Generic CR Research Articles

Faulty diagnostics model in e-commerce using AI

Risk administration has increased due to increased online orders to avoid sales invoices. Failure to pay a bill within 90 days of receipt constitutes overdue payment. The credit rating is used to determine the probability that consumers will fail. The CR has been thoroughly studied and several learning algorithms have been proposed. The main goal is to create a CR model for the role of Risk Solution Services (RSS), an industry standard for predicting the parameters of customer default in e-commerce risk assessment. The risk assessment included a general concept, exclusion criteria, and details of the ordering process. The most recent design should operate both independently and in conjunction with the NRC Primary Risk Audit as it is intended to replace the overall screening risk assessment. This article is about a CR implementation of Genetic Programming (GP) with Artificial Intelligence (AI). The dataset includes RSS-enabled purchase requisitions. The results show that the GP pre-risk control model goes beyond the generic CR model in terms of classification accuracy. A system with more discriminatory capacity is produced by combining GP models with the NRC's major risk assessment.

The 1-jet determination of stationary discs attached to generic CR submanifolds

The existence of a non-defective stationary disc attached to a non-degenerate model quadric in \({\mathbb {C}}^N\) is a necessary condition to ensure the unique 1-jet determination of the lifts of a key family of stationary discs [6]. In this paper, we give an elementary proof of the equivalence when the model quadric is strongly pseudoconvex, recovering a result of Tumanov [14]. Our proof is based on the explicit expression of stationary discs, and opens up a conjecture for the unique 1-jet determination to hold when the model is not necessarily strongly pseudoconvex.

Stationary discs and finite jet determination for CR mappings in higher codimension

We discuss stationary discs for generic CR manifolds and apply them to the problem of finite jet determination for CR mappings. We prove that a C2-smooth CR diffeomorphism of two C4-smooth strictly pseudoconvex Levi generating CR manifolds is uniquely determined by its 2-jet at a given point. A new key element of the proof is the existence of non-defective stationary discs.

Distributional boundary values of holomorphic functions on product domains

We show that holomorphic functions of polynomial growth on domains with corners have distributional boundary values in an appropriate sense, provided the corners are generic CR manifolds. We also prove an analog of the Bochner–Hartogs theorem for these boundary values for the simplest such domains, the product domains.

On the c ∞ version of the reflection principle for mappings between CR manifolds

We prove a smooth version of the classical Schwarz reflection principle for CR mappings between an abstract CR manifold $M$ and a generic CR manifold embedded in euclidean complex space. As a consequence of our results, we settle a conjecture of X. Huang formulated in 1994.

Condition R and holomorphic mappings of domains with generic corners

A piecewise smooth domain is said to have generic corners if the corners are generic CR manifolds. It is shown that a biholomorphic mapping from a piecewise smooth pseudoconvex domain with generic corners in complex Euclidean space that satisfies Condition R to another domain extends as a smooth diffeomorphism of the respective closures if and only if the target domain is also piecewise smooth with generic corners and satisfies Condition R. Further it is shown that a proper map from a domain with generic corners satisfying Condition R to a product domain of the same dimension extends continuously to the closure of the source domain in such a way that the extension is smooth on the smooth part of the boundary. In particular, the existence of such a proper mapping forces the smooth part of the boundary of the source to be Levi degenerate.

Local Geometry of Levi-Forms Associated with the Existence of Complex Submanifolds and the Minimality of Generic CR Manifolds

Let M be a generic CR manifold in \(\Bbb{C}^{m+d}\) of codimension d, locally given as the common zero set of real-valued functions r1,…,rd. Given an integer δ=1,…,d, we find a necessary and sufficient condition for M to contain a real submanifold of codimension δ with the same CR structure. We also find a necessary and sufficient condition and several sufficient conditions for M to admit a complex submanifold of complex dimension n, for any n=1,…,m. We use the method of prolongation of an exterior differential system. The conditions are systems of partial differential equations on r1,…,rd of third order.

