The Graded Cobordism Group of Codimension-One Immersions

The cobordism group N(Mⁿ) of codimension-one immersions in the n-manifold Mⁿ has a natural filtration induced by any cellular decomposition. The problem addressed in this paper is the explicit computation of the graded group gr*N(Mⁿ). We introduce some new invariants for immersions enlightening the Atiyah–Hirzebruch spectral sequence associated to N(M), which are of combinatorial-geometric nature. Explicit computations are developed for n 7, and the group structure is also investigated for orientable 4-manifolds.     

Symmetry in the vanishing of Ext over stably symmetric algebras

A Frobenius algebra over a field k is called symmetric if the Nakayama automorphism is an inner automorphism. A stably symmetric algebra is defined to be a generalization of a symmetric k-algebra. In this paper we will study symmetry in the vanishing of Ext for such algebras R, namely, for all finitely generated R-modules M and N, for all i0 if and only if for all i0. We show that a certain class of noetherian stably symmetric Gorenstein algebras, such as the group algebra of a finite group and the exterior algebra Λ(kⁿ) when n is odd, have this symmetry using Serre duality. We also show that every exterior algebra Λ(kⁿ), whether n is even or odd, has this symmetry for graded modules using Koszul duality.     

Cohomology of compact hyperkähler manifolds and its applications

We announce the structure theorem for theH ²(M)-generated part of cohomology of a compact hyperkähler manifold. This computation uses an action of the Lie algebra so(4,n–2) wheren=dimH ²(M) on the total cohomology space ofM. We also prove that every two points of the connected component of the moduli space of holomorphically symplectic manifolds can be connected with so-called twistor lines — projective lines holomorphically embedded in the moduli space and corresponding to the hyperkähler structures. This has interesting implications for the geometry of compact hyperkähler manifolds and of holomorphic vector bundles over such manifolds.     

On a conjecture by Erdös-Straus

Two algorithms have been constructed. The first is intended for obtaining such residue-classes represented by the numberN to the given modulusM, that for the primen N (modM) equation (1) is solvable in natural numbersx,y,z. Particularly, whenM=120120 (see Table 2) we obtain 198 suchN, i.e. the hypothesis indicated below is true with a probability greater than 0.99835. The second algorithm is intended for testing the conjecture by Erdös-Straus when 10⁷ <n10⁸.     

Discretization of Compact Riemannian Manifolds Applied to the Spectrum of Laplacian

For κ ⩾ 0 and r₀ > 0 let ℳ(n, κ, r₀) be the set of all connected, compact n-dimensional Riemannian manifolds (Mⁿ, g) with Ricci (M, g) ⩾ −(n−1) κ g and Inj (M) ⩾ r₀. We study the relation between the kth eigenvalue λ_k(M) of the Laplacian associated to (Mⁿ,g), Δ = −div(grad), and the kth eigenvalue λ_k(X) of a combinatorial Laplacian associated to a discretization X of M. We show that there exist constants c, C > 0 (depending only on n, κ and r₀) such that for all M ∈ ℳ(n, κ, r₀) and X a discretization of for all k < |X|. Then, we obtain the same kind of result for two compact manifolds M and N ∈ ℳ(n, κ, r₀) such that the Gromov–Hausdorff distance between M and N is smaller than some η > 0. We show that there exist constants c, C > 0 depending on η, n, κ and r₀ such that for all . Mathematics Subject Classification (2000): 58J50, 53C20 Supported by Swiss National Science Foundation, grant No. 20-101 469     

A Sharp Inequality of Markov Type for Polynomials Associated with Laguerre Weight

The best possible constant A_n in an inequality of Markov type

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, where ·_[0, ∞) denotes the sup-norm on the half real line [0, ∞) and p_n is an arbitrary polynomial of degree at most n, is determined in terms of the weighted Chebyshev polynomials associated with the Laguerre weight e^−x on [0, ∞).     

