Nondegenerate Function Research Articles

Lagrange geometry on tangent manifolds

Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a nondegenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange geometry. In this paper, we study a generalization which consists of replacing the tangent bundle by a general tangent manifold, and the Lagrangian by a family of compatible, local, Lagrangian functions. We give several examples and find the cohomological obstructions to globalization. Then, we extend the connections used in Finsler and Lagrange geometry, while giving an index‐free presentation of these connections.

The Brockett Problem for Linear Discrete Control Systems

The Brockett problem for linear discrete control systems is studied. A method for designing time-varying stabilizing feedback is developed. The Brockett problem for second-order systems with scalar feedback and nondegenerate transfer function is shown to have a positive solution.

Generalized transversely projective structure on a transversely holomorphic foliation

The results of Biswas (2000) are extended to the situation of transversely projective foliations. In particular, it is shown that a transversely holomorphic foliation defined using everywhere locally nondegenerate maps to a projective space ℂℙn, and whose transition functions are given by automorphisms of the projective space, has a canonical transversely projective structure. Such a foliation is also associated with a transversely holomorphic section of N⊗−k for each k ∈ [3, n + 1], where N is the normal bundle to the foliation. These transversely holomorphic sections are also flat with respect to the Bott partial connection.

HYPERSURFACE EXCEPTIONAL SINGULARITIES

This paper studies hypersurface exceptional singularities in [Formula: see text] defined by non-degenerate function. For each canonical hypersurface singularity, there exists a weighted homogeneous singularity such that the former is exceptional if and only if the latter is exceptional. So we study the weighted homogeneous case and prove that the number of weights of weighted homogeneous exceptional singularities are finite. Then we determine all exceptional singularities of the Brieskorn type of dimension 3.

On a Normal Approximation of U-Statistics

We consider U-statistics of order 2 constructed upon independent identically distributed random variables $X_1,\ldots,X_n$ with values in a measurable space $(\mathfrak{X,B})$. For U-statistics with a nondegenerate kernel and canonical functions $g: \mathfrak{X}\mapsto$ {\bf R} and $h: \mathfrak{X}^2\mapsto$ {\bf R}, we investigate a problem on the estimation of the rate of convergence in the central limit theorem. The result obtained implies that the estimate of order $n^{-1/2}$ depends only on the third moment {\bf E}$|g(X_1)|^3$ and the weak moment $\sup_{x > 0}(x^{5/3} {\bf P}\{|h(X_1,\,X_2)| > x\})$ of order~${\frac{5}{3}}$.

Infinitely Many Solutions for a Floquet-Type BVP with Superlinearity Indefinite in Sign

By the application of a technique developed by G. J. Butler we find infinitely many solutions of the BVP[formula]where q:[0,ω]→R is allowed to change sign, g:R→R is superlinear and g(x)x>0 for all x≠0, and Λ:R2→R2 is a continuous, positively homogeneous, and nondegenerate map. At first we apply the main result to obtain solutions with a prescribed large number of zeros when Λ is the rotation of a fixed angle λ; second, we find infinitely many subharmonic solutions of any order and, again, solutions with a prescribed large number of zeros for the periodic problem associated to the equation ẍ+cẋ+q(t)g(x)=0, with q and g as above and c∈R.

Polynomial expansions of boolean functions in images of nonhomogeneous operators

To represent the functions in terms of which expansions are constructed, the so-called dp-, pt -, and dt -operators are used. The existence of polynomial expansions of Boolean functions in functions obtained by applying heterogeneous operators of the above-mentioned types to a nondegenerate Boolean function is proved. Some methods of finding coefficients of these expansions are also considered.

Monodromy of functions on isolated cyclic quotients

We study two important invariants of the monodromy of a function on an isolated cyclic quotient ( C n/G,0) , where G is a finite cyclic group: the Lefschetz number and the zeta-function. Our approach relies on a certain “good” toric modification of C n inducing a toric resolution of the cyclic quotient. We prove that the Lefschetz number has a sum decomposition into Lefschetz numbers of well-defined weighted-homogeneous “pieces” of the initial function, the weights depending only on the group action. We define a class of nondegenerate functions and prove for them a zeta-function formula, using Varchenko's approach via the Newton polyhedron.

