An Effective Conservation Result for Nonstandard Arithmetic

We prove that a nonstandard extension of arithmetic is effectively conservative over Peano arithmetic by using an internal version of a definable ultrapower. By the same method we show that a certain extension of the nonstandard theory with a saturation principle has the same proof‐theoretic strength as second order arithmetic, where comprehension is restricted to arithmetical formulas.     

On arithmetic complexity of certain constructive logics

A constructive arithmetical theory is an arbitrary set of closed arithmetical formulas that is closed with respect to derivability in an intuitionsitic arithmetic with the Markov principle and the formal Church thesis. For each arithmetical theory T there is a corresponding logic L(T) consisting of closed predicate formulas in which all arithmetic instances belong to T. For so-called internally enumerable constructive arithmetical theories with the property of existentiality, it is proved that the logic L(T) is II₁ ^T -@#@ complete. This implies, for example, that the logic of traditional constructivism is II₂ ⁰-complete.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 94–104, July, 1992.     

Hyperfinite construction of G-expectation

The hyperfinite G-expectation is a nonstandard discrete analogue of G-expectation (in the sense of Robinsonian nonstandard analysis). A lifting of a continuous-time G-expectation operator is defined as a hyperfinite G-expectation which is infinitely close, in the sense of nonstandard topology, to the continuous-time G-expectation. We develop the basic theory for hyperfinite G-expectations and prove an existence theorem for liftings of (continuous-time) G-expectation. For the proof of the lifting theorem, we use a new discretization theorem for the G-expectation (also established in this paper, based on the work of Dolinsky et al. [Weak approximation of G-expectations, Stoch. Process. Appl. 122(2) (2012), pp. 664–675]).     

A generalization of a conservativity theorem for classical versus intuitionistic arithmetic

A basic result in intuitionism is Π⁰₂‐conservativity. Take any proof p in classical arithmetic of some Π⁰₂‐statement (some arithmetical statement ?x.?y.P(x, y), with P decidable). Then we may effectively turn p in some intuitionistic proof of the same statement. In a previous paper [1], we generalized this result: any classical proof p of an arithmetical statement ?x.?y.P(x, y), with P of degree k, may be effectively turned into some proof of the same statement, using Excluded Middle only over degree k formulas. When k = 0, we get the original conservativity result as particular case. This result was a by‐product of a semantical construction. J. Avigad of Carnegie Mellon University, found a short, direct syntactical derivation of the same result, using H. Friedman's A‐translation. His proof is included here with his permission. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A remark on uniform spaces with invariant nonstandard hulls

Let (X , Γ) be a uniform space with its uniformity generated by a set of pseudo‐metrics Γ. Let the symbol ? denote the usual infinitesimal relation on *X , and define a new infinitesimal relation ≈ on *X by writing x ≈ y whenever *? (x, p ) ? *? (y, p ) for each ? ∈ Γ and each p ∈ X . We call (X , Γ) an S‐space if the relations ? and ≈ coincide on fin(*X ). S ‐spaces are interesting because their nonstandard hulls have representations within Nelson's internal set theory (IST, [5]). This was shown in [1], where it was also observed that the class of uniform spaces that have invariant nonstandard hulls is contained in the class of S ‐spaces. The question of whether there are S ‐spaces that do not have invariant nonstandard hulls was left open in [1]. In this note we show that when the uniformity of an S ‐space is given by a single pseudometric, the space has invariant nonstandard hulls. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interpolation in superintuitionistic predicate logics with equality

It is shown that the Craig interpolation property and the Beth property are preserved under passage from a superintuitionistic predicate logic to its extension via standard axioms for equality, and under adding formulas of pure equality as new axioms. We find an infinite independent set of formulas which, though not equivalent to formulas of pure equality, may likewise be added as new axiom schemes without loss of the interpolation, or Beth, property. The formulas are used to construct a continuum of logics with equality, which are intermediate between the intuitionistic and classical ones, having the interpolation property. Moreover, an equality-free fragment of the logics constructed is an intuitionistic predicate logic, and formulas of pure equality satisfy all axioms of the classical predicate logic. Supported by RFFR grant No. 96-01-01552. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 543–561, September–October, 1997.     

Reverse mathematics and order theoretic fixed point theorems

The theory of countable partially ordered sets (posets) is developed within a weak subsystem of second order arithmetic. We within \(\mathsf {RCA_0}\) give definitions of notions of the countable order theory and present some statements of countable lattices equivalent to arithmetical comprehension axiom over \(\mathsf {RCA_0}\). Then we within \(\mathsf {RCA_0}\) give proofs of Knaster–Tarski fixed point theorem, Tarski–Kantorovitch fixed point theorem, Bourbaki–Witt fixed point theorem, and Abian–Brown maximal fixed point theorem for countable lattices or posets. We also give Reverse Mathematics results of the fixed point theory of countable posets; Abian–Brown least fixed point theorem, Davis’ converse for countable lattices, Markowski’s converse for countable posets, and arithmetical comprehension axiom are pairwise equivalent over \(\mathsf {RCA_0}\). Here the converses state that some fixed point properties characterize the completeness of the underlying spaces.     