An adaptive fast direct solver for boundary integral equations in two dimensions

We describe an algorithm for the rapid direct solution of linear algebraic systems arising from the discretization of boundary integral equations of potential theory in two dimensions. The algorithm is combined with a scheme that adaptively rearranges the parameterization of the boundary in order to minimize the ranks of the off-diagonal blocks in the discretized operator, thus obviating the need for the user to supply a parameterization r of the boundary for which the distance ‖r(s)−r(t)‖ between two points on the boundary is related to their corresponding distance |s−t| in the parameter space. The algorithm has an asymptotic complexity of , where N is the number of nodes in the discretization. The performance of the algorithm is illustrated with several numerical examples.     

On Mean Convergence of Lagrange Interpolation for General Arrays

For n1, let {x_jn}ⁿ_j=1 be n distinct points in a compact set K and letL_n[·] denote the corresponding Lagrange interpolation operator. Let v be a suitably restricted function on K. What conditions on the array {x_jn}_1jn, n1 ensure the existence of p>0 such that lim_n→∞ (f−L_n[f]) v_Lp(K)=0 for very continuous f: K→ ? We show that it is necessary and sufficient that there exists r>0 with sup_n1 π_nv_Lr(K) ∑ⁿ_j=1 (1/|π′_n| (x_jn))<∞. Here for n1, π_n is a polynomial of degree n having {x_jn}ⁿ_j=1 as zeros. The necessity of this condition is due to Ying Guang Shi.     

Tighter Bounds on the Solution of a Divide-and-Conquer Maximin Recurrence

LetM(n) be defined by the recurrencewherefis an arbitrary nondecreasing function andM(1) is given. The recurrenceM(n) is a divide-and-conquer maximin recurrence, which occurs in a variety of problems in the analysis of algorithms. In this paper, a new upper bound onM(n) is first derived. The derived bound is smaller than the one proposed previously by Li and Reingold. It is at most two times the exact solution ofM(n). Using the bound, we further show thatM(n) ≤ 2E(n), whereE(n) is defined by the recurrenceE(n) = E(⌊n/2⌋) + E(⌈n/2⌉) + f(⌊n/2⌋). From this result, we can conclude that a divide-and-conquer algorithm whose time complexity is expressed asM(n) is as efficient as a divide-and-conquer algorithm whose time complexity is expressed asE(n).     

Direct algorithm for computation of derivatives of the Daubechies basis functions

We extend the direct algorithm for computing the derivatives of the compactly supported Daubechies N-vanishing-moment basis functions. The method yields exact values at dyadic rationals for the nth derivative (0  n  N − 1) of the basis functions, when it exists. Example results are shown for the first derivatives of the basis functions from the Daubechies N-vanishing-moment extremal phase orthonormal family (for N = 3, 4, and 5), and the CDF(2, N) spline-based biorthogonal family (for N = 6, 8, and 10).     

Pinching theorems of hypersurfaces in a unit sphere

Let Mn be a complete hypersurface in Sn+1(1) with constant mean curvature. Assume that Mn has n−1 principal curvatures with the same sign everywhere. We prove that if RicM≤C₋(H), either S?S₊(H) or RicM?0 or the fundamental group of Mn is infinite, then S is constant, S=S₊(H) and Mn is isometric to a Clifford torus with . These rigidity theorems are still valid for compact hypersurface without constancy condition on the mean curvature.     

Erdős–Szekeres-Type Theorems for Segments and Noncrossing Convex Sets

A family of convex sets is said to be in convex position, if none of its members is contained in the convex hull of the others. It is proved that there is a function N(n) with the following property. If is a family of at least N(n) plane convex sets with nonempty interiors, such that any two members of have at most two boundary points in common and any three are in convex position, then has n members in convex position. This result generalizes a theorem of T. Bisztriczky and G. Fejes Tóth. The statement does not remain true, if two members of may share four boundary points. This follows from the fact that there exist infinitely many straight-line segments such that any three are in convex position, but no four are. However, there is a function M(n) such that every family of at least M(n) segments, any four of which are in convex position, has n members in convex position.     

