Infinite games in the Cantor space and subsystems of second order arithmetic

In this paper we study the determinacy strength of infinite games in the Cantor space and compare them with their counterparts in the Baire space. We show the following theorems: 1. RCA₀ ? ‐Det* ? ‐Det* ? WKL₀. 2. RCA₀ ? ( )2‐Det* ? ACA₀. 3. RCA₀ ? ‐Det* ? ‐Det* ? ‐Det ? ‐Det ? ATR₀. 4. For 1 < k < ω, RCA₀ ? ( )_k ‐Det* ? ( )_{k –1}‐Det. 5. RCA₀ ? ‐Det* ? ‐Det. Here, Det* (respectively Det) stands for the determinacy of infinite games in the Cantor space (respectively the Baire space), and ( )_k is the collection of formulas built from formulas by applying the difference operator k – 1 times. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some More Conservation Results on the Baire Category Theorem

In this paper, we generalize a result of Brown and Simpson [1] to prove that RCA₀+Π⁰_∞‐BCT is conservative over RCA₀ with respect to the set of formulae in the form ∃!Xφ(X), where φ is arithmetical. We also consider the conservation of Π⁰_0∞‐BCT over Σ^b₁‐NIA+∇^b₁‐CA.     

Determinacy of Wadge classes and subsystems of second order arithmetic

In this paper we study the logical strength of the determinacy of infinite binary games in terms of second order arithmetic. We define new determinacy schemata inspired by the Wadge classes of Polish spaces and show the following equivalences over the system RCA₀*, which consists of the axioms of discrete ordered semi‐rings with exponentiation, Δ₁⁰ comprehension and Π₀⁰ induction, and which is known as a weaker system than the popularbase theory RCA₀: 1. Bisep(Δ₁⁰, Σ₁⁰)‐Det* ? WKL₀, (1) 2. Bisep(Δ₁⁰, Σ₂⁰)‐Det* ? ATR₀ + Σ₁¹ induction, (2) 3. Bisep(Σ₁⁰, Σ₂⁰)‐Det* ? Sep(Σ₁⁰, Σ₂⁰)‐Det* ? Π₁¹‐CA₀, (3) 4. Bisep(Δ₂⁰, Σ₂⁰)‐Det* ? Π₁¹‐TR₀, (4) where Det* stands for the determinacy of infinite games in the Cantor space (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Jordan curve theorem and the Schönflies theorem in weak second-order arithmetic

In this paper, we show within ${\mathsf{RCA}_0}In this paper, we show within that both the Jordan curve theorem and the Sch?nflies theorem are equivalent to weak K?nig’s lemma. Within , we prove the Jordan curve theorem using an argument of non-standard analysis based on the fact that every countable non-standard model of has a proper initial part that is isomorphic to itself (Tanaka in Math Logic Q 43:396–400, 1997).      

Periodic solutions for a kind of second order neutral functional differential equations

In this paper, criteria are established for the existence of periodic solutions to a second order sublinear neutral differential equation. Our method is based on careful a priori estimation and continuation theorem, and our sublinear condition is an improvement of the boundedness condition in some recent results.     

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.     

The Wiener integral with respect to second order processes with stationary increments

We prove that for any second order stochastic process X with stationary increments with continuous paths and continuous variance function, there exists a tempered measure μ (for which we give an explicit expression) related with the domain of the Wiener integral with respect to X as follows: the space of tempered distributions f such that the Fourier transform of f is square integrable with respect to μ is always a dense subset of the domain of the Wiener integral. Moreover, we provide sufficient conditions on μ in order that the domain of the integral is exactly this space of distributions. We apply our results to the fractional Brownian motion. In particular, it is proved that the domain of the Wiener integral with respect to the fractional Brownian motion with Hurst parameter H>1/2 contains distributions that are not given by locally integrable functions, this fact was suggested by Pipiras and Taqqu (2000) in [5]. We have also considered the example of the process given by Ornstein and Uhlenbeck as a model for the position of a Brownian particle.     

Convergence analysis of some first order and second order time accurate gradient schemes for semilinear second order hyperbolic equations