Fragments of Arithmetic and true sentences

By a theorem of R. Kaye, J. Paris and C. Dimitracopoulos, the class of the Π_n+1‐sentences true in the standard model is the only (up to deductive equivalence) consistent Π_n+1‐theory which extends the scheme of induction for parameter free Π_n+1‐formulas. Motivated by this result, we present a systematic study of extensions of bounded quantifier complexity of fragments of first‐order Peano Arithmetic. Here, we improve that result and show that this property describes a general phenomenon valid for parameter free schemes. As a consequence, we obtain results on the quantifier complexity, (non)finite axiomatizability and relative strength of schemes for Δ_n+1‐formulas. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modified realizability and predicate logic

Semantics of predicate formulas based on the notion of modified realizability for arithmetic formulas and interpretations of the language of arithmetic in all finite types are considered. For a number of natural constructive interpretations, the corresponding predicate logic of modified realizability is proved to be nonarithmetical. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 259–269, February, 1997. This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00416 and by the International Science Foundation under grant No. NFQ000. Translated by V. N. Dubrovsky     

Periodic solutions preserved by nonstandard finite-difference schemes for the Lotka–Volterra system: a different approach

The Lotka–Volterra predator–prey system x′ = x ? xy, y′ = ? y+xy is a good differential equation system for testing numerical methods. This model gives rise to mutually periodic solutions surrounding the positive fixed point (1,1), provided the initial conditions are positive. Standard finite-difference methods produce solutions that spiral into or out of the positive fixed point. Previously, the author [Roeger, J. Diff. Equ. Appl. 12(9) (2006), pp. 937–948], generalized three different classes of nonstandard finite-difference methods that when applied to the predator–prey system produced periodic solutions. These methods preserve weighted area; they are symplectic with respect to a noncanonical structure and have the property that the computed points do not spiral. In this paper, we use a different approach. We apply the Jacobian matrix procedure to find a fourth class of nonstandard finite-difference methods. The Jacobian matrix method gives more general nonstandard methods that also produce periodic solutions for the predator–prey model. These methods also preserve the positivity property of the solutions.     

Quasi‐completeness and functions without fixed‐points

We prove a completeness criterion for quasi‐reducibility and generalize it to higher levels of the arithmetical hierarchy. As an application of the criterion we obtain Q‐completeness of the set of all pairs (x, n) such that the prefix‐free Kolmogorov complexity of x is less than n. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Alpha-theory: An elementary axiomatics for nonstandard analysis

The methods of nonstandard analysis are presented in elementary terms by postulating a few natural properties for an infinite “ideal” number . The resulting axiomatic system, including a formalization of an interpretation of Cauchy's idea of infinitesimals, is related to the existence of ultrafilters with special properties, and is independent of ZFC. The Alpha-Theory supports the feeling that technical notions such as superstructure, ultrapower and the transfer principle are definitely not needed in order to carry out calculus with actual infinitesimals.     

The arithmetic of cuts in models of arithmetic

We present a number of results on the structure of initial segments of models of Peano arithmetic with the arithmetic operations of addition, subtraction, multiplication, division, exponentiation and logarithm. Each of the binary operations introduced is defined in two dual ways, often with quite different results, and we attempt to systematise the issues and show how various calculations may be carried out. To understand the behaviour of addition and subtraction we introduce a notion of derivative on cuts, analogous to differentiation in the calculus. Multiplication, division and other operations are described by higher order versions of derivative. The work here is presented as important preliminary work related to a nonstandard measure theory of non‐definable bounded subsets of a model of Peano arithmetic.     

