Affine Tangent Research Articles

Singularities of helix surfaces in Euclidean 3-space

In this paper we study helix surfaces whose unit normals make a constant angle with a fixed direction. As an application of the singularity theory, we classify the generic singularities of helix surfaces, which are cuspidaledges and swallowtails. These singularities are deeply related to the order of contact between the generating curves of helix surfaces and the affine tangent bundles over unit spheres. We also show that the unit normals and rulings of helix surfaces are spherical Legendrian dual to each other. Moreover, we investigate the singularities of helix surfaces from the viewpoint of opening map-germ.

On widetilde{J}-tangent Affine Hyperspheres

In this paper we study widetilde{J}-tangent affine hyperspheres, where widetilde{J} is the canonical para-complex structure on mathbb {R}^{2n+2}. The main purpose of this paper is to give a classification of widetilde{J}-tangent affine hyperspheres of an arbitrary dimension with an involutive distribution mathcal {D}. In particular, we classify all such hyperspheres in the 3-dimensional case. We also show that there is a direct relation between widetilde{J}-tangent affine hyperspheres and Calabi products. As an application we obtain certain classification results. In particular, we show that, with one exception, all odd dimensional proper flat affine hyperspheres are, after a suitable affine transformation, widetilde{J}-tangent. Some examples of widetilde{J}-tangent affine hyperspheres are also given.

Ordinary Isolated Singularities of Algebraic Varieties

Let A be the local ring, at a singular point P, of an algebraic variety $$V\subset {\mathbb {A}}^{r+1}_k$$ of multiplicity $$e=e(A)>1$$. If V is a curve in Orecchia (Can Math Bull 24:423–431, 1981) P was said to be an ordinary singularity when V has e (simple) tangents at P or equivalently when the projectivized tangent cone $$\mathrm{{Proj}}(G(A))$$ of V at P is reduced (in which case consists of e points). In this paper, we show that the definition of ordinary singularity has a natural extension to higher dimensional varieties, in the case in which P is an isolated singularity and the normalization $$\overline{A}$$ of A is regular. In fact, we define P to be an ordinary singularity if the projectivized tangent cone $$\mathrm{{Proj}}(G(A))$$ of V at P is reduced, i.e. is a variety in $${\mathbb {P}}^{r}_k$$. We prove that an ordinary singularity has multilinear projectivized tangent cone that is a union of e linear varieties $$L_1,\ldots ,L_e$$. In the case in which $$L_1,\ldots ,L_e$$ are in generic position, we show that the affine tangent cone is also reduced and then multilinear. Finally, we show how to construct wide classes of parametric varieties with regular normalization at a singular isolated ordinary point.

On 3 -dimensional e J -tangent centro-affine hypersurfaces and e J -tangent affine hyperspheres with some null-directions

Let $\widetilde{J}$ be the canonical para-complex structure on $\mathbb{R}^4$. In this paper we study $3$-dimensional centro-affine hypersurfaces with a $\widetilde{J}$-tangent centro-affine vector field (sometimes called $\widetilde{J}$-tangent centro-affine hypersurfaces) as well as $3$-dimensional $\widetilde{J}$-tangent affine hyperspheres with the property that at least one null-direction of the second fundamental form coincides with either $\mathcal{D}^+$ or $\mathcal{D}^-$. The main purpose of this paper is to give a full local classification of the above mentioned hypersurfaces. In particular, we prove that every nondegenerate centro-affine hypersurface of dimension $3$ with a $\widetilde{J}$-tangent centro-affine vector field which has two null-directions $\mathcal{D}^+$ and $\mathcal{D}^-$ must be both an affine hypersphere and a hyperquadric. Some examples of these hypersurfaces are also given.

