Formal Vector Fields Research Articles

Extending discrete geometric singular perturbation theory to non-hyperbolic points

Abstract We extend the recently developed discrete geometric singular perturbation theory to the non-normally hyperbolic regime. Our primary tool is the Takens embedding theorem, which provides a means of approximating the dynamics of particular maps with the time-1 map of a formal vector field. First, we show that the so-called reduced map, which governs the slow dynamics near slow manifolds in the normally hyperbolic regime, can be locally approximated by the time-1 map of the reduced vector field which appears in continuous-time geometric singular perturbation theory. In the non-normally hyperbolic regime, we show that the dynamics of fast-slow maps with a unipotent linear part can be locally approximated by the time-1 map induced by a fast-slow vector field in the same dimension, which has a nilpotent singularity of the corresponding type. The latter result is used to describe (i) the local dynamics of two-dimensional fast-slow maps with non-normally singularities of regular fold, transcritical and pitchfork type, and (ii) dynamics on a (potentially high-dimensional) local center manifold in n-dimensional fast-slow maps with regular contact or fold submanifolds of the critical manifold. In general, our results show that the dynamics near a large and important class of singularities in fast-slow maps can be described via the use of formal embedding theorems which allow for their approximation by the time-1 map of a fast-slow vector field featuring a loss of normal hyperbolicity.

Geometrical modelling of neuronal clustering and development

The dynamic geometry of neuronal development is an essential concept in theoretical neuroscience. We aimed to design a mathematical model which outlines stepwise in an innovative form and designed to model neuronal development geometrically and modelling spatially the neuronal-electrical field interaction. We demonstrated flexibility in forming the cell and its nucleus to show neuronal growth from inside to outside that uses a fractal cylinder to generate neurons (pyramidal/sphere) in form of mathematically called ‘surface of revolution’. Furthermore, we verified the effect of the adjacent neurons on a free branch from one-side, by modelling a ‘normal vector surface’ that represented a group of neurons. Our model also indicated how the geometrical shapes and clustering of the neurons can be transformed mathematically in the form of vector field that is equivalent to the neuronal electromagnetic activity/electric flux. We further simulated neuronal-electrical field interaction that was implemented spatially using Van der Pol oscillator and taking Laplacian vector field as it reflects biophysical mechanism of neuronal activity and geometrical change. In brief, our study would be considered a proper platform and inspiring modelling for next more complicated geometrical and electrical constructions.

English

We construct a class of quantum field theories depending on the data of a holomorphic Poisson structure on a piece of the underlying spacetime. The main technical tool relies on a characterization of deformations and anomalies of such theories in terms of the Gelfand-Fuchs cohomology of formal Hamiltonian vector fields. In the case that the Poisson structure is non-degenerate such theories are topological in a certain weak sense, which we refer to as de Rham topological. While the Lie algebra of translations acts in a homotopically trivial way, we will show that the space of observables of such a theory does not define an E_n-algebra. Additionally, we will highlight a conjectural relationship to theories of supergravity in four and five dimensions.

ВИЗУАЛИЗАЦИЯ РАСПРЕДЕЛЕНИЯ СПИРАЛЬНОЙ НАМАГНИЧЕННОСТИ ФЕРРОМАГНИТНОГО СЛОЯ СВЕРХПРОВОДЯЩЕГО СПИНОВОГО ВЕНТИЛЯ

In this paper, we investigate the switch of a magnet-based superconducting spin valve. As a magnet, we consider the MnSi compound characterized by a complex magnetic structure with its magnetization in the form of a spiral helicoid. A model is given for a double-layered superconducting spin valve based on the spiral magnet with an adjoining superconducting thin layer. As shown previously, critical temperature of such a superconducting film depends on the direction of the spiral magnet. Calculations have been performed for magnetic dynamics of this structure, which in prospect can serve as the basis for creating memory or logic elements of low temperature nanoelectronics. Several problems have been treated with regards to mathematical simulation of switching the spin valve magnetization, research on the spin valve structure using electromagnetic simulation software to change magnetization and rotation of the spiral magnetic vector under the magnetic-field pulse, construction and analysis of 3D magnetization distributions in spiral magnets to trace the process of remagnetization. In this work, we use a Matlab-based software environment and tools for constructing distribution plots of vector fields. It is shown that the direction of the spiral magnet can be switch by pulsed magnetic field. Research has been done on the resultant magnetization distributions visualized in the form of vector fields.

