Normal Form Results Research Articles

Normal form of equivariant maps in infinite dimensions

Local normal form theorems for smooth equivariant maps between infinite-dimensional manifolds are established. These normal form results are new even in finite dimensions. The proof is inspired by the Lyapunov–Schmidt reduction for dynamical systems and by the Kuranishi method for moduli spaces. It uses a slice theorem for Fréchet manifolds as the main technical tool. As a consequence, the abstract moduli space obtained by factorizing a level set of the equivariant map with respect to the group action carries the structure of a Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient by a compact group of the zero set of a smooth map. The general results are applied to the moduli space of anti-self-dual instantons, the Seiberg–Witten moduli space and the moduli space of pseudoholomorphic curves.

Nonlinear vibration of axially moving simply-supported circular cylindrical shell

This paper investigates the nonlinear free and forced vibration of axially moving simply-supported thin circular cylindrical shells in the sub-harmonic region, considering the effect of viscous structure damping. Motion equations of axially moving circular cylindrical shells in cylindrical coordinates are derived with the aid of the Hamilton principle and employing the Donnell–Mushtari nonlinear shells’ theory. Three nonhomogeneous nonlinear partial differential equilibrium equations concerning displacements across the three directions of cylindrical coordinate are reduced to two equations as a result of using the definition of the airy function. The particular and private solution of airy stress function is obtained utilizing the assumption of the form of the displacement in the radial direction based on the simply-supported boundary conditions and combination of axisymmetric and asymmetric driven and companion modes. By gaining the airy function, the third motion equation is discretized using the Galerkin method into the set of coupled nonlinear nonhomogeneous ordinary differential equations. The perturbation method and Runge–Kutta 4th order are employed as a semi-analytical and a numerical solution method, which in practice leads to the prediction of the nonlinear frequencies and amplitudes of various modes of vibration at different velocities and external excitation fluctuation domain and frequency. The bifurcation analysis of the velocity and external excitation parameters in sub-harmonic regions shows the changes in the instability condition of the system. It causes the appearance of the pitchfork bifurcation and the Neimark–Sacker bifurcation points. The accuracy of both, the direct normal form and numerical method results are compared against each other and validated in the particular case in the absence of the velocity with available data. The system would be more stable at a higher speed near 1:1 external resonance due to the activation of more asymmetric vibration modes and the growth of the nonlinear softening characteristic in the absence of companion vibration modes.

First-order concatenation theory with bounded quantifiers

We study first-order concatenation theory with bounded quantifiers. We give axiomatizations with interesting properties, and we prove some normal-form results. Finally, we prove a number of decidability and undecidability results.

Questions and Dependency in Intuitionistic Logic

In recent years, the logic of questions and dependencies has been investigated in the closely related frameworks of inquisitive logic and dependence logic. These investigations have assumed classical logic as the background logic of statements, and added formulas expressing questions and dependencies to this classical core. In this paper, we broaden the scope of these investigations by studying questions and dependency in the context of intuitionistic logic. We propose an intuitionistic team semantics, where teams are embedded within intuitionistic Kripke models. The associated logic is a conservative extension of intuitionistic logic with questions and dependence formulas. We establish a number of results about this logic, including a normal form result, a completeness result, and translations to classical inquisitive logic and modal dependence logic.

Vector potential normal form classification for completely integrable solenoidal nilpotent singularities

We introduce an sl2-invariant family of polynomial vector fields with an irreducible nilpotent singularity. In this paper, we are concerned with characterization and normal form classification of these vector fields. We show that the family is a Lie subalgebra and each vector field from this family is volume-preserving, completely integrable, and rotational. All such vector fields share a common quadratic invariant. We provide a Poisson structure for the Lie subalgebra from which the second invariant for each vector field can be readily derived. We show that each vector field from this family can be uniquely characterized by two alternative representations which can be found in applications: one uses a vector potential while the other uses two functionally independent Clebsch potentials. Our normal form results are designed to preserve these structures and representations.

Deformations of Lie Groupoids

AbstractWe study deformations of Lie groupoids by means of the cohomology which controls them. This cohomology turns out to provide an intrinsic model for the cohomology of a Lie groupoid with values in its adjoint representation. We prove several fundamental properties of the deformation cohomology including Morita invariance, a van Est theorem, and a vanishing result in the proper case. Combined with Moser’s deformation arguments for groupoids, we obtain several rigidity and normal form results.