A micromechanical model for the grain size dependent super-elasticity degeneration of NiTi shape memory alloys

Recent experiments show that super-elasticity degeneration occurs during the cyclic deformation of polycrystalline NiTi SMAs and the extent of such degeneration depends strongly on the grain size of the polycrystalline aggregate. Thus, in this work, a micromechanical model is constructed to describe the grain size dependent super-elasticity degeneration of polycrystalline NiTi SMAs. The polycrystalline aggregate is modeled as a composite material, i.e., each grain-interior (GI) phase is assumed to be a spherical inclusion embedded in a matrix of grain-boundary (GB) phase. For GI phase, two inelastic deformation processes, i.e., martensite transformation and transformation induced plasticity, and their interaction are considered, simultaneously; for the GB phase, its plastic deformation is described by a unified visco-plastic model. To describe the interaction between the inclusion and matrix and calculate the macroscopic overall performances of the modeled composite material, a modified incremental Mori–Tanaka's homogenization method is proposed by introducing the Eshelby's tensor of a spherical inclusion embedded in a finite spherical domain and the affine tangent modulus and affine strain increment. Numerical algorithm is developed to implement the proposed micromechanical model. A new and simple affine linearization method is proposed to linearize the constitutive equations of GI and GB phases. The proposed micromechanical model is verified by comparing the simulated results with the corresponding experimental ones. It is shown that the grain size dependent super-elasticity degeneration of NiTi SMAs can be well captured. Moreover, the calculated results indicate that when the average grain size is large (>160 nm), the super-elasticity degeneration originates from the interaction between the martensite transformation and dislocation slipping in GI phase; while, when the average grain size is small (<160 nm), the super-elasticity degeneration is dominated by the plastic deformation of GB phase and the stress redistribution between GI and GB phase.

Discrete Time Term Structure Theory and Consistent Recalibration Models

We develop a theory and applications of forward characteristic processes in discrete time following the ideas of a seminal paper of Kallsen and Kruhner [Finance Stoch., 19 (2015), pp. 583--615]. Particular emphasis is given to the dynamics of volatility surfaces which can be readily formulated and implemented from our chosen discrete point of view. In mathematical terms we provide an algorithmic answer to the following question: describe a rich, still tractable class of discrete time stochastic processes, whose marginal distributions are given at an initial time and which are free of arbitrage. In terms of mathematical finance we can construct models with predescribed (implied) volatility surface at initial time and a quite general volatility surface dynamics along time. In terms of the works of Carmona and Nadtochiy [Finance Stoch., 13 (2009), pp. 1--48; Finance Stoch., 16 (2012), pp. 63--104] we analyze the dynamics of tangent affine models. We believe that the discrete approach due to its technical sim...

$J$-tangent affine hyperspheres with an involutive contact distribution

$J$-tangent affine hyperspheres with an involutive contact distribution

Properties of an affine transport equation and its holonomy

An affine transport equation was used recently to study properties of angular momentum and gravitational-wave memory effects in general relativity. In this paper, we investigate local properties of this transport equation in greater detail. Associated with this transport equation is a map between the tangent spaces at two points on a curve. This map consists of a homogeneous (linear) part given by the parallel transport map along the curve plus an inhomogeneous part, which is related to the development of a curve in a manifold into an affine tangent space. For closed curves, the affine transport equation defines a "generalized holonomy" that takes the form of an affine map on the tangent space. We explore the local properties of this generalized holonomy by using covariant bitensor methods to compute the generalized holonomy around geodesic polygon loops. We focus on triangles and "parallelogramoids" with sides formed from geodesic segments. For small loops, we recover the well-known result for the leading-order linear holonomy ($\sim$ Riemann $\times$ area), and we derive the leading-order inhomogeneous part of the generalized holonomy ($\sim$ Riemann $\times$ area$^{3/2}$). Our bitensor methods let us naturally compute higher-order corrections to these leading results. These corrections reveal the form of the finite-size effects that enter into the holonomy for larger loops; they could also provide quantitative errors on the leading-order results for finite loops.

An algorithm based on rolling to generate smooth interpolating curves on ellipsoids

We present an algorithm to generate a smooth curve interpolating a set of data on an $n$-dimensional ellipsoid, which is given in closed form. This is inspired by an algorithm based on a rolling and wrapping technique, described in [11] for data on a general manifold embedded in Euclidean space. Since the ellipsoid can be embedded in an Euclidean space, this algorithm can be implemented, at least theoretically. However, one of the basic steps of that algorithm consists in rolling the ellipsoid, over its affine tangent space at a point, along a curve. This would allow to project data from the ellipsoid to a space where interpolation problems can be easily solved. However, even if one chooses to roll along a geodesic, the fact that explicit forms for Euclidean geodesics on the ellipsoid are not known, would be a major obstacle to implement the rolling part of the algorithm. To overcome this problem and achieve our goal, we embed the ellipsoid and its affine tangent space in $\mathbb{R}^{n+1}$ equipped with an appropriate Riemannian metric, so that geodesics are given in explicit form and, consequently, the kinematics of the rolling motion are easy to solve. By doing so, we can rewrite the algorithm to generate a smooth interpolating curve on the ellipsoid which is given in closed form.