Minimax detection of a signal for nondegenerate loss functions, and extremal convex problems

We study a wide class of minimax problems of signal detection under nonparametric alternatives and a modification of these problems for a special class of loss functions. Under rather general assumptions, we obtain the exact asymptotics (of Gaussian type) for the minimax error probabilities and the structure of asymptotically minimax tests. The methods are based on a reduction of the problems under consideration to extremal problems of minimization of a certain Hilbert norm on a convex set of sequences of probability measures on the real line. These extremal problems are investigated in a paper by I. A. Suslina for alternatives having the type of lq-ellipsoids with lp-balls removed. Bibliography: 16 titles.

On congruent holomorphic mappings into a Hermitian symmetric space

It is one of basic problems in differential geometry to find geometric conditions that determine the congruence class of a submanifold of a given manifold. For example, a general hypersurface of n-dimensional Euclidean space R n is determined up to congruence by its first and second fundamental forms. It is another classical result that for a certain generic hypersurfaces of R n (n > 4), the first fundamental form alone is sufficient to determine the congruence class. On the other hand, we know some remarkable facts in the complex-analytic case. First, any complex submanifold of a complex space form has the metric rigidity, i.e., it is determined up to congruence by its first fundamental form alone (the rigidity theorem of Calabi [1]). Secondly, Green [2] asserts even if the ambient space 5 is a general Kahler manifold with real-analytic Kahler metric, a certain generic holomorphic mapping (a nondegenerate mapping in his terminology) into it has the local metric rigidity: There is actually shown the existence of such a finite-dimensional family of local real-analytic hypersurfaces of S that a holomorphic mapping into S has the local metric rigidity unless its image lies in any of them. Only one cannot find any geometric conditions that assure the nondegeneracy for a mapping. In this article, we shall give local differential-geometric conditions that determine the congruence class of a holomorphic mapping into a Hermitian symmetric space under considerably general settings. For a technical reason, we have to restrict ourselves to the case where mappings are infinίtesimally in our sense. In particular we cannot deal with totally geodesic complex submanifolds. But our results extend Calabi's rigidity theorem to a direction different from [2], since our infinitesimally condition is equivalent to the full condition used in [1] in the case of complex space forms.

Locally Lagrange-Symplectic Manifolds

The symplectic forms which have a compatible tangent structure locally look like the symplectic form of a nondegenerate Lagrangian function. For this reason, the corresponding manifold is said to be locally Lagrange-symplectic. In the paper, we give some elementary properties of these manifolds, and discuss their symplectic reduction.

Sums of Lyapunov exponents for some polynomial maps of ${\bf C}^2$

We give a formula for the sum of the Lyapunov exponents of a nondegenerate polynomial map $f$ of ${\bf C}^{2}$ close to $(z,w)\to (z^d,w^d)$. The formula only involves the behavior of $f$ at infinity. In particular, it follows that the sum only depends on the homogeneous part of $f$ of degree $d$.

A characterization of complex projective space up to biholomorphic isometry

Obata [3] gave a characterization up to isometry of the standard sphere sn in terms of the Hessian operator on a complete Riemannian manifold. With the convention Hess u = Vgrad u, his result says that if M is a complete Riemannian manifold which admits a nondegenerate function u such that Hess u = -u- Id then M is isometric to the standard sphere. Obata goes on to prove related results in conformal geometry which take advantage of the existence of a function whose Hessian has a special form. Other authors have also obtained strong geometric properties of a Riemannian manifold by exploiting the existence of a function u with Hess u = f - Id for some function f. In particular, this last equation implies that M is a warped product. For a proof and related results, see, for example, Osgood and Stowe [5]. In the complex case, a characterization of cn up to isometry was given by Stoll [6] via the complex Monge-Amp~re operator. Stoll's result says that if M is a complex manifold which admits a strictly plurisubharmonic exhaustion r: M --> [0, ~) such that (dd c log r) n -- 0, then (M, r) = (C", Iz12). In other words M, with hermitian metric given by the Kahler form ddCr, is biholomorphically isometric to C". Obata also showed that a complete, connected and simply connected Kahler manifold is isometric to the complex projective space ]?n if and only if it admits a solution to a certain linear system of third order differential equations [4]. Blair [1] subsequently showed that in some cases this characterization of ]?n follows from a corresponding result for Riemannian manifolds and indicated that one would not expect a characterization of ]?n by a Hessian equation analogous to that which Obata used to characterize S n. In this paper we give a complex analog of Obata's theorem [3]. We characterize complex projective space up to biholomorphic isometry by the existence of a solution to a system of second order equations. Since ]?n with the Fubini-Study metric is not a warped product, there does not exist a nontrivial function u on ]?n whose Hessian is a multiple of the identity. However, ]?n with one point deleted is the hyperplane section bundle over I? n- 1 and the fibers of this bundle are totally geodesic complex lines. Thus, there is a relationship between the natural metric structure and the line bundle structure of ~n. This relationship provides the motivation for the construction of a function u on ]?n

