Normal Form Theorem Research Articles

A more general general proof theory

In this paper it is suggested to generalize our understanding of general (structural) proof theory and to consider it as a general theory of two kinds of derivations, namely proofs and dual proofs. The proposal is substantiated by (i) considerations on assertion, denial, and bi-lateralism, (ii) remarks on compositionality in proof-theoretic semantics, and (iii) comments on falsification and co-implication. The main formal result of the paper is a normal form theorem for the natural deduction proof system N2Int of the bi-intuitionistic logic 2Int. The proof makes use of the faithful embedding of 2Int into intuitionistic logic with respect to validity and shows that conversions of dual proofs can be sidestepped.

On dual pairs in Dirac geometry

In this note we discuss dual pairs in Dirac geometry. We show that this notion appears naturally when studying the problem of pushing forward a Dirac structure along a surjective submersion, and we prove a Dirac-theoretic version of Libermann's theorem from Poisson geometry. Our main result is an explicit construction of strong self-dual pairs for Dirac structures. This theorem not only recovers the global construction of symplectic realizations from [Crainic-Marcut 2011], but allows for a more conceptual understanding of it, yielding a simpler and more natural proof. As an application of the main theorem, we present a different approach to the recent normal form theorem around Dirac transversals from [Bursztyn-Lima-Meinrenken 2016].

Normal forms for Poisson maps and symplectic groupoids around Poisson transversals

Poisson transversals are submanifolds in a Poisson manifold which intersect all symplectic leaves transversally and symplectically. In this communication, we prove a normal form theorem for Poisson maps around Poisson transversals. A Poisson map pulls a Poisson transversal back to a Poisson transversal, and our first main result states that simultaneous normal forms exist around such transversals, for which the Poisson map becomes transversally linear, and intertwines the normal form data of the transversals. Our second result concerns symplectic integrations. We prove that a neighborhood of a Poisson transversal is integrable exactly when the Poisson transversal itself is integrable, and in that case we prove a normal form theorem for the symplectic groupoid around its restriction to the Poisson transversal, which puts all structure maps in normal form. We conclude by illustrating our results with examples arising from Lie algebras.

Axiomatizations for downward XPath on data trees

We give sound and complete axiomatizations for XPath with data tests by ‘equality’ or ‘inequality’, and containing the single ‘child’ axis. This data-aware logic predicts over data trees, which are tree-like structures whose every node contains a label from a finite alphabet and a data value from an infinite domain. The language allows us to compare data values of two nodes but cannot access the data values themselves (i.e. there is no comparison by constants). Our axioms are in the style of equational logic, extending the axiomatization of data-oblivious XPath, by B. ten Cate, T. Litak and M. Marx. We axiomatize the full logic with tests by ‘equality’ and ‘inequality’, and also a simpler fragment with ‘equality’ tests only. Our axiomatizations apply both to node expressions and path expressions. The proof of completeness relies on a novel normal form theorem for XPath with data tests.

Rigidity of infinitesimal momentum maps

In this paper we prove rigidity theorems for Poisson Lie group actions on Poisson manifolds. In particular, we prove that close infinitesimal momentum maps associated to Poisson Lie group actions are equivalent using a normal form theorem for SCI spaces. When the Poisson structure of the acted manifold is integrable, this yields rigidity also for lifted actions to the symplectic groupoid.

RETRACTED ARTICLE: New applications of Schr\xf6dinger type inequalities to the existence and uniqueness of Schr\xf6dingerean equilibrium

As new applications of Schrödinger type inequalities appearing in Jiang (J. Inequal. Appl. 2016:247, 2016), we first investigate the existence and uniqueness of a Schrödingerean equilibrium. Next we propose a tritrophic Hastings-Powell model with two different Schrödingerean time delays. Finally, the stability and direction of the Schrödingerean Hopf bifurcation are also investigated by using the center manifold theorem and normal form theorem.

The normal form theorem around Poisson transversals

We prove a normal form theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds), which simultaneously generalizes Weinstein's symplectic neighborhood theorem from symplectic geometry and Weinstein's splitting theorem. Our approach turns out to be essentially canonical, and as a byproduct, we obtain an equivariant version of the latter theorem.