To be or not to be constructive,that is not the question

In the early twentieth century, L.E.J. Brouwer pioneered a new philosophy of mathematics, called intuitionism. Intuitionism was revolutionary in many respects but stands out – mathematically speaking – for its challenge of Hilbert’s formalist philosophy of mathematics and rejection of the law of excluded middle from the ‘classical’ logic used in mainstream mathematics. Out of intuitionism grew intuitionistic logic and the associated Brouwer–Heyting–Kolmogorov interpretation by which ‘there exists $x$’ intuitively means ‘an algorithm to compute $x$ is given’. A number of schools of constructive mathematics were developed, inspired by Brouwer’s intuitionism and invariably based on intuitionistic logic, but with varying interpretations of what constitutes an algorithm. This paper deals with the dichotomy between constructive and non-constructive mathematics, or rather the absence of such an ‘excluded middle’. In particular, we challenge the ‘binary’ view that mathematics is either constructive or not. To this end, we identify a part of classical mathematics, namely classical Nonstandard Analysis, and show it inhabits the twilight-zone between the constructive and non-constructive. Intuitively, the predicate ‘$x$ is standard’ typical of Nonstandard Analysis can be interpreted as ‘$x$ is computable’, giving rise to computable (and sometimes constructive) mathematics obtained directly from classical Nonstandard Analysis. Our results formalise Osswald’s longstanding conjecture that classical Nonstandard Analysis is locally constructive. Finally, an alternative explanation of our results is provided by Brouwer’s thesis that logic depends upon mathematics.     

On the Additive Group Structure of the Nonstandard Models of the Theory of Integers

Let $\hat \mathbb{Z}$ denote the inverse limit of all finite cyclic groups. Let F, G and H be abelian groups with H ≤ G. Let FβH denote the abelian group (F × H, +^β), where +^βis defined by (a, x) +^β (b, y) = (a + b, x + y + β(a) + β(b) — β(a + b)) for a certain β : F → G linear mod H meaning that β(0) = 0 and β(a) + β(b) — β(a + b) ∈ H for all a, b in F. In this paper we show that the following hold: (1) The additive group of any nonstandard model ℤ* of the ring ℤ is isomorphic to (ℤ*₊/H)βH for a certain β : ℤ*₊/H → $\hat \mathbb{Z}$ linear mod H. (2) $\hat \mathbb{Z}$ is isomorphic to (ℤ₊/H )βH for some β : $\hat \mathbb{Z}$/H →ℚ linear mod H, though $\hat \mathbb{Z}$ is not the additive group of any model of Th(ℤ, +, ×) and the exact sequence H → $\hat \mathbb{Z}$ → $\hat \mathbb{Z}$/H is not splitting.     

A Reflection Principle and an Orthogonal Decomposition in Clifford Algebras

In this article we investigate spaces of functions defined in a domain Ω ? R with values in the Clifford algebra R _n. According to an inner product an orthogonal decomposition is proved. By this decomposition, we obtain a subspace A ₂(Ω) of regular functions with respect to the Dirac operator. In the orthogonal complement the Dirac equation with homogeneous boundary values is solvable. The decomposition can be proved in two ways: by a reflection principle and by Sobolev's regularity theorem. It will turn out, that the existence of the orthogonal decomposition and Sobolev's theorem is equivalent. So also a reflection principle will be proved, which describes the jump behavior of a Cauchy type integral. By the reflection principle, a countable dense subset of A ₂(Ω) can be obtained. Further considerations lead to a minimal generating system, by which the Bergman kernel function can be obtained. As a conclusion we also obtain Runge's theorem.     

Infinitely many solutions for polyharmonic equations of p(x)-Laplace type

We show the existence of infinitely many weak solutions to a class of quasilinear elliptic p(x)-polyharmonic Kirchhoff equations via the mountain pass principle without the (AR) condition. Furthermore, we obtain infinitely many solutions to this equation based on the genus theory, introduced by Krasnoselskii and the abstract critical point theorem (a variant of Ljusternik-Schnirelman theory) under Cerami condition.     

Sequentially lower complete spaces and Ekeland’s variational principle

By using sequentially lower complete spaces (see [Zhu, J., Wei, L., Zhu, C. C.: Caristi type coincidence point theorem in topological spaces. J. Applied Math., 2013, ID 902692 (2013)]), we give a new version of vectorial Ekeland’s variational principle. In the new version, the objective function is defined on a sequentially lower complete space and taking values in a quasi-ordered locally convex space, and the perturbation consists of a weakly countably compact set and a non-negative function p which only needs to satisfy p(x, y) = 0 iff x = y. Here, the function p need not satisfy the subadditivity. From the new Ekeland’s principle, we deduce a vectorial Caristi’s fixed point theorem and a vectorial Takahashi’s non-convex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. By considering some particular cases, we obtain a number of corollaries, which include some interesting versions of fixed point theorem.     

某些新广义函数类(英文)

给出由非标准离散函数及其差商所定义的新广义函数的某些类,它们密切联系于通常的直至某阶为连续可微的函数．周的一个深刻的定理被用来建立这些类与非标准Ｓｏｂｏｌｅｖ空间之间的关系．     

Noyaux du transfert automorphe de Langlands et formules de Poisson non linéaires

The main purpose of this paper is to show that some type of explicit nonlinear Poisson formulas, which is implied by Langlands’ functoriality principle, allows to build “kernels” of automorphic transfer. So, Langlands’ functoriality principle is equivalent to these nonlinear Poisson formulas.