Complete normal forms for real hypersurfaces in $$\mathbb {C}^3$$ at 2-nondegenerate points of Levi non-uniform rank zero

On two-symbol complete EOL forms

Abnormal sperm count and motility on semen analysis are not sufficiently predictive of abnormal Kruger morphology

In this paper, we study real hypersurfaces $M$ in ${ C}2$ at points $p\in M$ of infinite type. The degeneracy of $M$ at $p$ is assumed to be the least possible, namely such that the Levi form vanishes to first order in the CR transversal direction. A new phenomenon, compared to known normal forms in other cases, is the presence of resonances as roots of a universal polynomial in the 7-jet of the defining function of $M$. The main result is a complete (formal) normal form at points $p$ with no resonances. Remarkably, our normal form at such infinite type points resembles closely the Chern-Moser normal form at Levi-nondegenerate points. For a fixed hypersurface, its normal forms are parametrized by $S1\times { R}^*$, and as a corollary we find that the automorphisms in the stability group of $M$ at $p$ without resonances are determined by their 1-jets at $p$. In the last section, as a contrast, we also give examples of hypersurfaces with arbitrarily high resonances that possess families of distinct automorphisms whose jets agree up to the resonant order.

We consider the problem of describing the local biholomorphic equivalence class of a real-analytic hypersurface $M$ at a distinguished point $p_0\in M$ by giving a normal form for such objects. In order for the normal form to carry useful information about the biholomorphic equivalence class, we shall require that the transformation to normal form is unique modulo some finite dimensional group. A classical result due to Chern--Moser gives such a normal form for Levi nondegenerate hypersurfaces. The main results in this paper concern real-analytic hypersurfaces $M$ in $\Bbb C^3$ at certain Levi degenerate points $p_0\in M$, namely points at which $M$ is 2-nondegenerate. We give a partial normal form for all such $(M,p_0)$, i.e. a normal form for the data associated with 2-nondegeneracy. We also give a complete formal normal form for such $(M,p_0)$ under the additional condition that the Levi form has rank one at $p_0$. This result, combined with a recent theorem due to the author, M. S. Baouendi, and L. P. Rothschild stating that formal equivalences between real-analytic finitely nondegenerate hypersurfaces converge, gives a description of the biholomorphic equivalence class of a real-analytic hypersurface in $\Bbb C^3$ at a point of 2-nondegeneracy where the rank of the Levi form is one.

A complete normal form for everywhere Levi–degenerate hypersurfaces in [formula omitted

In Stróżyna E and Żołądek H (2015 Moscow Math. J. 15 141) a complete formal normal forms for germs of two-dimensional holomorphic vector fields with nilpotent singularity was obtained. That classification is quite nontrivial (7 cases), but it can be divided into general types like in the case of the elementary singularities. One could expect that also the analytic properties of the normal forms for the nilpotent singularities are analogous to the case of the elementary singularities. This is really true. In the cases analogous to the focus and the node the normal form is analytic. In the case analogous to the nonresonant saddle the normal form is often nonanalytic due to the small divisors phenomenon. In the cases analogous to the resonant saddles (including saddle-nodes) the normal form is nonanalytic due to bad properties of some homological operators associated with the first nontrivial term in the orbital normal form.

Applying methods of CR-geometry, we give a solution to the local equivalence problem for second order (smooth or analytic) ordinary differential equations. We do so by presenting a complete normal form (which is smooth or analytic respectively) for this class of ordinary differential equations (ODEs). The normal form is optimal in the sense that it is defined up to the automorphism group of the model (flat) ODE y ″ = 0. For a generic ODE, we also provide a unique (up to a discrete group action) normal form. By doing so, we give a solution to a problem which remained unsolved since the work of Arnold (1988). As another application of the normal form, we obtain distinguished curves associated with a differential equation that we call chains due to their analogy with the chains defined by Chern and Moser (Acta Math. 7;133:219–271).