Fuzzy Turing machines: Normal form and limitative theorems

A normal form for fuzzy Turing machines is proposed and examined. This normal form is arithmetical in nature since the truth values are substituted by n-ples of natural numbers and the operation interpreting the conjunction becomes a sort of truncated sum. Also, some of the results in the paper enable us to emphasize the inadequacy of the notion of fuzzy Turing machine for fuzzy computability, i.e. that this notion is not a good candidate for a ‘Church thesis’ in the fuzzy mathematics framework.

Complex dynamic of plankton–fish interaction with quadratic harvesting and time delay

The present paper deals with a food chain model consisting of three species phytoplankton, zooplankton and fish. We have divided the present paper into two parts. In the first part, we have assumed that the fish population is harvested using a non-linear harvesting function. Considering this rate of harvesting ‘E’ as a control parameter, we have estimated different ranges of harvesting parameter for maintaining the sustainability in the plankton ecosystem. Moreover, the bifurcation analysis of the system is carried out using normal form theorem by taking ‘E’ as bifurcation parameter. In the second part, a digestion delay corresponding to zooPlankton–fish interaction is introduced for more realistic consideration of the real world problem. Taking harvesting parameter in the stability range, the effect of time delay on the given system is investigated. This research demonstrate that for a certain range of delay, system enters into the excited state with the existence of stability switches which seems new findings for the Plankton–fish system. Explicit results are derived for stability and direction of the bifurcating periodic solution by using normal form theory and center manifold arguments. To validate our analytical findings numerical simulations are also executed.

The Morse–Bott–Kirwan condition is local

Kirwan identified a condition on a smooth function under which the usual techniques of Morse-Bott theory can be applied to this function. We prove that if a function satisfies this condition locally then it also satisfies the condition globally. As an application, we use the local normal form theorem to recover Kirwan's result that the norm-square of a momentum map satisfies Kirwan's condition.

On Paraconsistent Weak Kleene Logic: Axiomatisation and Algebraic Analysis

Paraconsistent Weak Kleene logic (PWK) is the 3-valued logic with two designated values defined through the weak Kleene tables. This paper is a first attempt to investigate PWK within the perspective and methods of abstract algebraic logic (AAL). We give a Hilbert-style system for PWK and prove a normal form theorem. We examine some algebraic structures for PWK, called involutive bisemilattices, showing that they are distributive as bisemilattices and that they form a variety, $${\mathcal{IBSL}}$$IBSL, generated by the 3-element algebra WK; we also prove that every involutive bisemilattice is representable as the Plonka sum over a direct system of Boolean algebras. We then study PWK from the viewpoint of AAL. We show that $${\mathcal{IBSL}}$$IBSL is not the equivalent algebraic semantics of any algebraisable logic and that PWK is neither protoalgebraic nor selfextensional, not assertional, but it is truth-equational. We fully characterise the deductive filters of PWK on members of $${\mathcal{IBSL}}$$IBSL and the reduced matrix models of PWK. Finally, we investigate PWK with the methods of second-order AAL--we describe the class $${\mathsf{Alg}}$$Alg(PWK) of PWK-algebras, algebra reducts of basic full generalised matrix models of PWK, showing that they coincide with the quasivariety generated by WK--which differs from $${\mathcal{IBSL}}$$IBSL--and explicitly providing a quasiequational basis for it.

Approximation of small-amplitude weakly coupled oscillators by discrete nonlinear Schrödinger equations

Small-amplitude weakly coupled oscillators of the Klein–Gordon lattices are approximated by equations of the discrete nonlinear Schrödinger type. We show how to justify this approximation by two methods, which have been very popular in the recent literature. The first method relies on a priori energy estimates and multi-scale decompositions. The second method is based on a resonant normal form theorem. We show that although the two methods are different in the implementation, they produce equivalent results as the end product. We also discuss the applications of the discrete nonlinear Schrödinger equation in the context of existence and stability of breathers of the Klein–Gordon lattice.

Bifurcation analysis of a discrete S I R S ${SIRS}$ epidemic model with standard incidence rate

Discrete epidemic models are popularly used to detect the pathogenesis, spreading, and controlling of the diseases. The three-dimensional discrete ${SIRS}$ epidemic models are more suitable than the two-dimensional discrete models to describe the spreading characters of the diseases. In this paper, the complex dynamical behaviors of a three-dimensional discrete ${SIRS}$ epidemic model with standard incidence rate are discussed. We choose the time step size parameter as a bifurcation parameter, the existence, stability, and direction of Hopf bifurcation are proved by using the normal form theorem and bifurcation theory. Moreover, the numerical simulations not only illustrate our results, but they also exhibit the complex dynamical behaviors, such as the invariant cycle, period-7 orbits and period-12 orbits with more than one attractors and chaotic sets. The flip bifurcation caused by the step size parameter is also obtained by a numerical simulation. Most importantly, when the adequate contact rate and the death rate of the infective individuals are chosen as the bifurcation parameters, there also exist a Hopf bifurcation, a flip bifurcation, chaos, and strange attractors. These results provide significant information for the disease controlling when there appear complex dynamical behaviors in the epidemic model.

