Abstract. We present a complete set of criteria for determining A-types of plane-to-plane map-germs of corank one with A-codimension ≤ 6, which provides a new insight into the A-classification theory from the viewpoint of recognition problem. As an application to generic differential geometry, we discuss about projections of smooth surfaces in 3-space. IntroductionWe revisit the A-classification of local singularities of plane-to-plane maps. Here A denotes the group of diffeomorphism germs of source and target planes preserving the origin. The classification has been achieved by J. H. Rieger, M. A. S. Ruas [12,13,15] -for instance, Table 1 below shows the list of all corank one mapgerms with A-codimension ≤ 6. When we apply the classification to some specific geometric situation, it often becomes a cumbersome task to detect which A-type a given map-germ belongs to, that is referred to as "A-recognition problem" (cf.[5]). In fact, Rieger's algorithm frequently uses Mather's Lemma to reduce the jet to some nicer form, at which the coordinate changes are not explicitly given (dotted lines in the recognition trees Fig. 1-5 in [12] indicate such processes). To fill up the process is not easy: the task is essentially related to deeper understanding on a filtered structure of the A-tangent space of the germ, as T. Gaffney pointed out in an earlier work [5].In this paper, we present a complete set of criteria for detecting A-types of corank one germs with A-codimension ≤ 6 (Theorem 3.1). That is a useful package consisting of two-phased criteria, which would easily be implemented in computer. The first one is about geometric conditions on 'specified jets' for topological Atypes in terms of intrinsic derivatives [20,16,17,11,8], and the second is about algebraic conditions on Talyor coefficients of germs with some specified jets, which are obtained by describing explicitly all the required coordinate changes of source and target of map-germs which are hidden in the classification process (Proposition 3.3, 3.5, 3.6, and 3.8).For example, look at the cases of the butterfly (x, xy + y 5 ± y 7 ) and the elder butterfly (x, xy +y 5 ), which are combined into a single topological A-type. Suppose that a map-germ f = (f 1 , f 2 ) : R 2 , 0 → R 2 , 0 with corank one is given. Put λ(x, y) := ∂(f1,f2) ∂(x,y) , and take an arbitrary vector field η := η 1 (x, y) ∂ ∂x + η 2 (x, y) ∂ ∂y near the origin of the source space so that η spans ker df on λ = 0. Denote η k g := η(η k−1 g). We show that the corresponding weighted homogeneous specified 2010 Mathematics Subject Classification. 57R45, 53A05, 53A15.

The purpose of this paper is to understand generic behavior of constraint functions in optimization problems relying on singularity theory of smooth mappings. To this end, we will focus on the subgroup K[G] of the Mather's group K, whose action to constraint map-germs preserves the corresponding feasible set-germs (i.e. the set consisting of points satisfying the constraints). We will classify map-germs with small stratum K[G] e -codimensions, and calculate the codimensions of the K[G]-orbits of jets represented by germs in the classification lists and those of the complements of these orbits. Applying these results and a variant of the transversality theorem, we will show that families of constraint mappings whose germ at any point in the corresponding feasible set is K[G]-equivalent to one of the normal forms in the classification list compose a residual set in the entire space of constraint mappings with at most 4-parameters. These results enable us to quantify genericity of given constraint mappings, and thus evaluate to what extent known test suites are generic.

No abstract

There is a unique A-moduli stratum of plane-to-plane germs which forms an open dense subset in the K-orbit of I2,3 : (x 2 + y 3 , xy). We describe explicitly the bifurcation diagram of its topologically Aeversal unfolding. Two geometric applications to parabolic crosscaps and parabolic umbilic are presented.2000 Mathematics Subject Classification. 57R45, 53A05, 53A15.

This study demonstrated the usefulness of radiomic features based on the Hessian index of differential topology for the prediction of prognosis prior to treatment in head-and-neck (HN) cancer patients. The Hessian index, which can indicate tumor heterogeneity with convex, concave, and other points (saddle points), was calculated as the number of negative eigenvalues of the Hessian matrix at each voxel on computed tomography (CT) images. Three types of signatures were constructed in a training cohort (n = 126), one type each from CT conventional features, Hessian index features, and combined features from the conventional and index feature sets. The prognostic value of the signatures were evaluated using statistically significant difference (p value, log-rank test) to compare the survival curves of low- and high-risk groups. In a test cohort (n = 68), the p values of the models built with conventional, index, combined features, and clinical variables were 2.95 $$\times$$ × 10–2, 1.85 $$\times$$ × 10–2, 3.17 $$\times$$ × 10–2, and 1.87 $$\times$$ × 10–3, respectively. When the features were integrated with clinical variables, the p values of conventional, index, and combined features were 3.53 $$\times$$ × 10–3, 1.28 $$\times$$ × 10–3, and 1.45 $$\times$$ × 10–3, respectively. This result indicates that index features could provide more prognostic information than conventional features and further increase the prognostic value of clinical variables in HN cancer patients.