Formal Global AKSZ Gauge Observables and Generalized Wilson Surfaces

We consider a construction of observables by using methods of supersymmetric field theories. In particular, we give an extension of AKSZ-type observables constructed in Mnev (Lett Math Phys 105:1735–1783, 2015) using the Batalin–Vilkovisky structure of AKSZ theories to a formal global version with methods of formal geometry. We will consider the case where the AKSZ theory is “split” which will give an explicit construction for formal vector fields on base and fiber within the formal global action. Moreover, we consider the example of formal global generalized Wilson surface observables whose expectation values are invariants of higher-dimensional knots by using BF field theory. These constructions give rise to interesting global gauge conditions such as the differential quantum master equation and further extensions.

Invariant ideals of local analytic and formal vector fields

The objective of this paper is to analyze analytic invariant sets of analytic ordinary differential equations (ODEs). For this purpose we introduce semi-invariants and invariant ideals as well as the notion of vector fields in Poincaré-Dulac normal form (PDNF). We prove that all invariant ideals of a vector field in PDNF are already invariant for its semi-simple linear part. Additionally, this paper provides a natural characterization of invariant ideals via semi-invariants.

AST419 - Chiral Differential Operators via Batalin-Vilkovisky Quantization

We show that the local observables of the curved beta gamma system encode the sheaf of chiral differential operators using the machinery of the book Factorization algebras in quantum field theory, by Kevin Costello and the second author, which combines renormalization, the Batalin-Vilkovisky formalism, and factorization algebras. Our approach is in the spirit of deformation quantization via Gelfand-Kazhdan formal geometry. We begin by constructing a quantization of the beta gamma system with an n-dimensional formal disk as the target. There is an obstruction to quantizing equivariantly with respect to the action of formal vector fields on the target disk, and it is naturally identified with the first Pontryagin class in Gelfand-Fuks cohomology. Any trivialization of the obstruction cocycle thus yields an equivariant quantization with respect to an extension of formal vector fields by the closed 2-forms on the disk. By results in the book listed above, we then naturally obtain a factorization algebra of quantum observables, which has an associated vertex algebra easily identified with the formal beta gamma vertex algebra. Next, we introduce a version of Gelfand-Kazhdan formal geometry suitable for factorization algebras, and we verify that for a complex manifold with trivialized first Pontryagin class, the associated factorization algebra recovers the vertex algebra of CDOs on the complex manifold.

Versal normal form for nonsemisimple singularities

The theory of versal normal form has been playing a role in normal form since the introduction of the concept by V.I. Arnol'd in [1,2]. But there has been no systematic use of it that is in line with the semidirect character of the group of formal transformations on formal vector fields, that is, the linear part should be done completely first, before one computes the nonlinear terms. In this paper, we address this issue by giving a complete description of a first order calculation in the case of the two- and three-dimensional irreducible nilpotent cases, which is then followed up by an explicit almost symplectic calculation to find the transformation to versal normal form in a particular fluid dynamics problem and in the celestial mechanics L4 problem.

Periodic and Quasi-Periodic Solutions for Reversible Unbounded Perturbations of Linear Schrödinger Equations

In this paper, we consider a new class of derivative nonlinear Schrodinger equations with reversible nonlinearities of the form $$\begin{aligned} \mathrm {i}u_t+u_{xx}+|u_x|^{4}u=0,\quad (t, x)\in {\mathbb {R}}\times {\mathbb {T}}. \end{aligned}$$ We obtain real analytic, linearly stable periodic solutions and quasi-periodic ones with two basic frequencies via infinite dimensional Kolmogorov–Arnold–Moser (KAM) theory for reversible systems. By investigating the gauge invariance and the compact form of vector fields, in our KAM iterative procedure, we remove the usual Diophantine restrictions on tangential frequencies and only use the Melnikov non-resonance conditions. In the proof, we also use Birkhoff normal form techniques due to the lack of external parameters in the equation above.

Correlated electron beam microbunching and shot-noise characterization with near- and far-field optical transition radiation

In this work we demonstrate how to analyze the dynamics of the correlated longitudinal current noise above and below the Poissonian shot-noise level in an e-beam, by calculating the emitted transition radiation (OTR) using a complex vector field formulation which is exact in the near and far fields. We simulate a non-zero emittance electron beam, uncorrelated in the transverse cross section, but with different types of longitudinal correlations, resulting in coherent optical transition radiation. We show how this method is applied on the opposite case (suppression), by tracking simulated beam particles that correlate due to quarter-plasma oscillation during drift, and show how the OTR tracks the dynamics of the electron beam noise level.

