Arbitrary Finite Order Research Articles

On Two-Dimensional Diffeomorphisms with Homoclinic Orbits to Nonhyperbolic Fixed Points

We consider two-dimensional diffeomorphisms with homoclinic orbits to nonhyperbolic fixed points. We assume that the point has arbitrary finite order degeneracy and is either of saddle-node or weak saddle type. We consider two cases when the homoclinic orbit is transversal and when a quadratic homoclinic tangency takes place. In the first case we give a complete description of orbits entirely lying in a small neighborhood of the homoclinic orbit. In the second case we study main bifurcations in one-parameter families that split generally the homoclinic tangency but retain the degeneracy type of the fixed point.

Weak solvability of the initial-boundary value problem for inhomogeneous incompressible Kelvin–Voigt fluid motion model of arbitrary finite order

Weak solvability of the initial-boundary value problem for inhomogeneous incompressible Kelvin–Voigt fluid motion model of arbitrary finite order

An improved estimator of Shannon entropy with applications to systems with memory

We investigate the memory properties of discrete sequences built upon a finite number of states. We find that the block entropy can reliably determine the memory for systems modeled as Markov chains of arbitrary finite order. Further, we provide an entropy estimator that remarkably gives accurate results when correlations are present. To illustrate our findings, we calculate the memory of daily precipitation series at different locations. Our results are in agreement with existing methods being at the same time valid in the undersampled regime and independent of model selection.

О fc-ниль-хороших кольцах формальных матриц

In 2018, Abdolyusefi, Ashrafi, and Chen gave a definition of a 2-nil-good ring element in their work, generalizing the notion of a graceful ring element introduced two years earlier by Kalugeryan and Lam, as well as the definition of a 2-nil-good ring. In the same work, it was shown that the Morita context ring, i.e. a formal matrix ring of the second order is 2-nil-good if the rings over which it is considered are themselves 2-nil-good. In this paper, we generalize further, defining k-nil-good elements and k-nil-good rings, and state a condition under which a formal matrix ring of an arbitrary finite order is k-nil-good.

Inference for partially linear additive higher-order spatial autoregressive model with spatial autoregressive error and unknown heteroskedasticity

This article extends spatial autoregressive model with spatial autoregressive disturbances (SARAR(1,1)) which is the most popular spatial econometric model to the case of an arbitrary finite number of nonparametric additive terms and spatial autoregressive models with spatial autoregressive disturbances of arbitrary finite order (SARAR(R,S)). We propose a sieve two-stage least squares (S2SLS) regression and generalized method of moments (GMM) procedure of the high-order spatial autoregressive parameters of the disturbance process. Under some sufficient conditions, we show that the proposed estimator for the finite dimensional parameter is consistent and asymptotically normally distributed. We show that each proposed estimator for the additive terms is consistent and also asymptotically normally distributed at a rate slower than Consistent estimators for the asymptotic variances of the proposed estimators are provided. In addition, using asymptotic properties to make statistical inference for the parametric and additive is also considered. Monte Carlo evidence suggests that the estimation procedure performs reasonably well in small samples and the proposed approach has some practical value. The proposed method is applied to analyzing factors which affect haze pollution.

Glassy dynamics from generalized mode-coupling theory: existence and uniqueness of solutions for hierarchically coupled integro-differential equations

Generalized mode-coupling theory (GMCT) is a first-principles-based and systematically correctable framework to predict the complex relaxation dynamics of glass-forming materials. The formal theory amounts to a hierarchy of infinitely many coupled integro-differential equations, which may be approximated using a suitable finite-order closure relation. Although previous studies have suggested that finite-order GMCT leads to well-defined solutions, and that the hierarchy converges as the closure level increases, no rigorous and general result in this direction is known. Here we unambiguously establish the existence and uniqueness of solutions to generic, schematic GMCT hierarchies that are closed at arbitrary finite order. We consider two types of commonly invoked closure approximations, namely mean-field and exponential closures. We also distinguish explicitly between overdamped and underdamped glassy dynamics, corresponding to hierarchies of first-order and second-order integro-differential equations, respectively. We find that truncated GMCT hierarchies closed under an exponential closure conform to previously developed mathematical theories, both in the overdamped and underdamped case, such that the existence of a unique solution can be readily inferred. Self-consistent mean-field closures, however, of which the well-known standard-MCT closure approximation is a special case, warrant additional arguments for mathematical rigor. We demonstrate that the existence of a priori bounds on the solution is sufficient to also prove that unique solutions exist for such self-consistent hierarchies. To complete our analysis, we present simple arguments to show that these a priori bounds must exist, motivated by the physical interpretation of the GMCT solutions as density correlation functions. Overall, our work contributes to the theoretical justification of GMCT for studies of the glass transition, placing this hierarchical framework on a firmer mathematical footing.