On the Norm of the Metric Projections

LetXbe a Banach space. GivenMa subspace ofXwe denote withP_Mthe metric projection ontoM. We defineπ(X) sup{P_M: Ma proximinal subspace ofX}. In this paper we give a bound forπ(X). In particular, whenX=L_p, we obtain the inequality P_M2^|2/p−1|, for every subspaceMofL_p. We also show thatπ(X)=π(X*).     

An n log n Algorithm for Online BDD Refinement

Binary decision diagrams are in widespread use in verification systems for the canonical representation of finite functions. Here we consider multivalued BDDs, which represent functions of the form : ^ν →  , where is a finite set of leaves. We study a rather natural online BDD refinement problem: a partition of the leaves of several shared BDDs is gradually refined, and the equivalence of the BDDs under the current partition must be maintained in a discriminator table. We show that it can be solved in O(n log n) time if n bounds both the size of the BDDs and the total size of update operations. Our algorithm is based on an understanding of BDDs as the fixed points of an operator that in each step splits and gathers nodes. We apply our algorithm to show that automata BDD-represented transition functions can be minimized in time O(n · log n), where n is the total number of BDD nodes representing the automaton. This result is not an instance of Hopcroft's classical minimization algorithm, which breaks down for BDD-represented automata because of the BDD path compression property.     

Some Applications of Critical Point Theory of Distance Functions on Riemannian Manifolds

In this paper, we use the theory of critical points of distance functions to study the rigidity and topology of Riemannian manifolds with sectional curvature bounded below. We prove that an n-dimensional complete connected Riemannian manifold M with sectional curvature K _M 1 is isometric to an n-dimensional Euclidean unit sphere if M has conjugate radius bigger than /2 and contains a geodesic loop of length 2. We also prove that if M is an n(3)-dimensional complete connected Riemannian manifold with K _M 1 and radius bigger than /2, then any closed connected totally geodesic submanifold of dimension not less than two of M is homeomorphic to a sphere.     

Minimal Morse functions on a pair of manifolds

The existence theorem for a minimal Morse function on a pair of manifolds (M ⁿ,N ^k), wheren-k 3,k 6, is proved.Translated from Ukrainskii Matematiccheskii Zhurnal, Vol. 45, No. 1, pp. 143–144, January, 1993.     

General absolute summability factor theorems involving quasi-power-increasing sequences

In this paper a general theorem concerning the |A,δ|_k summability methods has been proved, which generalizes two results of Şevli and Leindler [H. Şevli and L. Leindler, On the absolute summability factors of infinite series involving quasi-power-increasing sequences, Computers and Mathematics with Applications 57 (2009), 702–709]. We obtain sufficient conditions for ∑a_nλ_n to be summable |A,δ|_k, k1, 0δ<1/k, by using quasi-power-increasing sequences.     

Conformal capacities and conformally invariant functions on Riemannian manifolds

This paper develops the theory of conformal invariants initiated inJ. Differential Geom 8 (1973), 487–510 for a Riemannian manifoldM with dimensionn2. We construct and study four conformally invariant functions M, M, M, M resp. depending on 4, 3 or 2 points onM, defined as extremal capacities for condensers associated with those points. These functions have similarities with the classical invariants onS ⁿ ,R ⁿ orH ⁿ . Their properties, and especially their continuity, are efficient tools for solving some problems of conformal geometry in the large.     

The index of maximum ambiguous density for irreducible non-powerful sign pattern matrices

Let INS_n,p be the set of n×n irreducible non-powerful (generalized) sign pattern matrices with period p, and let AINS_n,p. In this paper, we introduce a new parameter called the index of maximum ambiguous density of A. Furthermore, the generalized index of maximum ambiguous density of A, which generalizes the concept of the index of maximum ambiguous density, is introduced. Moreover, some bounds on these indices are obtained, and we exhibit a system of gaps in the set of the index of maximum ambiguous density for AINS_n,p. Finally, the index and the generalized index of maximum ambiguous density for irreducible non-powerful zero-symmetric sign pattern matrices are discussed.