An abstract version of the Morse index theorem and its application to hypersurfaces of constant mean curvature

In [5] Smale generalized the Morse index theorem (originally proved by Morse in [3]) to elliptic partial differential systems in several independent variables. Smale's result was used by Simons in [4] to obtain the index theorem for minimal submanifolds. The purpose of this paper is to give an abstract version of the Morse index theorem and use it to prove an index theorem for hypersurfaces of constant mean curvature. This was sugested by Barbosa and do Carmo in [1].     

Convexity Theorem for Subharmonic Functions with Respect to the Kolmogorov Operator

In this note, a version of the classical Riesz theorem for convexity of mean of subharmonic function on a annulus is given for the Kolmogorov's operator. Further, potential theoretical interpretations of a `mean value measure' are listed. Mathematics Subject Classifications (2000): 31B35, 31C05, 31C15.     

Quantitative Results on Algebraic Independence of Some Numbers with Good Approximations

In this paper, we shall establish transcendence measures of several numbers with good approximations by algebraic numbers. This generalizes the results of Caveny (Proc. Amer. Math. Soc. 120 (1994) 349–357) and Tubbs (Proc. Amer. Math. Soc. 109 (1990) 297–304), and gives a quantitative version of Ably's result (C.R. Acad. Sci. Paris 1(313), série I (1991) 653–655).     

DANZER-GRüNBAUM'S THEOREM REVISITED

In this paper we prove some stronger versions of Danzer-Grünbaum's theorem including the following stability-type result. For 0 < α < 14π/27 the maximum number of vertices of a convex polyhedron in E ³ such that all angles between adjacent edges are bounded from above by α is 8. One of the main tools is the spherical geometry version of Pál's theorem. This revised version was published online in August 2006 with corrections to the Cover Date.     

Two complexity results on c-optimality in experimental design

Finding a c-optimal design of a regression model is a basic optimization problem in statistics. We study the computational complexity of the problem in the case of a finite experimental domain. We formulate a decision version of the problem and prove its NP\boldsymbol{\mathit{NP}}-completeness. We provide examples of computationally complex instances of the design problem, motivated by cryptography. The problem, being NP\boldsymbol{\mathit{NP}}-complete, is then relaxed; we prove that a decision version of the relaxation, called approximate c-optimality, is P-complete. We derive an equivalence theorem for linear programming: we show that the relaxed c-optimality is equivalent (in the sense of many-one LOGSPACE-reducibility) to general linear programming.     

On the D-affinity of the flag variety in type B2

The flag varieties in characteristic 0 are well-known to be D-affine. In positive characteristic, however, only those in type A ₁ and A ₂ have been proved to be so. In this paper we will show in type B ₂ the cohomology vanishing of the first term in the p-filtration of the sheaf of differential operators on the flag variety. This is a necessary condition for the variety to be D-affine. Received: 7 February 2000 / Revised version: 30 June 2000     

Representing a Functional Curve by Curves with Fewer Peaks

In this paper, we study the problems of (approximately) representing a functional curve in 2-D by a set of curves with fewer peaks. Representing a function (or its curve) by certain classes of structurally simpler functions (or their curves) is a basic mathematical problem. Problems of this kind also find applications in applied areas such as intensity-modulated radiation therapy (IMRT). Let f\bf f be an input piecewise linear functional curve of size n. We consider several variations of the problems. (1) Uphill–downhill pair representation (UDPR): Find two nonnegative piecewise linear curves, one nondecreasing (uphill) and one nonincreasing (downhill), such that their sum exactly or approximately represents f\bf f. (2) Unimodal representation (UR): Find a set of unimodal (single-peak) curves such that their sum exactly or approximately represents f\bf f. (3) Fewer-peak representation (FPR): Find a piecewise linear curve with at most k peaks that exactly or approximately represents f\bf f. Furthermore, for each problem, we consider two versions. For the UDPR problem, we study its feasibility version: Given ε>0, determine whether there is a feasible UDPR solution for f\bf f with an approximation error ε; its min-ε version: Compute the minimum approximation error ε ^∗ such that there is a feasible UDPR solution for f\bf f with error ε ^∗. For the UR problem, we study its min-k version: Given ε>0, find a feasible solution with the minimum number k ^∗ of unimodal curves for f\bf f with an error ε; its min-ε version: given k>0, compute the minimum error ε ^∗ such that there is a feasible solution with at most k unimodal curves for f\bf f with error ε ^∗. For the FPR problem, we study its min-k version: Given ε>0, find one feasible curve with the minimum number k ^∗ of peaks for f\bf f with an error ε; its min-ε version: given k≥0, compute the minimum error ε ^∗ such that there is a feasible curve with at most k peaks for f\bf f with error ε ^∗. Little work has been done previously on solving these functional curve representation problems. We solve all the problems (except the UR min-ε version) in optimal O(n) time, and the UR min-ε version in O(n+mlog m) time, where m<n is the number of peaks of f\bf f. Our algorithms are based on new geometric observations and interesting techniques.     