Long time stability of small finite gap solutions of the cubic nonlinear Schrödinger equation on [formula omitted

In this paper we study long time stability of a class of nontrivial, quasi-periodic solutions depending on one spacial variable of the cubic defocusing non-linear Schrödinger equation on the two dimensional torus. We prove that these quasi-periodic solutions are orbitally stable for finite but long times, provided that their Fourier support and their frequency vector satisfy some complicated but explicit condition, which we show holds true for most solutions.The proof is based on a normal form result. More precisely we expand the Hamiltonian in a neighborhood of a quasi-periodic solution, we reduce its quadratic part to diagonal constant coefficients through a KAM scheme, and finally we remove its cubic terms with a step of nonlinear Birkhoff normal form. The main difficulty is to impose second and third order Melnikov conditions; this is done by combining the techniques of reduction in order of pseudo-differential operators with the algebraic analysis of resonant quadratic Hamiltonians.

Long time stability of small-amplitude Breathers in a mixed FPU-KG model

In the limit of small couplings in the nearest neighbor interaction, and small total energy, we apply the resonant normal form result of a previous paper of ours to a finite but arbitrarily large mixed Fermi–Pasta–Ulam Klein–Gordon chain, i.e., with both linear and nonlinear terms in both the on-site and interaction potential, with periodic boundary conditions. An existence and orbital stability result for Breathers of such a normal form, which turns out to be a generalized discrete nonlinear Schrodinger model with exponentially decaying all neighbor interactions, is first proved. Exploiting such a result as an intermediate step, a long time stability theorem for the true Breathers of the KG and FPU–KG models, in the anti-continuous limit, is proven.

Tensors, !-graphs, and non-commutative quantum structures

Categorical quantum mechanics (CQM) and the theory of quantum groups rely heavily on the use of structures that have both an algebraic and co-algebraic component, making them well-suited for manipulation using diagrammatic techniques. Diagrams allow us to easily form complex compositions of (co)algebraic structures, and prove their equality via graph rewriting. One of the biggest challenges in going beyond simple rewriting-based proofs is designing a graphical language that is expressive enough to prove interesting properties (e.g. normal form results) about not just single diagrams, but entire families of diagrams. One candidate is the language of !-graphs, which consist of graphs with certain subgraphs marked with boxes (called !-boxes) that can be repeated any number of times. New !-graph equations can then be proved using a powerful technique called !-box induction. However, previously this technique only applied to commutative (or cocommutative) algebraic structures, severely limiting its applications in some parts of CQM and (especially) quantum groups. In this paper, we fix this shortcoming by offering a new semantics for non-commutative !-graphs using an enriched version of Penrose's abstract tensor notation.

Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part I. Finite-dimensional discretization

We consider discretized Hamiltonian PDEs associated with a Hamiltonian function that can be split into a linear unbounded operator and a regular nonlinear part. We consider splitting methods associated with this decomposition. Using a finite-dimensional Birkhoff normal form result, we show the almost preservation of the actions of the numerical solution associated with the splitting method over arbitrary long time and for asymptotically large level of space approximation, provided the Sobolev norm of the initial data is small enough. This result holds under generic non-resonance conditions on the frequencies of the linear operator and on the step size. We apply these results to nonlinear Schrödinger equations as well as the nonlinear wave equation.

Tangency Bifurcations of Global Poincaré Maps

One tool to analyze the qualitative behavior of a periodic orbit of a vector field in $\mathbb{R}^n$ is to consider the Poincaré return map to an $(n-1)$-dimensional section. The image under the Poincaré map of a point on this section that lies near the periodic orbit is obtained by following the flow of the vector field until the next (local) intersection. It is well known that the Poincaré map defined on a section transverse to a periodic orbit is a diffeomorphism locally near the periodic orbit. However, in practice one often considers the Poincaré map not only locally but also on a much larger global and typically unbounded section. Generically, there are then points where the flow is tangent to the section, and these give rise to discontinuities of the Poincaré map. In fact, the orbits of some points may not even return to the section, in which case the Poincaré map is not defined at all. In this paper we study tangency bifurcations of invariant manifolds of Poincaré maps on global sections of vector fields in $\mathbb{R}^2$ and $\mathbb{R}^3$. At such a bifurcation the manifold becomes tangent to the section, which results in a qualitative change of the Poincaré map while the underlying flow itself does not undergo a bifurcation. Using tools from singularity theory, we present normal forms of the codimension-one tangency bifurcations in the neighborhood of the respective tangency point. The study of these bifurcations is motivated by and illustrated with the examples of the (unforced) Van der Pol oscillator and a system modeling a semiconductor laser with optical injection. Finally, we present a framework for the generalization of our normal-form results to arbitrary dimension and codimension.