An Infeasible Path-Following Method for Monotone Complementarity Problems

We propose an infeasible path-following method for solving the monotone complementarity problem. This method maintains positivity of the iterates and uses two Newton steps per iteration---one with a centering term for global convergence and one without the centering term for local superlinear convergence. We show that every cluster point of the iterates is a solution, and if the underlying function is affine or is sufficiently smooth and a uniform nondegenerate function on $\Re_{++}^n$, then the convergence is globally Q-linear. Moreover, if every solution is strongly nondegenerate, the method has local quadratic convergence. The iterates are guaranteed to be bounded when either a Slater-type feasible solution exists or when the underlying function is an R0-function.

Unconditional limit theorems from conditional limit theorems

The method of proof developed here may be used to obtain unconditional limit theorems from conditional limit theorems in a variety of settings. It is known that given two samples with the same arbitrary nondegenerate common distribution function, the conditional distribution of the sum of the tied midranks of one sample within the pooled ordered sample, centered on its conditional expectation, and normed by the square root of its conditional variance, given the values of its numbers of ties, is asymptotically normal as the two sample sizes tend to infinity, provided that the proportions of ties obey a certain constraint. In this note the unconditional distribution of this same statistic is shown to be asymptotically normal also.

Harmonicity of gradient mappings of level surfaces in a real affine space

In our previous paper [4] we have investigated level surfaces of a non-degenerate function ϕ in a real affine space An+1 by using the gradient vector field\(\tilde E\). We gave characterizations of ϕ by means of the shape operatorS, the transversal connection τ, and studied the difference between\(\tilde E\) and the affine normal. In particular we showed that a graph defined by a non-degenerate function satisfiesS=0 and τ=0. In this paper we consider harmonic gradient mappings of level surfaces and apply these results to a certain problem which is similar to the affine Bernstein problem conjectured by S. S. Chern [3].

A local version of Levenberg's theorem on determining measures and Leja's polynomial condition

A local version of Levenberg's theorem characterizing Leja's polynomial condition in terms of determining measures is given. This yields a very short proof of the invariance of Leja's property under non-degenerate holomorphic maps.

Domains of Attraction of the Max-Semistable Laws under Linear and Power Normalizations

We consider the domains of attraction of the univariate semi-stable laws in the scheme of cumulative maxima of independent identically distributed random variables and obtain necessary and sufficient conditions for a nondegenerate distribution function to belong to the domain of attraction of a max-semistable law under linear and power normalization. These theorems include the well-known criteria for belonging to the domains of attraction of max-stable laws as a special case; examples are given. The relation between the domains of attraction under linear and power normalization is shown.

Level surfaces of non-degenerate functions inR n+1

We study level surfaces of non-degenerate functions inR n+1. Such level surfaces are non-degenerate in the sense of affine differential geometry. In affine differential geometry, the affine normal plays an important role for the study of a non-degenerate hypersurface. In this note, being motivated by Koszul's work we take a canonical vector field $$\tilde E$$ for level surfaces of a non-degenerate function φ and give certain characterizations of φ when $$\tilde E$$ is transversal, by the shape operatorS, the transversal connection τ, and consider the difference between $$\tilde E$$ and the affine normalĒ.

PSEUDOFORMS WITH NEGATIVE CURVATURE ON . II

On the projective space with hyperplanes in general position removed, a pseudoform with negative holomorphic sectional curvature is constructed in the class of -nondegenerate mappings ().