Hopf-cyclic cohomology of the Connes–Moscovici Hopf algebras with infinite dimensional coefficients

We discuss a new strategy for the computation of the Hopf-cyclic cohomology of the Connes–Moscovici Hopf algebra \(\mathcal{H}_n\). More precisely, we introduce a multiplicative structure on the Hopf-cyclic complex of \(\mathcal{H}_n\), and we show that the van Est type characteristic homomorphism from the Hopf-cyclic complex of \(\mathcal{H}_n\) to the Gelfand–Fuks cohomology of the Lie algebra \(W_n\) of formal vector fields on \({\mathbb {R}}^n\) respects this multiplicative structure. We then illustrate the machinery for \(n=1\).

Dimension of the SLE Light Cone, the SLE Fan, and {{rm SLE}_kappa(rho)} for {kappa in (0,4)} and {rho in}{big[{tfrac{kappa}{2}}-4,-2big)}

Suppose that h is a Gaussian free field (GFF) on a planar domain. Fix {kappa in (0,4)}. The {{rm SLE}_kappa} light cone L{(theta)} of h with opening angle {theta in [0,pi]} is the set of points reachable from a given boundary point by angle-varying flow lines of the (formal) vector field {e^{ih/chi}}, {chi = tfrac{2}{sqrt{kappa}} - tfrac{sqrt{kappa}}{2}}, with angles in {[-tfrac{theta}{2},tfrac{theta}{2}]}. We derive the Hausdorff dimension of L{(theta)}.If {theta =0} then L{(theta)} is an ordinary {{rm SLE}_{kappa}} curve (with {kappa < 4}); if {theta = pi} then L{(theta)} is the range of an {{rm SLE}_{kappa'}} curve ({kappa' = 16/kappa > 4}). In these extremes, this leads to a new proof of the Hausdorff dimension formula for {{rm SLE}}.We also consider {{rm SLE}_kappa(rho)} processes, which were originally only defined for {rho > -,2}, but which can also be defined for {rho leq -2} using Lévy compensation. The range of an {{rm SLE}_kappa(rho)} is qualitatively different when {rho leq -2}. In particular, these curves are self-intersecting for {kappa < 4} and double points are dense, while ordinary {{rm SLE}_kappa} is simple. It was previously shown (Miller and Sheffield in Gaussian free field light cones and {{rm SLE}_kappa(rho)}, 2016) that certain {{rm SLE}_kappa(rho)} curves agree in law with certain light cones. Combining this with other known results, we obtain a general formula for the Hausdorff dimension of {{rm SLE}_kappa(rho)} for all values of {rho}.Finally, we show that the Hausdorff dimension of the so-called {{rm SLE}_kappa}fan is the same as that of ordinary {{rm SLE}_kappa}.

Orbital Linearization of Smooth Completely Integrable Vector Fields

The main purpose of this paper is to prove the smooth local orbital linearization theorem for smooth vector fields which admit a complete set of first integrals near a nondegenerate singular point. The main tools used in the proof of this theorem are the formal orbital linearization theorem for formal integrable vector fields, the blowing-up method, and the Sternberg-Chen isomorphism theorem for formally-equivalent smooth hyperbolic vector fields.

Specific features of the flow structure in a reactive type turbine stage

The results of experimental studies of the gas dynamics for a reactive type turbine stage are presented. The objective of the studies is the measurement of the 3D flow fields in reference cross sections, experimental determination of the stage characteristics, and analysis of the flow structure for detecting the sources of kinetic energy losses. The integral characteristics of the studied stage are obtained by averaging the results of traversing the 3D flow over the area of the reference cross sections before and behind the stage. The averaging is performed using the conservation equations for mass, total energy flux, angular momentum with respect to the axis z of the turbine, entropy flow, and the radial projection of the momentum flux equation. The flow parameter distributions along the channel height behind the stage are obtained in the same way. More thorough analysis of the flow structure is performed after interpolation of the experimentally measured point parameter values and 3D flow velocities behind the stage. The obtained continuous velocity distributions in the absolute and relative coordinate systems are presented in the form of vector fields. The coordinates of the centers and the vectors of secondary vortices are determined using the results of point measurements of velocity vectors in the cross section behind the turbine stage and their subsequent interpolation. The approach to analysis of experimental data on aerodynamics of the turbine stage applied in this study allows one to find the detailed space structure of the working medium flow, including secondary coherent vortices at the root and peripheral regions of the air-gas part of the stage. The measured 3D flow parameter fields and their interpolation, on the one hand, point to possible sources of increased power losses, and, on the other hand, may serve as the basis for detailed testing of CFD models of the flow using both integral and local characteristics. The comparison of the numerical and experimental results, as regards local characteristics, using statistical methods yields the quantitative estimate of their agreement.