Improving the lazy Krivine machine

Krivine presents the $\mathcal {K}$ machine, which produces weak head normal form results. Sestoft introduces several call-by-need variants of the ${\mathcal {K}}$ machine that implement result sharing via pushing update markers on the stack in a way similar to the TIM and the STG machine. When a sequence of consecutive markers appears on the stack, all but the first cause redundant updates. Improvements related to these sequences have dealt with either the consumption of the markers or the removal of the markers once they appear. Here we present an improvement that eliminates the production of marker sequences of length greater than one. This improvement results in the ${\mathcal {C}}$ machine, a more space and time efficient variant of ${\mathcal {K}}$ . We then apply the classic optimization of short-circuiting operand variable dereferences to create the call-by-need ${\mathcal {S}}$ machine. Finally, we combine the two improvements in the ${\mathcal {CS}}$ machine. On our benchmarks this machine uses half the stack space, performs one quarter as many updates, and executes between 27% faster and 17% slower than our ? variant of Sestoft's lazy Krivine machine. More interesting is that on one benchmark ?, ${\mathcal {S}}$ , and ${\mathcal {C}}$ consume unbounded space, but ${\mathcal {CS}}$ consumes constant space. Our comparisons to Sestoft's Mark 2 machine are not exact, however, since we restrict ourselves to unpreprocessed closed lambda terms. Our variant of his machine does no environment trimming, conversion to deBruijn-style variable access, and does not provide basic constants, data type constructors, or the recursive let. (The Y combinator is used instead.)

Analysis of factors affecting power system normal form results

This paper discusses the experience in the use of normal form (NF) theory to study nonlinear modal behavior in stressed power systems. Making use of NF theory, we examine the use of several analytical indices to quantify system nonlinear modal interaction and discuss major factors affecting NF results and interpretation. Specifically, the influence of the initial conditions and the effect of strong modal resonance on the normal form analysis is studied. Based on this understanding, heuristic criteria for guessing initial conditions area devised for assessing model adequacy. New insights into the nature of system oscillations are offered and criteria for characterizing nonlinear effects are discussed. The analysis of factors affecting NF results is illustrated on a four-machine two-area test power network. Numerical calculations are presented and discussed to check the accuracy of the method analysis.

Bifurcations of periodic orbits with spatio-temporal symmetries

Motivated by recent analytical and numerical work on two- and three-dimensional convection with imposed spatial periodicity, we analyse three examples of bifurcations from a continuous group orbit of spatio-temporally symmetric periodic solutions of partial differential equations. Our approach is based on centre manifold reduction for maps, and is in the spirit of earlier work by Iooss (1986) on bifurcations of group orbits of spatially symmetric equilibria. Two examples, two-dimensional PWs and three-dimensional alternating pulsating waves (APW), have discrete spatio-temporal symmetries characterized by the cyclic groups , n = 2 (PW) and n = 4 (APW). These symmetries force the Poincaré return map to be the nth iterate of a map : . The group orbits of PW and APW are generated by translations in the horizontal directions and correspond to a circle and a two-torus, respectively. An instability of PWs can lead to solutions that drift along the group orbit, while bifurcations with Floquet multiplier (FM) +1 of APWs do not lead to drifting solutions. The third example we consider, alternating rolls, has the spatio-temporal symmetry of APWs as well as being invariant under reflections in two vertical planes. This leads to the possibility of a doubling of the marginal FM and of bifurcation to two distinct types of drifting solutions. We conclude by proposing a systematic way of analysing steady-state bifurcations of periodic orbits with discrete spatio-temporal symmetries, based on applying the equivariant branching lemma to the irreducible representations of the spatio-temporal symmetry group of the periodic orbit, and on the normal form results of Lamb J S W (1996 Local bifurcations in k-symmetric dynamical systems Nonlinearity 9 537-57). This general approach is relevant to other pattern formation problems, and contributes to our understanding of the transition from ordered to disordered behaviour in pattern-forming systems.

