Algebraic proof theory for substructural logics: Cut-elimination and completions

We carry out a unified investigation of two prominent topics in proof theory and order algebra: cut-elimination and completion, in the setting of substructural logics and residuated lattices.We introduce the substructural hierarchy — a new classification of logical axioms (algebraic equations) over full Lambek calculus FL, and show that a stronger form of cut-elimination for extensions of FL and the MacNeille completion for subvarieties of pointed residuated lattices coincide up to the level N₂ in the hierarchy. Negative results, which indicate limitations of cut-elimination and the MacNeille completion, as well as of the expressive power of structural sequent calculus rules, are also provided.Our arguments interweave proof theory and algebra, leading to an integrated discipline which we call algebraic proof theory.     

Spike solutions of a nonlinear Schrödinger equation with degenerate potential

We study the existence of positive solutions to the elliptic equation ε²Δu(x,y)−V(y)u(x,y)+f(u(x,y))=0 for (x,y) in an unbounded domain subject to the boundary condition u=0 whenever is nonempty. Our potential V depends only on the y variable and is a bounded or unbounded domain which may coincide with . The positive parameter ε is tending to zero and our solutions u_ε concentrate along minimum points of the unbounded manifold of critical points of V.     

Fixed points of asymptotic contractions

Let be a contractive gauge function in the sense that φ is continuous, φ(s)<s for s>0, and if f:M→M satisfies d(f(x),f(y))?φ(d(x,y)) for all x,y in a complete metric space (M,d), then f always has a unique fixed point. It is proved that if T:M→M satisfies

     

Modular metric spaces, I: Basic concepts

The notion of a modular is introduced as follows. A (metric) modular on a set X is a function w:(0,∞)×X×X→[0,∞] satisfying, for all x,y,z∈X, the following three properties: x=y if and only if w(λ,x,y)=0 for all λ>0; w(λ,x,y)=w(λ,y,x) for all λ>0; w(λ+μ,x,y)≤w(λ,x,z)+w(μ,y,z) for all λ,μ>0. We show that, given x₀∈X, the set X_w={x∈X:lim_λ→∞w(λ,x,x₀)=0} is a metric space with metric , called a modular space. The modular w is said to be convex if (λ,x,y)?λw(λ,x,y) is also a modular on X. In this case X_w coincides with the set of all x∈X such that w(λ,x,x₀)<∞ for some λ=λ(x)>0 and is metrizable by . Moreover, if or , then ; otherwise, the reverse inequalities hold. We develop the theory of metric spaces, generated by modulars, and extend the results by H. Nakano, J. Musielak, W. Orlicz, Ph. Turpin and others for modulars on linear spaces.     

Existence and stability of solutions with periodically moving weak internal layers

We consider the periodic parabolic differential equation under the assumption that ε is a small positive parameter and that the degenerate equation f(u,x,t,0)=0 has two intersecting solutions. We derive conditions such that there exists an asymptotically stable solution up(x,t,ε) which is T-periodic in t, satisfies no-flux boundary conditions and tends to the stable composed root of the degenerate equation as ε→0.     

Additivity of Jordan maps on standard Jordan operator algebras

Let A be a standard Jordan operator algebra on a Hilbert space of dimension >1 and B be an arbitrary Jordan algebra. In this note, we prove that if a bijection ?:A→B satisfies

     

Characterizations of Lie derivations of triangular algebras

Let T be a triangular algebra over a commutative ring R. In this paper, under some mild conditions on T, we prove that if δ:T→T is an R-linear map satisfying

δ([x,y])=[δ(x),y]+[x,δ(y)]     

Gauges of Baire class one functions

Let K be a compact metric space and be a bounded Baire class one function. We proved that for any ε>0 there exists an upper semicontinuous positive function δ of f with finite oscillation index and |f(x)−f(y)|<ε whenever d(x,y)<min{δ(x),δ(y)}.     

Asymptotics of instability zones of the Hill operator with a two term potential

Let γn denote the length of the nth zone of instability of the Hill operator Ly=−y^″−[4tαcos2x+2α²cos4x]y, where α≠0, and either both α, t are real, or both are pure imaginary numbers. For even n we prove: if t, n are fixed, then for α→0

     

A minimal classical sequent calculus free of structural rules

Gentzen’s classical sequent calculus has explicit structural rules for contraction and weakening. They can be absorbed (in a right-sided formulation) by replacing the axiom P,¬P by Γ,P,¬P for any context Γ, and replacing the original disjunction rule with Γ,A,B implies Γ,A∨B.This paper presents a classical sequent calculus which is also free of contraction and weakening, but more symmetrically: both contraction and weakening are absorbed into conjunction, leaving the axiom rule intact. It uses a blended conjunction rule, combining the standard context-sharing and context-splitting rules: Γ,Δ,A and Γ,Σ,B implies Γ,Δ,Σ,A∧B. We refer to this system as minimal sequent calculus.We prove a minimality theorem for the propositional fragment : any propositional sequent calculus S (within a standard class of right-sided calculi) is complete if and only ifS contains (that is, each rule of is derivable in S). Thus one can view as a minimal complete core of Gentzen’s .     