Topology of Double Points of Rank Zero on Threefolds in C⁴

Classification of Isolated Double Points of Rank Zero on Primals in <i>S_n </i>

In this work, inspired by the technique of the complete transversal, used for the classification of plane branches, developed by Hefez, A. and Hernandes, M., as well as Bruce, J.W., Kirk, N.P. and du Plesis, A.A., study the singularities of applications, we establish a classification of vector fields through their normal forms. In the case of vector fields with non zero linear part in $(\mathbb{C}^{2}, 0) $ and nilpotent fields in $(\mathbb {C}^{n}, 0), n\geq 2$ we recover the classical normal forms for those fields, and we provide a formal normal form different from Takens in dimension 2. Likewise, we obtain the normal form for the vector fields in $(\mathbb{C},0)$ of any multiplicity.

Distributive Normal Forms in First-Order Logic

We introduce new invariant tensors in CR structures which can be viewed as higher order Levi forms. Using the second and third order tensors, we give a complete formal normal form (in the sense of Chern-Moser) for a real hypersurface at a generic Levi degeneracy. (We say that M has a generic Levi degeneracy at p if the Levi determinant vanishes at p but its differential does not, and the set of Levi degenerate points of M is transverse to the Levi null space at p.) By applying a convergence theorem for formal mappings due to the author, Baouendi, and Rothschild, we conclude that the above mentioned formal normal form provides a complete set of biholomorphic invariants for real-analytic hypersurfaces at generic Levi degeneracies.

The Analytic and Formal Normal Form for the Nilpotent Singularity

Lawvere showed that generalised metric spaces are categories enriched over $[0, \infty]$, the quantale of the positive extended reals. The statement of enrichment is a quantitative analogue of being a preorder. Towards seeking a logic for quantitative metric reasoning, we investigate three $[0,\infty]$-valued propositional logics over the Lawvere quantale. The basic logical connectives shared by all three logics are those that can be interpreted in any quantale, viz finite conjunctions and disjunctions, tensor (addition for the Lawvere quantale) and linear implication (here a truncated subtraction); to these we add, in turn, the constant $1$ to express integer values, and scalar multiplication by a non-negative real to express general affine combinations. Quantitative equational logic can be interpreted in the third logic if we allow inference systems instead of axiomatic systems. For each of these logics we develop a natural deduction system which we prove to be decidably complete w.r.t. the quantale-valued semantics. The heart of the completeness proof makes use of the Motzkin transposition theorem. Consistency is also decidable; the proof makes use of Fourier-Motzkin elimination of linear inequalities. Strong completeness does not hold in general, even (as is known) for theories over finitely-many propositional variables; indeed even an approximate form of strong completeness in the sense of Pavelka or Ben Yaacov -- provability up to arbitrary precision -- does not hold. However, we can show it for theories axiomatized by a (not necessarily finite) set of judgements in normal form over a finite set of propositional variables when we restrict to models that do not map variables to $\infty$; the proof uses Hurwicz's general form of the Farkas' Lemma.

In this paper, the scattering and spectral theory of [math] is developed, where [math] is the Laplacian with respect to a scattering metric [math] on a compact manifold [math] with boundary and [math] is real; this extends our earlier results in the two-dimensional case. Included in this class of operators are perturbations of the Laplacian on Euclidean space by potentials homogeneous of degree zero near infinity. Much of the particular structure of geometric scattering theory can be traced to the occurrence of radial points for the underlying classical system. In this case the radial points correspond precisely to critical points of the restriction, [math] , of [math] to [math] and under the additional assumption that [math] is Morse a functional parameterization of the generalized eigenfunctions is obtained.\n¶ The main subtlety of the higher dimensional case arises from additional complexity of the radial points. A normal form near such points obtained by Guillemin and Schaeffer is extended and refined, allowing a microlocal description of the null space of [math] to be given for all but a finite set of “threshold” values of the energy; additional complications arise at the discrete set of “effectively resonant” energies. It is shown that each critical point at which the value of [math] is less than [math] is the source of solutions of [math] . The resulting description of the generalized eigenspaces is a rather precise, distributional, formulation of asymptotic completeness. We also derive the closely related [math] and time-dependent forms of asymptotic completeness, including the absence of [math] channels associated with the nonminimal critical points. This phenomenon, observed by Herbst and Skibsted, can be attributed to the fact that the eigenfunctions associated to the nonminimal critical points are “large” at infinity; in particular they are too large to lie in the range of the resolvent [math] applied to compactly supported functions.