Local normal vector field formulation for periodic scattering problems formulated in the spectral domain.

We present two adapted formulations, one tailored to isotropic media and one for general anisotropic media, of the normal vector field framework previously introduced to improve convergence near arbitrarily shaped material interfaces in spectral simulation methods for periodic scattering geometries. The adapted formulations enable the definition and generation of the normal vector fields to be confined to a region of prolongation that includes the material interfaces but is otherwise limited. This allows for a more flexible application of geometrical transformations like rotation and translation per scattering object in the unit cell. Moreover, these geometrical transformations enable a cut-and-connect strategy to compose general geometries from elementary building blocks. The entire framework gives rise to continuously parameterized geometries.

Stability data, irregular connections and tropical curves

We study a class of meromorphic connections ∇(Z) on P1, parametrised by the central charge Z of a stability condition, with values in a Lie algebra of formal vector fields on a torus. Their definition is motivated by the work of Gaiotto, Moore and Neitzke on wall-crossing and three-dimensional field theories. Our main results concern two limits of the families ∇(Z) as we rescale the central charge Z → RZ. In the R → 0 conformal limit we recover a version of the connections introduced by Bridgeland and Toledano Laredo (and so the Joyce holomorphic generating functions for enumerative invariants), although with a different construction yielding new explicit formulae. In the R → ∞ large complex structure limit the connections ∇(Z) make contact with the Gross-Pandharipande-Siebert approach to wall-crossing based on tropical geometry. Their flat sections display tropical behaviour, and also encode certain tropical/relative Gromov-Witten invariants.

Doubly-Resonant Saddle-Nodes in $$\mathbb {C}^{3}$$ C 3 and the Fixed Singularity at Infinity in the Painlevé Equations: Formal Classification

In this work we consider formal singular vector fields in $$\mathbb {C}^{3}$$ with an isolated and doubly-resonant singularity of saddle-node type at the origin. Such vector fields come from irregular two-dimensional systems with two opposite non-zero eigenvalues, and appear for instance when studying the irregular singularity at infinity in Painleve equations , $$j\in \left\{ I,II,III,IV,V\right\} $$ , for generic values of the parameters. Under generic assumptions we give a complete formal classification for the action of formal diffeomorphisms (by changes of coordinates) fixing the origin and fibered in the independent variable x. We also identify all formal isotropies (self-conjugacies) of the normal forms. In the particular case where the flow preserves a transverse symplectic structure, e.g. for Painleve equations, we prove that the normalizing map can be chosen to preserve the transverse symplectic form.

Intersections of SLE Paths: the double and cut point dimension of SLE

We compute the almost-sure Hausdorff dimension of the double points of chordal mathrm {SLE}_kappa for kappa > 4, confirming a prediction of Duplantier–Saleur (1989) for the contours of the FK model. We also compute the dimension of the cut points of chordal mathrm {SLE}_kappa for kappa > 4 as well as analogous dimensions for the radial and whole-plane mathrm {SLE}_kappa (rho ) processes for kappa > 0. We derive these facts as consequences of a more general result in which we compute the dimension of the intersection of two flow lines of the formal vector field e^{ih/chi }, where h is a Gaussian free field and chi > 0, of different angles with each other and with the domain boundary.

Formal Connections for Families of Star Products

We define the notion of a formal connection for a smooth family of star products with fixed underlying symplectic structure. Such a formal connection allows one to relate star products at different points in the family. This generalizes the formal Hitchin connection, which was introduced by the first author. We establish a necessary and sufficient condition that guarantees the existence of a formal connection, and we describe the space of formal connections for a family as an affine space modelled on the formal symplectic vector fields. Moreover, we showthat if the parameter space has trivial first cohomology group, any two flat formal connections are related by an automorphism of the family of star products.

CHARACTERISTIC CLASSES OF FLAGS OF FOLIATIONS AND LIE ALGEBRA COHOMOLOGY

We prove the conjecture by Feigin, Fuchs and Gelfand describing the Lie algebra cohomology of formal vector fields on an $n$-dimensional space with coefficients in symmetric powers of the coadjoint representation. We also compute the cohomology of the Lie algebra of formal vector fields that preserve a given flag at the origin. The latter encodes characteristic classes of flags of foliations and was used in the formulation of the local Riemann-Roch Theorem by Feigin and Tsygan. Feigin, Fuchs and Gelfand described the first symmetric power and to do this they had to make use of a fearsomely complicated computation in invariant theory. By the application of degeneration theorems of appropriate Hochschild-Serre spectral sequences we avoid the need to use the methods of FFG, and moreover we are able to describe all the symmetric powers at once.