Related papers: Criteria for Morin singularities into higher dimen…
We define an equivalence relation called A-isotopy between finitely determined map-germs, which is a strengthened version of A-equivalence. We consider the number of A-isotopy classes of equidimensional Morin singularities, and some other…
We give criteria for Morin singularities for germs of maps into lower dimensions. As an application, we study the bifurcation of Lefschetz singularities.
We introduce the notion of curvature parameters for singular plane curves with finite multiplicities and define the notion of curvatures for them. We then provide criteria to determine their singularity types for A-simple singularities. As…
We introduce a system of invariants of isotopy classes of Morse polynomials ${\mathbb R}^2 \to {\mathbb R}^1$, prove its completeness for polynomials of degrees $\leq 4$, calculate all 71 possible values of these invariants for the case of…
We prove a formula for Thom polynomials of Morin (or A_d) singularities in any codimension. We use a combination of the test-curve method of Porteous, and the localization methods in equivariant cohomology. Our formulas are independent of…
We establish a connection between Morin singularities and stable homotopy groups of spheres. This connection allows us to describe how the images of singularity strata behave around the image of a more complicated stratum.
Algebraic surfaces in the complex projective space with a high number of A-type singularities have been presented in a recent paper. We extend the construction in order to obtain lower bounds for the maximal number of A singularities for…
We give an attempt to build a classification of planar integral point sets. For two obtained classes, we provide general constructions of upper bounds for minimal diameter of integral point sets in higher dimensions of certain cardinality.…
We establish an interesting connection between Morin singularities and stable homotopy groups of spheres. We apply this connection to computations of cobordism groups of certain singular maps. The differentials of the spectral sequence…
We obtain the Assouad dimensions of Moran sets under suitable condition. Using the homogeneous set, we also study the Assouad dimensions of Cantor-like sets.
First, this paper presents a systematic procedure for constructing criteria for singularities of curves of finite multiplicities in $\boldsymbol{R}^N$. Based on this method, we provide explicit criteria for singularities of multiplicities…
We give simple criteria for the singularities appearing on surfaces codimension less than or equal to two. As applications, we give conditions for codimension two singularities that appear in ruled surfaces and center maps of surfaces in…
q-oscillator models are considered in two and higher dimensions and their symmetries are explored. New symmetries are found for both isotropic and anisotropic cases. Applications to the spectra of triatomic molecules and superdeformed…
We give useful and simple criteria for determining D_4 singularities of wave fronts. As an application, we investigate behaviors of singular curvatures of cuspidal edges near D_4^+ singularities.
We give criteria for sharksfin and deltoid singularities from the plane into the plane. We also give geometric meanings for the conditions in the criterion of a sharksfin. As applications, we investigate such singularities appearing on an…
We study exceptional quotient singularities. In particular, we prove an exceptionality criterion in terms of the $\alpha$-invariant of Tian, and utilize it to classify four-dimensional and five-dimensional exceptional quotient…
Some soliton equation in 2+1 dimensions and their 1+1 and/or dimensional integrable reductions are considered.
We introduce a concept of porosity for measures and study relations between dimensions and porosities for two classes of measures: measures on $R^n$ which satisfy the doubling condition and strongly porous measures on $R$.
The dimensions of sets of matrices of various types, with specified eigenvalue multiplicities, are determined. The dimensions of the sets of matrices with given Jordan form and with given singular value multiplicities are also found. Each…
We develop Morita theory of monoids in a closed symmetric monoidal category, in the context of enriched category theory.