Engel theorem through singularities

We prove a singular version of the Engel theorem. We prove a normal form theorem for germs of holomorphic singular Engel systems with good conditions on its singular set. As an application, we prove that there exists an integral analytic curve passing through the singular points of the system. Also, we prove that a globally decomposable Engel system on a four dimensional projective space has singular set with atypical codimension.

Dynamics in a diffusive predator–prey system with a constant prey refuge and delay

In this paper, a diffusive predator–prey system with a constant prey refuge and time delay subject to Neumann boundary condition is considered. Local stability and Turing instability of the positive equilibrium are studied. The effect of time delay on the model is also obtained, including locally asymptotical stability and existence of Hopf bifurcation at the positive equilibrium. And the properties of Hopf bifurcation are determined by center manifold theorem and normal form theorem of partial functional differential equations. Some numerical simulations are carried out.

Effective Control and Bifurcation Analysis in a Chaotic System with Distributed Delay Feedback

We discuss the dynamic behavior of a new Lorenz-like chaotic system with distributed delayed feedback by the qualitative analysis and numerical simulations. It is verified that the equilibria are locally asymptotically stable whenα∈(0,α0)and unstable whenα∈(α0,∞); Hopf bifurcation occurs whenαcrosses a critical valueα0by choosingαas a bifurcation parameter. Meanwhile, the explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived by normal form theorem and center manifold argument. Furthermore, regardingαas a bifurcation parameter, we explore variation tendency of the dynamics behavior of a chaotic system with the increase of the parameter valueα.

The Impact of Hybrid Quarantine Strategies and Delay factor on Viral Prevalence in Computer Networks

Recently, the quarantine approach, which has been applied to infectious disease control, is widely regarded as an effective measure to suppress viral spread in computer networks. Hence, in order to prevent the spread of computer virus in network, and consider the latent period of a latent computer, a new delayed epidemic model of computer virus with hybrid quarantine strategies is presented. By regarding the delay as bifurcating parameter and analyzing the associated characteristic equation, the dynamical behaviors, including local asymptotical stability in which the virus spreading can be controlled, and furthermore, local Hopf bifurcation occurs in the system, which implies computer virus is out of control, are investigated. By applying the normal form and center manifold theorem, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also determined. Some numerical simulations are provided to support our theoretical results which also imply that hybrid quarantine strategies can inhibit viral spread effectively, and make the model be asymptotically stable.

The stability of nonlinear Schrödinger equations with a potential in high Sobolev norms revisited

We consider the nonlinear Schrödinger equations with a potential on $\mathbb T^d$. For almost all potentials, we show the almost global stability in very high Sobolev norms. We apply an iteration of the Birkhoff normal form, as in the formulation introduced by Bourgain [4]. This result reproves a dynamical consequence of the infinite dimensional Birkhoff normal form theorem by Bambusi and Grebert [2].

Two-sided context specifications in formal grammars

In a recent paper (M. Barash, A. Okhotin, “An extension of context-free grammars with one-sided context specifications”, Inform. and Comput., 2014), the authors introduced an extension of the context-free grammars equipped with an operator for referring to the left context of the substring being defined. This paper proposes a more general model, in which context specifications may be two-sided, that is, both the left and the right contexts can be specified by the corresponding operators. The paper gives the definitions, presents several examples of grammars and establishes a basic normal form theorem. This normal form, in particular, leads to a simple parsing algorithm working in time O(n4), where n is the length of the input string.

Hopf bifurcation and global stability of a delayed predator–prey model with prey harvesting

In this paper, we consider a delayed prey–predator model with modified Leslie–Gower term and Michaelis–Menten type prey harvesting. Regarding time delay as a bifurcation parameter, and using the normal form and center manifold theorem of functional differential equations and partial differential equations, we obtain the existence, stability and direction of periodic solutions of functional differential equations and partial differential equations, respectively. Numerical simulations are carried out to depict our theoretical analysis.