Completeness for flat modal fixpoint logics

This paper exhibits a general and uniform method to prove axiomatic completeness for certain modal fixpoint logics. Given a set Γ of modal formulas of the form γ(x,p₁,…,p_n), where x occurs only positively in γ, we obtain the flat modal fixpoint language L_?(Γ) by adding to the language of polymodal logic a connective ?_γ for each γ∈Γ. The term ?_γ(φ₁,…,φ_n) is meant to be interpreted as the least fixed point of the functional interpretation of the term γ(x,φ₁,…,φ_n). We consider the following problem: given Γ, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language L_?(Γ) on Kripke structures. We prove two results that solve this problem.First, let be the logic obtained from the basic polymodal by adding a Kozen-Park style fixpoint axiom and a least fixpoint rule, for each fixpoint connective ?_γ. Provided that each indexing formula γ satisfies a certain syntactic criterion, we prove this axiom system to be complete.Second, addressing the general case, we prove the soundness and completeness of an extension of . This extension is obtained via an effective procedure that, given an indexing formula γ as input, returns a finite set of axioms and derivation rules for ?_γ, of size bounded by the length of γ. Thus the axiom system is finite whenever Γ is finite.     

An elliptic system with bifurcation parameters on the boundary conditions

In this paper we consider the elliptic system Δu=a(x)upvq, Δv=b(x)urvs in Ω, a smooth bounded domain, with boundary conditions , on ∂Ω. Here λ and μ are regarded as parameters and p,s>1, q,r>0 verify (p−1)(s−1)>qr. We consider the case where a(x)?0 in Ω and a(x) is allowed to vanish in an interior subdomain Ω₀, while b(x)>0 in . Our main results include existence of nonnegative nontrivial solutions in the range 0<λ<λ₁?∞, μ>0, where λ₁ is characterized by means of an eigenvalue problem, and the uniqueness of such solutions. We also study their asymptotic behavior in all possible cases: as both λ,μ→0, as λ→λ₁<∞ for fixed μ (respectively μ→∞ for fixed λ) and when both λ,μ→∞ in case λ₁=∞.     

Monotone method for singular BVP in the presence of upper and lower solutions

In this paper we examine existence of monotone approximations of solutions of singular boundary value problem -(p(x)y^′(x))^′=q(x)f(x,y,py^′) for 0<x?b and lim_x→0⁺p(x)y^′(x)=0,α₁y(b)+β₁p(b)y^′(b)=γ₁. Under quite general conditions on f(x,y,py^′) we show that solution of the singular two point boundary value problem is unique. Here is allowed to have integrable singularity at x=0 and we do not assume .     

On uniqueness for nonlinear elliptic equation involving the Pucci's extremal operator

In this article we study uniqueness of positive solutions for the nonlinear uniformly elliptic equation in RN, limr→∞u(r)=0, where denotes the Pucci's extremal operator with parameters 0<λ?Λ and p>1. It is known that all positive solutions of this equation are radially symmetric with respect to a point in RN, so the problem reduces to the study of a radial version of this equation. However, this is still a nontrivial question even in the case of the Laplacian (λ=Λ). The Pucci's operator is a prototype of a nonlinear operator in no-divergence form. This feature makes the uniqueness question specially challenging, since two standard tools like Pohozaev identity and global integration by parts are no longer available. The corresponding equation involving is also considered.     

Positive integral operators in unbounded domains

We study positive integral operators in with continuous kernel k(x,y). We show that if the operator is compact and Hilbert-Schmidt. If in addition k(x,x)→0 as |x|→∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and is trace class. Replacing the first assumption by the stronger then and the bilinear series converges also in L¹. Sharp norm bounds are obtained and Mercer's theorem is derived as a special case.     

Differentiation of sets in measure

Suppose F(ε), for each ε∈[0,1], is a bounded Borel subset of Rd and F(ε)→F(0) as ε→0. Let A(ε)=F(ε)?F(0) be symmetric difference and P be an absolutely continuous measure on Rd. We introduce the notion of derivative of F(ε) with respect to ε, dF(ε)/dε=dA(ε)/dε, such that

     

Lattice-theoretic properties of algebras of logic

In the theory of lattice-ordered groups, there are interesting examples of properties — such as projectability — that are defined in terms of the overall structure of the lattice-ordered group, but are entirely determined by the underlying lattice structure. In this paper, we explore the extent to which projectability is a lattice-theoretic property for more general classes of algebras of logic. For a class of integral residuated lattices that includes Heyting algebras and semi-linear residuated lattices, we prove that a member of such is projectable iff the order dual of each subinterval [a,1]$[a,1]$ is a Stone lattice. We also show that an integral GMV algebra is projectable iff it can be endowed with a positive Gödel implication. In particular, a ΨMV or an MV algebra is projectable iff it can be endowed with a Gödel implication. Moreover, those projectable involutive residuated lattices that admit a Gödel implication are investigated as a variety in the expanded signature. We establish that this variety is generated by its totally ordered members and is a discriminator variety.