The [formula omitted] closing lemma for generic [formula omitted] endomorphisms

Given a compact m-dimensional manifold M and 1⩽r⩽∞, consider the space Cr(M) of self mappings of M. We prove here that for every map f in a residual subset of C1(M), the C1 closing lemma holds. In particular, it follows that the set of periodic points is dense in the nonwandering set of a generic C1 map. The proof is based on a geometric result asserting that for generic Cr maps the future orbit of every point in M visits the critical set at most m times.

A Cr closing lemma for a class of symplectic diffeomorphisms

We prove a Cr closing lemma for a class of partially hyperbolic symplectic diffeomorphisms. We show that for a generic Cr symplectic diffeomorphism, r = 1, 2, …, with a two-dimensional centre and close to a product map with a bunching condition, the set of all periodic points is dense.

Global approximation of CR functions on Bloom-Graham model graphs in ℂⁿ

We define a class of generic CR submanifolds of C n \mathbb {C}^n of real codimension d d , 1 ≤ d ≤ n 1\leq d\leq n , called the Bloom-Graham model graphs, whose graphing functions are partially decoupled in their dependence on the variables in the real directions. We prove a global version of the Baouendi-Treves CR approximation theorem for Bloom-Graham model graphs with a polynomial growth assumption on their graphing functions.

Foliation by graphs of CR mappings and a nonlinear Riemann-Hilbert problem for smoothly bounded domains

Let S be a generic C-infinity smooth CR manifold in C^n, n > 1, and let M be a generic C-infinity CR submanifold of S X C^m. We prescribe conditions on M so that it is the disjoint union of graphs of CR maps f:S-->C^m. We also consider the special case where S is the boundary of a bounded, smoothly bounded open set D in C^n, n > 1. In this case we obtain conditions which guarantee that such an f above extends to be analytic on D. This provides a solution to a particular nonlinear Riemann-Hilbert boundary value problem for analytic functions.

Formules Locales de Type Martinelli - Bochner - Koppelman sur des Variétés CR

We construct local formulas of Martinelli - Bochner - Koppelman type on q - concave generic CR - manifolds. We apply these results to study the tangentiel Cauchy–Riemann equations and the Hartogs - Bochner phenomenon on such manifolds.

Germs of local automorphisms of real-analytic CR structures and analytic dependence on $k$-jets

The topic of the paper is the study of germs of local holomor- phisms f between C n and C n ' such that f(M) � Mand df(T c M) = T c M ' for MC n and M ' � C n ' generic real-analytic CR submanifolds of arbitrary codimensions. It is proved that for M minimal and Mfinitely nondegener- ate, such germs depend analytically on their jets. As a corollary, an analytic structure on the set of all germs of this type is obtained. ' be connected locally closed real-analytic submanifolds, x ∈ M,x ' ∈ Mbe arbitrary points. The complex tangent subspace TxM ∩iTxM will be denoted by T c x M. M is a CR manifold, if dimT c x M is constant. In this case dimCRM := dimCT c M is called the CR dimension and codimCRM := dimIRTM − dimIRT c M the CR codimension. A CR submanifold M ⊂ C n is called generic, if TM +iTM = C n . Suppose for the moment that M,M ' ⊂ C n are generic real-analytic CR subman- ifolds of the same CR dimension and the same CR codimension. Baouendi, Ebenfelt and Rothschild found optimal nondegeneracy conditions on M and Msuch that a germ at x of a local biholomorphism f (between some neighborhoods in C n ) with f(M) ⊂ M ' ,f(x) = x ' , is uniquely determined by itsk-jetj k xf, wherek is an integer

Global holomorphic approximations of CR functions on CR manifolds

Under some geometric and analytic conditions on a generic CR submanifold , it is proved that there exists a neighborhood Ω of with the following property: For any strongly pseudoconvex domain any smooth CR function can be uniformly approximated on by functions holomorphic on Dbar:.

Famiglie di dischi analitici con bordo su sottovarietàCR non generiche

In this work the authors prove that all sufficiently small analytic discs in the tangent space of a non generic embedded CR manifold M′⊂CN (M′ at least of class C4) can be lifted uniquely to analytic discs in CN with boundaries on M′ Moreover if M′ is real analytic or C∞ then real analytic or C∞ discs lift without loss of derivatives. If M′ is of class CK then there is a 1+ε loss of derivatives in the lifting. A stability result is also proven.