Relativized logspace and generalized quantifiers over finite ordered structures

AbstractWe here examine the expressive power of first order logic with generalized quantifiers over finite ordered structures. In particular, we address the following problem: Given a familyQof generalized quantifiers expressing a complexity classC, what is the expressive power of first order logic FO(Q) extended by the quantifiers inQ? From previously studied examples, one would expect that FO(Q) capturesLC, i.e., logarithmic space relativized to an oracle inC. We show that this is not always true. However, after studying the problem from a general point of view, we derive sufficient conditions onCsuch that FO(Q) capturesLC. These conditions are fulfilled by a large number of relevant complexity classes, in particular, for example, byNP. As an application of this result, it follows that first order logic extended by Henkin quantifiers capturesLNP. This answers a question raised by Blass and Gurevich [Ann. Pure Appl. Logic, vol. 32, 1986]. Furthermore we show that for many familiesQof generalized quantifiers (including the family of Henkin quantifiers), each FO(Q)-formula can be replaced by an equivalent FO(Q)-formula with only two occurrences of generalized quantifiers. This generalizes and extends an earlier normal-form result by I. A. Stewart [Fundamenta Inform, vol. 18, 1993].

THE STRUCTURE AND COMPLEXITY OF MINIMAL NFA’S OVER A UNARY ALPHABET

Many difficult open problems in theoretical computer science center around non-determinism. We study the fundamental problem of converting a given deterministic finite automaton (DFA) to a minimal nondeterministic finite automaton (NFA). Despite extensive work on finite automata, this fundamental problem has remained open. Recently, we studied this problem and showed that this (and related) problems are computationally hard.11 Herewe study the restriction of this problem to the case when the input DFA is over a one-letter alphabet. Even in this restricted case the problem is computationally hard even though our evidence of hardness is different from (and is weaker than) the standard ones such as NP-hardness. Let A → B denote the problem of converting a finite automaton of type A to a minimal finite automaton of type B. Our main result is that DFA → NFA , when the input is a unary cyclic DFA (a DFA whose graph is a simple cycle), is in NP but not in P unless NP ⊆ DTIME (nO( log n)). Our work was also motivated by the problem of finding structurally simple “normal forms” of NFA’s over a unary alphabet. We present some normal forms for minimal NFA’s over a unary alphabet and present an application to lower bounds on the size complexity of an NFA. In fact, the normal form result is used in a nontrivial manner to show the NP membership result stated above.

Subset languages of Petri nets part I: The relationship to string languages and normal forms

In this paper we are interested in language theoretical properties of Petri nets. However, in this research we consider a step to consist of concurrent firing of a set of transitions. In this way a firing sequence will be a sequence of subsets of the set of all transitions. The set of all such subset firing sequences (or the set of all those subset firing sequences that lead to one of the finite number of final markings) forms the subset language of a given net (to consider such a set to be a language one has to consider subsets to be letters of the alphabet used to record behaviour of nets). The aim of the paper is to present a systematic research exploring the above point of view. First of all we point out several differences between the standard ‘string language’ approach and the ‘subset language’ approach. Then we proceed to the investigation of properties of Petri nets as generators of subset languages. We provide here some normal form results (and notice that those results and their proofs are quite different from the case when one considers Petri nets as generators of string languages).

On finite grammar forms

This paper discusses finite grammar forms. In particular, results concerning successor and nonsuccessor families, as well as normal form (with respect to minimality) results are obtained. Interconnections with recent work concerning graph families are also pointed out.

On calculation of sound fields around three dimensional objects by integral equation methods

This paper is concerned with the use of integral equation methods in numerical calculations of exterior sound fields around scattering objects. The objects investigated are a rigid body with edges and vertices, a rigid plate, and an absorbing body. For each case integral equation solutions are developed, derived from Green's formula, and its normal derivative form, and the numerical results are found to agree with measurements. The method of limiting conductance is tested and it is found that the normal derivative form results show better agreement with measurements than those from the ordinary formulation. Other problems which arose in the calculations are also considered: e.g., evaluation of the normal derivatives of the double layer potential, treatment of edges and vertices of three dimensional objects, and the relative merits of the two formulations in practical applications.

On non context-free grammar forms

A grammar formF defines via a so-calledinterpretation mechanism, a family of languages (F). In this paper we establish that for many grammar formsF, (the family of context-free languages) implies (F)= RE (the family of recursively enumerable sets). We conjecture that this is also true in general. Because of this, it seems necessary to use restricted interpretations for non context-free grammar forms, a form giving then rise to a family We compare the obvious alternatives for restricting interpretations and focus attention on two promising alternatives, Q (F) and st(F) and their combination Q, st(F). Using st-interpretations, surprising families can be generated and strong normal form results can be obtained. Closure results and decidability results are also given. UnderQ, st-interpretation, it is possible to characterize a number of well-known families of languages between CF and RE, including the families of EOL, ETOL, matrix and scattered context languages.