This work is devoted to the convergence analysis of finite volume schemes for a model of semilinear second order hyperbolic equations. The model includes for instance the so‐called Sine‐Gordon equation which appears for instance in Solid Physics (cf. Fang and Li, Adv Math (China) 42 (2013), 441–457; Liu et al., Numer Methods Partial Differ Equ 31 (2015), 670–690). We are motivated by two works. The first one is Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043) where a recent class of nonconforming finite volume meshes is introduced. The second one is Eymard et al. (Numer Math 82 (1999), 91–116) where a convergence of a finite volume scheme for semilinear elliptic equations is provided. The mesh considered in Eymard et al. (Numer Math 82 (1999), 91–116) is admissible in the sense of Eymard et al. (Elsevier, Amsterdam, 2000, 723–1020) and a convergence of a family of approximate solutions toward an exact solution when the mesh size tends to zero is proved. This article is also a continuation of our previous two works (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1278–1321; Bradji, Numer Methods Partial Differ Equ 29 (2013), 1–39) which dealt with the convergence analysis of implicit finite volume schemes for the wave equation. We use as discretization in space the generic spatial mesh introduced in Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043), whereas the discretization in time is performed using a uniform mesh. Two finite volume schemes are derived using the discrete gradient of Eymard et al. (IMA J Numer Anal 30 (2010), 1009–1043). The unknowns of these two schemes are the values at the center of the control volumes, at some internal interfaces, and at the mesh points of the time discretization. The first scheme is inspired from the previous work (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1–39), whereas the second one (in which the discretization in time is performed using a Newmark method) is inspired from the work (Bradji, Numer Methods Partial Differ Equ 29 (2013), 1278–1321). Under the assumption that the mesh size of the time discretization is small, we prove the existence and uniqueness of the discrete solutions. If we assume in addition to this that the exact solution is smooth, we derive and prove three error estimates for each scheme. The first error estimate is concerning an estimate for the error between a discrete gradient of the approximate solution and the gradient of the exact solution whereas the second and the third ones are concerning the estimate for the error between the exact solution and the discrete solution in the discrete seminorm of and in the norm of . The convergence rate is proved to be for the first scheme and for the second scheme, where (resp. k) is the mesh size of the spatial (resp. time) discretization. The existence, uniqueness, and convergence results stated above do not require any relation between k and . The analysis presented in this work is also applicable in the gradient schemes framework. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 5–33, 2017     

Existence of homoclinic solution for the second order Hamiltonian systems

An existence theorem of homoclinic solution is obtained for a class of the nonautonomous second order Hamiltonian systems , ∀t∈R, by the minimax methods in the critical point theory, specially, the generalized mountain pass theorem, where L(t) is unnecessary uniformly positively definite for all t∈R, and W(t,x) satisfies the superquadratic condition W(t,x)/|x|²→+∞ as |x|→∞ uniformly in t, and need not satisfy the global Ambrosetti-Rabinowitz condition.     

On stability of the second order neutral differential equation

There exist a well-developed stability theory for neutral differential equations of the first order and only a few results on functional differential equations of the second order. One of the aims of this paper is to fill this gap. Explicit tests for stability of linear neutral delay differential equations of the second order are obtained.     

Periodic solutions of a second order forced sublinear differential equation with delay

In this paper, existence criteria are established for the existence of periodic solutions of a second order sublinear differential equation with delay and forcing function. Our method is based on careful a priori estimation and continuation theorems, and our sublinear condition is an improvement of the boundedness condition in some recent results.     

Fractional integration associated with second order divergence operators on R~n

The classical Hardy-Littlewood-Sobolev theorems for Riesz potentials (-Δ) -α/2 are extended to the generalised fractional integrals L-α/2 for 0 < α < n, where L =-div A is a uniformly complex elliptic operator with bounded measurable coefficients in Rn.     

Periodic solutions of second order discrete Hamiltonian systems with potential indefinite in sign

In this paper, an existence theorem is obtained for the periodic solutions of second-order discrete Hamiltonian systems with potential indefinite in sign by Linking theorem in the critical point theory.     

Nonstandard second-order arithmetic and Riemannʼs mapping theorem

In this paper, we introduce systems of nonstandard second-order arithmetic which are conservative extensions of systems of second-order arithmetic. Within these systems, we do reverse mathematics for nonstandard analysis, and we can import techniques of nonstandard analysis into analysis in weak systems of second-order arithmetic. Then, we apply nonstandard techniques to a version of Riemannʼs mapping theorem, and show several different versions of Riemannʼs mapping theorem.     

On sufficient second order optimality conditions in multiobjective optimization

A second order sufficient optimality criterion is presented for a multiobjective problem subject to a constraint given just as a set. To this aim, we first refine known necessary conditions in such a way that the sufficient ones differ by the replacement of inequalities by strict inequalities. Furthermore, we show that no relationship holds between this criterion and a sufficient multipliers rule, when the constraint is described by inequalities and equalities. Finally, improvements of this criterion for the unconstrained case are presented, stressing the differences with single-objective optimization     

Non-collision periodic solutions of second order singular dynamical systems

We study the existence of non-collision periodic solutions for second order singular dynamical systems. The repulsive case and the attractive case are dealt with using a unified topological approach. The proof is based on a well-known fixed point theorem for completely continuous operators, involving a new type of cone. We do not need to consider so-called strong force conditions. Moreover, for the repulsive case, the critical case can be covered. Recent results in the literature, even in the scalar case, are complemented, generalized and improved.