$C^1$-regularity for local graph representations of immersions

We consider immersions admitting uniform graph representations over the affine tangent space over a ball of fixed radius r>0. We show that for sufficiently small C^0-norm of the graph functions, each graph function is smooth with small C^1-norm.

Some geometric properties of hypersurfaces with constant $r$-mean curvature in Euclidean space

Let f: M → R m+1 be an isometrically immersed hypersurface. In this paper, we exploit recent results due to the authors to analyze the stability of the differential operator L r associated with the rth Newton tensor of f. This appears in the Jacobi operator for the variational problem of minimizing the r-mean curvature H r . Two natural applications are found. The first one ensures that under a mild condition on the integral of H r over geodesic spheres, the Gauss map meets each equator of S m infinitely many times. The second one deals with hypersurfaces with zero (r + 1)-mean curvature. Under similar growth assumptions, we prove that the affine tangent spaces f * T p M, p ∈ M, fill the whole R m+1 .

Estimates for maximal functions associated with hypersurfaces in ℝ3 and related problems of harmonic analysis

We study the boundedness problem for maximal operators $ \mathcal{M} $ associated with averages along smooth hypersurfaces S of finite type in 3-dimensional Euclidean space. For p > 2, we prove that if no affine tangent plane to S passes through the origin and S is analytic, then the associated maximal operator is bounded on $ {L^p}\left( {{\mathbb{R}^3}} \right) $ if and only if p > h(S), where h(S) denotes the so-called height of the surface S (defined in terms of certain Newton diagrams). For non-analytic S we obtain the same statement with the exception of the exponent p = h(S). Our notion of height h(S) is closely related to A. N. Varchenko’s notion of height h(ϕ) for functions ϕ such that S can be locally represented as the graph of ϕ after a rotation of coordinates. Several consequences of this result are discussed. In particular we verify a conjecture by E. M. Stein and its generalization by A. Iosevich and E. Sawyer on the connection between the decay rate of the Fourier transform of the surface measure on S and the Lp-boundedness of the associated maximal operator $ \mathcal{M} $, and a conjecture by Iosevich and Sawyer which relates the Lp-boundedness of $ \mathcal{M} $ to an integrability condition on S for the distance to tangential hyperplanes, in dimension 3. In particular, we also give essentially sharp uniform estimates for the Fourier transform of the surface measure on S, thus extending a result by V. N. Karpushkin from the analytic to the smooth setting and implicitly verifying a conjecture by V. I. Arnold in our context. As an immediate application of this, we obtain an $ {L^p}\left( {{\mathbb{R}^3}} \right) - {L^2}(S) $ Fourier restriction theorem for S.

A new line search inexact restoration approach for nonlinear programming

A new general scheme for Inexact Restoration methods for Nonlinear Programming is introduced. After computing an inexactly restored point, the new iterate is determined in an approximate tangent affine subspace by means of a simple line search on a penalty function. This differs from previous methods, in which the tangent phase needs both a line search based on the objective function (or its Lagrangian) and a confirmation based on a penalty function or a filter decision scheme. Besides its simplicity the new scheme enjoys some nice theoretical properties. In particular, a key condition for the inexact restoration step could be weakened. To some extent this also enables the application of the new scheme to mathematical programs with complementarity constraints.