Conservation laws and stability of higher derivative extended Chern-Simons

The higher derivative field theories are notorious for the stability problems both at classical and quantum level. Classical instability is connected with unboundedness of the canonical energy, while the unbounded energy spectrum leads to the quantum instability. For a wide class of higher derivative theories, including the extended Chern-Simons, other bounded conserved quantities which provide the stability can exist. The most general gauge invariant extended Chern-Simons theory of arbitrary finite order n admits (n − 1)-parameter series of conserved energy-momentum tensors. If the 00-component of the most general representative of this series is bounded, the theory is stable. The stability condition requires from the free extended Chern-Simons theory to describe the unitary reducible representation of the Poincaré group. The unstable theory corresponds to nonunitary representation.

On the transient (T) condition for random walk in mixing environment

We prove a ballistic strong law of large numbers and an invariance principle for random walks in strong mixing environments, under condition $(T)$ of Sznitman (cf. Ann. Probab. 29 (2001) 724–765). This weakens for the first time Kalikow’s ballisticity assumption on mixing environments and proves the existence of arbitrary finite order moments for the approximate regeneration time of F. Comets and O. Zeitouni (Israel J. Math. 148 (2005) 87–113). The main technical tool in the proof is the introduction of renormalization schemes, which had only been considered for i.i.d. environments.

Stable Interactions Between the Extended Chern-Simons Theory and a Charged Scalar Field with Higher Derivatives: Hamiltonian Formalism

The constrained Hamiltonian formalism for the extended higher derivative Chern–Simons theory of an arbitrary finite order is considered. It is shown that the n-th order theory admits an (n–1)-parametric series of conserved tensors. It is clarified that this theory admits a series of canonically non-equivalent Hamiltonian formulations, where a zero-zero component of any conserved tensor can be chosen as a Hamiltonian. The canonical Ostrogradski Hamiltonian is included into this series. An example of interactions with a charged scalar field is also given, which preserve the selected representative of the series of Hamiltonian formulations.

Testing for serial independence in vector autoregressive models

We consider tests for serial independence of arbitrary finite order for the innovations in vector autoregressive models. The tests are expressed as L2-type criteria involving the difference of the joint empirical characteristic function and the product of corresponding marginals. Asymptotic as well as Monte-Carlo results are presented.

SENR/NRPy+ : Numerical relativity in singular curvilinear coordinate systems

We report on a new open-source, user-friendly numerical relativity code package called SENR/NRPy+. Our code extends previous implementations of the BSSN reference-metric formulation to a much broader class of curvilinear coordinate systems, making it ideally-suited to modeling physical configurations with approximate or exact symmetries. In the context of modeling black hole dynamics, it is orders of magnitude more efficient than other widely used open-source numerical relativity codes. NRPy+ provides a Python-based interface in which equations are written in natural tensorial form and output at arbitrary finite difference order as highly efficient C code, putting complex tensorial equations at the scientist's fingertips without the need for an expensive software license. SENR provides the algorithmic framework that combines the C codes generated by NRPy+ into a functioning numerical relativity code. We validate against two other established, state-of-the-art codes, and achieve excellent agreement. For the first time we demonstrate--in the context of puncture, trumpet, and dual black hole evolutions--nearly exponential convergence of constraint violation and gravitational waveform errors to zero as the order of spatial finite difference derivatives is increased, while holding the spherical-like coordinate grids fixed at moderate resolution. Such behavior outside the horizons is remarkable, as numerical errors do not converge to zero inside horizons, and all points along the polar axis are coordinate singularities. The formulation addresses such coordinate singularities via cell-centered grids and a simple change of basis that analytically regularizes tensor components with respect to the coordinates. Future plans include extending this formulation to allow dynamical coordinate grids and bispherical-like distribution of points to efficiently capture orbiting compact binary dynamics.

Iterative Convex Overbounding Algorithms for BMI Optimization Problems

This paper considers Iterative Convex Overbounding (ICO) techniques for Bilinear Matrix Inequality (BMI) problems. It is very common for BMIs to be present in multi-objective control, as well as many other optimization problems. Theoretically, ICO techniques guarantee monotonic convergence to a local optimum, and do not require the introduction of conservatism or relaxations. In this work, we propose an update to ICO which allows for improved results and a new convergence path. We also illustrate that ICO techniques are extensible to problems in which initial feasible design points are not known. Finally, we illustrate that ICO can be extended to matrix polynomial inequalities of arbitrary finite order. These ideas are demonstrated on a simple example.

On bifurcation delay: An alternative approach using Geometric Singular Perturbation Theory

To explain the phenomenon of bifurcation delay, which occurs in planar systems of the form x˙=ϵf(x,z,ϵ), z˙=g(x,z,ϵ)z, where f(x,0,0)>0 and g(x,0,0) changes sign at least once on the x-axis, we use the Exchange Lemma in Geometric Singular Perturbation Theory to track the limiting behavior of the solutions. Using the trick of extending dimension to overcome the degeneracy at the turning point, we show that the limiting attracting and repulsion points are given by the well-known entry-exit function, and the minimum of z on the trajectory is of order exp⁡(−1/ϵ). Also we prove smoothness of the return map up to arbitrary finite order in ϵ.