Sur la reéduction modulo p des polynoômes absolument irreéductibles

 On the reduction modulo p of absolutely irreducible polynomials. Let K be a number field and F(X,Y) be an absolutely irreducible polynomial of K[X,Y]. In this note, using an effective version of Riemann-Roch theorem and a version of the implicit functions theorem, we calculate a positive number A such that if ℘ is prime ideal of the ring of integers of K with norm , then the reduction of F(X,Y) modulo ℘ is an absolutely irreducible polynomial. (Réu le 1 Février 1999; en forme finale 21 Septembre 1999)     

Locally symmetric minimal affine Lagrangian surfaces in C<Superscript>2</Superscript>

One of the basic facts known in the theory of minimal Lagrangian surfaces is that a minimal Lagrangian surface of constant curvature in C ² must be totally geodesic. In affine geometry the constancy of curvature corresponds to the local symmetry of a connection. In Opozda (Geom. Dedic. 121:155–166, 2006), we proposed an affine version of the theory of minimal Lagrangian submanifolds. In this paper we give a local classification of locally symmetric minimal affine Lagrangian surfaces in C ². Only very few of surfaces obtained in the classification theorems are Lagrangian in the sense of metric (pseudo-Riemannian) geometry. The research supported by the KBN grant 1 PO3A 034 26.     

Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds

We give a sufficient condition for a metric (homology) manifold to be locally bi-Lipschitz equivalent to an open subset in R ⁿ . The condition is a Sobolev condition for a measurable coframe of flat 1-forms. In combination with an earlier work of D. Sullivan, our methods also yield an analytic characterization for smoothability of a Lipschitz manifold in terms of a Sobolev regularity for frames in a cotangent structure. In the proofs, we exploit the duality between flat chains and flat forms, and recently established differential analysis on metric measure spaces. When specialized to R ⁿ , our result gives a kind of asymptotic and Lipschitz version of the measurable Riemann mapping theorem as suggested by Sullivan.     

On the construction of stable curvilinear pp version elements for mixed formulations of elasticity and Stokes flow

Summary. The use of mixed finite element methods is well-established in the numerical approximation of the problem of nearly incompressible elasticity, and its limit, Stokes flow. The question of stability over curved elements for such methods is of particular significance in the p version, where, since the element size remains fixed, exact representation of the curved boundary by (large) elements is often used. We identify a mixed element which we show to be optimally stable in both p and h refinement over curvilinear meshes. We prove optimal p version (up to ) and h version (p = 2, 3) convergence for our element, and illustrate its optimality through numerical experiments. Received August 25, 1998 / Revised version received February 16, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

Graded pseudo-valuation domains

Abstract

We introduce a graded version of pseudo-valuation rings for integral domains graded by a torsionless monoid, and we extend several results to the graded situation.     

