demonstrated. Sufficient conditions of the finite-time blow-up of these solutions, as well as the analytical estimates of the blow-up time, are obtained. A numerical method that allows to precise the blow-up moment for specified initial data is proposed.     

Noncomputability,unpredictability, undecidability,and unsolvability in economic and finance theories

We outline, briefly, the role that issues of the nexus between noncomputability and unpredictability, on the one hand, and between undecidability and unsolvability, on the other hand, have played in Computable Economics (CE). The mathematical underpinnings of CE are provided by (classical) recursion theory, varieties of computable and constructive analysis and aspects of combinatorial optimization. The inspiration for this outline was provided by Professor Graça's thought‐provoking recent article. © 2012 Wiley Periodicals, Inc. Complexity, 2012     

On Representing Concepts in Finite Models

We present a method of transferring Tarski's technique of classifying finite order concepts by means of truth‐definitions into finite mode theory. The other considered question is the problem of representability relations on words or natural numbers in finite models. We prove that relations representable in finite models are exactly those which are of degree ≤ o′. Finally, we consider theories of sufficiently large finite models. For a given theory T we define sl(T) as the set of all sentences true in almost all finite models for T. For theories of sufficiently large models our version of Tarski's technique becomes practically the same as the classica one. We investigate also degrees of undecidability for theories of sufficiently large finite models. We prove for some special theory ST that its degree is stronger than 0′ but still not more than Σ⁰₂.     

The minimum degree of recursively representable choice functions

We show that the minimum degree of Turing complexity of a recursively representable choice function is Õ″, the degree of a complete ∑₂ set in the Kleene-Mostowski hierarchy. A consequence of this result is that the complexity of such choice functions in this sense is bounded strictly above the degrees of R.E. subsets on $\text{N}$.     

On the Universal Splitting Property

We prove that if an incomplete computably enumerable set has the the universal splitting property then it is low₂. This solves a question from Ambos-Spies and Fejer [1] and Downey and Stob [7]. Some technical improvements are discussed.     

幂零矩阵的m次根

主要研究当A是幂零矩阵时,方程Xm=A的性质.我们可以得到一些关于方程Xm=A无解性与A自身的特点之间的关系.     

The Fractal Nature of Riem/Diff I

This paper is devoted to large scale aspects of the geometry of the space of isometry classes of Riemannian metrics, with a 2-sided curvature bound, on a fixed compact smooth manifold of dimension at least five. Using a mix of tools from logic/computer science, and differential geometry and topology, we study the diameter functional and its critical points, as well as their distribution (density) within the space and the structure of their neighborhoods.