On the Geometry of Rolling and Interpolation Curves on S n , SO n , and Grassmann Manifolds

We present a procedure to generate smooth interpolating curves on submanifolds, which are given in closed form in terms of the coordinates of the embedding space. In contrast to other existing methods, this approach makes the corresponding algorithm easy to implement. The idea is to project the prescribed data on the manifold onto the affine tangent space at a particular point, solve the interpolation problem on this affine subspace, and then project the resulting curve back on the manifold. One of the novelties of this approach is the use of rolling mappings. The manifold is required to roll on the affine subspace like a rigid body, so that the motion is described by the action of the Euclidean group on the embedding space. The interpolation problem requires a combination of a pullback/push forward with rolling and unrolling. The rolling procedure by itself highlights interesting properties and gives rise to a new, but simple, concept of geometric polynomial curves on manifolds. This paper is an extension of our previous work, where mainly the 2-sphere case was studied in detail. The present paper includes results for the n-sphere, orthogonal group SO n , and real Grassmann manifolds. In particular, we present the kinematic equations for rolling these manifolds along curves without slip or twist, and derive from them formulas for the parallel transport of vectors along curves on the manifold.

Legendrian Varieties

We investigate the geometry of Legendrian complex projective manifolds $X\subset\PP V$. By definition, this means $V$ is a complex vector space of dimension $2n+2$, endowed with a symplectic form, and the affine tangent space to $X$ at each point is a maximal isotropic subspace. We establish basic facts about their geometry and exhibit examples of inhomogeneous smooth Legendrian varieties, the first examples of such in dimension greater than one.

How tangents solve algebraic equations, or a remarkable geometry of discriminant varieties

Let D d,k denote the discriminant variety of degree d polynomials in one variable with at least one of its roots being of multiplicity ≥ k. We prove that the tangent cones to D d,k span D d,k − 1 thus, revealing an extreme ruled nature of these varieties. The combinatorics of the web of affine tangent spaces to D d,k in D d,k − 1 is directly linked to the root multiplicities of the relevant polynomials. In fact, solving a polynomial equation P(z) = 0 turns out to be equivalent to finding hyperplanes through a given point P ( z ) ∈ D d , I ≈ A d which are tangent to the discriminant hypersurface D d,2 . We also connect the geometry of the Viète map V d : A r o o t d → A c o e f d , given by the elementary symmetric polynomials, with the tangents to the discriminant varieties { D d,k }. Various d-partitions {μ} provide a refinement { D o μ} of the stratification of A c o e f d by the D d,k 's. Our main result, Theorem 7.1, describes an intricate relation between the divisibility of polynomials in one variable and the families of spaces tangent to various strata { D o μ}.

Spectral theorem for convex monotone homogeneous maps, and ergodic control

We consider convex maps f : R n→ R n that are monotone (i.e., that preserve the product ordering of R n ), and nonexpansive for the sup-norm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is nonempty, is isomorphic to a convex inf-subsemilattice of R n , whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group on n letters.

A Spectral Theorem for Convex Monotone Homogeneous Maps

We consider convex maps f : Rn → Rn that are monotone (i.e., that preserve the product ordering of Rn), and non-expansive for the sup-norm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is non-empty, is isomorphic to a convex inf-semilattice, whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This bound on the dimension is attained when f is piecewise affine. This yields in particular an uniqueness result for the bias vector of ergodic control problems. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group on n letters.

Affine-Invariant Distances, Envelopes and Symmetry Sets

Affine invariant symmetry sets of planar curves are introduced and studied in this paper. Two different approaches are investigated. The first one is based on affine invariant distances, and defines the symmetry set as the closure of the locus of points on (at least) two affine normals and affine-equidistant from the corresponding points on the curve. The second approach is based on affine bitangent conics. In this case the symmetry set is defined as the closure of the locus of centers of conics with (at least) 3-point contact with the curve at two or more distinct points on the curve. This is equivalent to conic and curve having, at those points, the same affine tangent, or the same Euclidean tangent and curvature. Although the two analogous definitions for the classical Euclidean symmetry set are equivalent, this is not the case for the affine group. We present a number of properties of both affine symmetry sets, showing their similarities with and differences from the Euclidean case. We conclude the paper with a discussion of possible extensions to higher dimensions and other transformation groups, as well as to invariant Voronoi diagrams.

On the nonparticle structure of gauge interactions

Gauge interactions incorporate nonparticle fields with nonunitary and nilpotent time development, analogue to the motion of a free mass pointp2/2m0. The nontransversal gauge and the «Lagrange multiplier» fields, together with the Faddeev-Popov nonparticle fields, represent the affine tangent structure of the gauge group. Nonadjoint representations of the affine Lie algebra use composite gauge fields. Their constituents without particle structure build a Hilbert space for the composites, different from the usual Fock space.