Cobordism of immersions of surfaces¶in non-orientable 3-manifolds

Some properties of non-orientable 3-manifolds are shown. In particular, for a connected, non-orientable 3-manifold M, the group of cobordism clases of immersions of surfaces in M is isomorphic to a group structure on the set H ₂(M,Z/2Z)×H ₁(M,Z/2Z)×Z/2Z. Received: 8 June 2000 / Revised version: 2 October 2000     

Abstract,weighted, and multidimensional Adamjan-Arov-Krein theorems,and the singular numbers of Sarason commutants

The classical Adamjan-Arov-Krein (A-A-K) theorem relating the singular numbers of Hankel operators to best approximations of their symbols by rational functions is given an abstract version. This provides results for Hankel operators acting in weightedH ²(T; ), as well as inH ²(T ^d ), and an A-A-K type extension of Sarason's interpolation theorem. In particular, it is shown that all compact Hankel operators inH ²(T ^d ) are zero.Author partially supported by NSF grant DMS89-11717.     

The Monodromy Conjecture for hyperplane arrangements

The Monodromy Conjecture asserts that if c is a pole of the local topological zeta function of a hypersurface, then exp(2πic) is an eigenvalue of the monodromy on the cohomology of the Milnor fiber. A stronger version of the conjecture asserts that every such c is a root of the Bernstein-Sato polynomial of the hypersurface. In this note we prove the weak version of the conjecture for hyperplane arrangements. Furthermore, we reduce the strong version to the following conjecture: −n/d is always a root of the Bernstein-Sato polynomial of an indecomposable essential central hyperplane arrangement of d hyperplanes in C ⁿ .     

Reflexivity of N-tuples of contractions with rich joint left essential spectrum

It will be shown that WOT-closed algebra generated by N-tuple of double commuting contractions, for which the polydiscD ^N is a spectral set and whose joint left essential spectrum is dominating for the algebraH (D ^N ) is reflexive. The second version of our main result, instead of double commutativity, uses the membership of the classC ₀.     

VORONOI POLYHEDRA OF UNIT BALL PACKINGS WITH SMALL SURFACE AREA

In this note, first, we give a very short new proof of the theorem which yields a lower bound for the surface area of Voronoi cells of unit ball packings in E ^d and implies Rogers' upper bound for the density of unit ball packings in E ^d for all d ≥ 2. Second we sharpen locally a classical result of Gauss by finding the locally smallest surface area Voronoi cells of lattice unit ball packings in E ³. This revised version was published online in August 2006 with corrections to the Cover Date.     

Polynomial diffeomorphisms of C2: V. critical points and Lyapunov exponents

There is an invariant measure μ, which is the pluri-complex version of the harmonic measure of the Julia set for polynomial maps of C.In this paper we give an integral formula for the Lyapunov exponents of a polynomial automorphism with respect to μ, analogous to the Brolin-Manning formula polynomial maps of C.Our formula relates the Lyapunov exponents to the value of a Green function at a type of critical point which we define in this paper. We show that these the critical points have a natural dynamical interpretation.     

Optimized look-ahead recurrences for adjacent rows in the Padé table

A new look-ahead algorithm for recursively computing Padé approximants is introduced. It generates a subsequence of the Padé approximants on two adjacent rows (defined by fixed numerator degree) of the Padé table. Its two basic versions reduce to the classical Levinson and Schur algorithms if no look-ahead is required. The new algorithm can be viewed as a combination of the look-ahead sawtooth and the look-ahead Levinson and Schur algorithms that we proposed before, but now the look-ahead step size is minimal (as in the sawtooth version) and the computational costs are as low as in the least expensive competing algorithms (including our look-ahead Levinson and Schur algorithms). The underlying recurrences link well-conditioned basic pairs,i.e., pairs of sufficiently different neighboring Padé forms.The algorithm can be used to solve Toeplitz systems of equationsTx = b. In this application it comes in several versions: anO(N ²) Levinson-type form, anO(N ²) Schur-type form, and a superfastO(N log² N) Schur-type version. As an option of the first two versions, the corresponding block LDU decompositions ofT ^–1 orT, respectively, can be found.     

Global Rank Axioms for Poset Matroids

An excellent introduction to the topic of poset matroids is due to Barnabei, Nicoletti and Pezzoli. In this paper, we investigate the rank axioms for poset matroids; thereby we can characterize poset matroids in a “global” version and a “pseudo-global” version. Some corresponding properties of combinatorial schemes are also obtained.