Geometry of Higgs bundles over elliptic curves related to automorphisms of simple Lie algebras, Calogero–Moser systems, and KZB equations

We construct twisted Calogero-Moser (CM) systems with spins as the Hitchin systems derived from the Higgs bundles over elliptic curves, where transitions operators are defined by an arbitrary finite order automorphisms of the underlying Lie algebras. In this way we obtain the spin generalization of the twisted D'Hoker- Phong and Bordner-Corrigan-Sasaki-Takasaki systems. As by product, we construct the corresponding twisted classical dynamical r-matrices and Knizhnik-Zamolodchikov-Bernard equations related to the automorphisms of the Lie algebras.

Twisted Γ-Lie algebras and their vertex operator representations

Let Γ be a generic subgroup of the multiplicative group C⁎ of nonzero complex numbers. We define a class of Lie algebras associated to Γ, called twisted Γ-Lie algebras, which are natural generalizations of the twisted affine Lie algebras. Starting from an arbitrary even sublattice Q of ZN and an arbitrary finite order isometry of ZN preserving Q, we construct a family of twisted Γ-vertex operators acting on generalized Fock spaces which afford irreducible representations for certain twisted Γ-Lie algebras. As an application, this recovers a number of known vertex operator realizations for infinite dimensional Lie algebras, such as twisted affine Lie algebras, extended affine Lie algebras of type A, trigonometric Lie algebras of series A and B, unitary Lie algebras, and BC-graded Lie algebras.

A New Asymptotic Perturbation Theory with Applications to Models of Massless Quantum Fields

Let H 0 and H I be a self-adjoint and a symmetric operator on a complex Hilbert space, respectively, and suppose that H 0 is bounded below and the infimum E 0 of the spectrum of H 0 is a simple eigenvalue of H 0 which is not necessarily isolated. In this paper, we present a new asymptotic perturbation theory for an eigenvalue E(λ) of the operator $${H(\lambda)\,:=\,H_0 + \lambda H_{I}\,(\lambda \in \mathbb{R} \setminus \{0\})}$$ satisfying lim λ → 0 E(λ) = E 0. The point of the theory is in that it covers also the case where E 0 is a non-isolated eigenvalue of H 0. Under a suitable set of assumptions, we derive an asymptotic expansion of E(λ) up to an arbitrary finite order of λ as λ → 0. We apply the abstract results to a model of massless quantum fields, called the generalized spin-boson model (Arai and Hirokawa in J Funct Anal 151:455–503, 1997) and show that the ground-state energy of the model has asymptotic expansions in the coupling constant λ as λ → 0.

Symplectic automorphisms of K3 surfaces of arbitrary finite order

Symplectic automorphisms of K3 surfaces of arbitrary finite order

On the equivalence of functional bases of differential invariants for nonconjugate subgroups of the local Lie groups of point transformations

We formulate and prove a criterion of the equivalence of functional bases of differential invariants of an arbitrary finite order k for nonconjugate subgroups of the local Lie groups of point transformations.

An Explicit Approach to Modeling Finite-Order Type Spaces and Applications

Author(s): Qin, Cheng-Zhong; Yang, Chun-Lei | Abstract: Every abstract type of a belief-closed type space corresponds to an infinite belief hierarchy. But only finite order of beliefs is necessary for most applications. As we demonstrate, many important insights from recent development in the theory of Bayesian games with higher-order uncertainty involve belief hierarchies of order 2.We start with characterizing order 2 and show that they form a convex set and contain the convex hull of both the naive and complete-information type spaces. We establish conditions for private-value heterogeneous naive priors to be embedded in order-2 consistent priors, so as to retro-fit the Harsanyi doctrine of having nature generate all fundamental uncertainties in a game at the very beginning.We then extend the notion of consistent priors to arbitrary finite order k. We define an abstract belief-closed space to be of order k if it can be mapped via a type morphism into the canonical representation of an order-k consistent prior. We show that order-k type spaces are those in which any two types of each player must be either identical implying one of them is redundant or separable by their order (k-1) belief hierarchies. Finite type spaces are always of finite orders. We consider or a type space and show that they are finite-order type spaces themselves. The condition of global stability under uncertainty ensures the convergence of the Bayesian-Nash equilibria with the projection type spaces to those with the original type space.By defining a total variation norm based on finite-order projections, we generalize Kajii and Morris's (1997) idea of equilibrium robustness to Bayesian games. We then establish the robustness of Bayesian-Nash equilibria that generalizes the robustness results of Monderer and Samet (1989) for complete-information games.We apply our framework of finite-order type spaces or consistent priors to review several important models in the literature and illustrate some new insights.

Stable classical groups and strongly torsion generated groups

Strongly torsion generated groups are those with a single normal generator, of arbitrary finite order. They are useful for realizing sequences of abelian groups as homology groups. Known examples include stable alternating groups and stable groups generated by elementary matrices. Here the class of such groups is extended, by consideration of other stable classical groups, including orthogonal and symplectic groups. Discussion of other