Gorenstein projective dimensions of complexes

We show that over a right coherent left perfect ring R, a complex C of left R-modules is Gorenstein projective if and only if C ^m is Gorenstein projective in R-Mod for all m ∈ ℤ. Basing on this we show that if R is a right coherent left perfect ring then Gpd(C) = sup{Gpd(C ^m )|m ∈ ℤ} where Gpd(−) denotes Gorenstein projective dimension.     

On the classification of perfect codes: side class structures

The side class structure of a perfect 1-error correcting binary code (hereafter referred to as a perfect code) C describes the linear relations between the coset representatives of the kernel of C. Two perfect codes C and C′ are linearly equivalent if there exists a non-singular matrix A such that AC = C′ where C and C′ are matrices with the code words of C and C′ as columns. Hessler proved that the perfect codes C and C′ are linearly equivalent if and only if they have isomorphic side class structures. The aim of this paper is to describe all side class structures. It is shown that the transpose of any side class structure is the dual of a subspace of the kernel of some perfect code and vice versa; any dual of a subspace of a kernel of some perfect code is the transpose of the side class structure of some perfect code. The conclusion is that for classification purposes of perfect codes it is sufficient to find the family of all kernels of perfect codes.     

On Intersection Problem for Perfect Binary Codes

The main result is that to any even integer q in the interval 0 ≤ q ≤ 2^{n+1-2log(n+1)}, there are two perfect codes C₁ and C₂ of length n = 2^m − 1, m ≥ 4, such that |C₁ ∩ C₂| = q.     

Optimal tactics for close support operations. II. Degraded communications with linear time-varying dynamics

The optimal decision rules for force commitments are obtained in mathematical studies of C³ (command, control, and communication). Recursive equations for the solution of the C³ problem are derived for a perturbation model with linear time-varying dynamics. Air and ground commanders are assumed to have perfect intelligence with degraded communication between them. Numerical results are given for several amphibious assaults.     

4类图完美匹配数目的递推求法

本文研究了4类特殊图完美匹配数目的显式表达式.利用划分,求和,再递推的方法分别给出了图3-n Z4,2-n(2-C6),2-n(2-K4)和3-n(C4-C6)的完美匹配数目的计算公式.     

Representation theorem for locally defined operators in the space of Whitney differentiable functions

Under the assumption that is perfect, a representation theorem for locally defined operators mapping the space C ^m (A) of Whitney differentiable functions into C ¹(A) is given and an open problem is presented.     

Groups with an almost regular involution

An involution j of a group G is said to be almost perfect in G if any two involutions in j^G whose product has infinite order are conjugated by a suitable involution in j^G. Let G contain an almost perfect involution j and |C_G(j)| < ∞. Then the following statements hold: (1) [j,G] is contained in an FC-radical of G, and |G: [j,G]| ⩽ |C_G(j)|; (2) the commutant of an FC-radical of G is finite; (3) FC(G) contains a normal nilpotent class 2 subgroup of finite index in G. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 360–368, May–June, 2007.     

Berge graphs with chordless cycles of bounded length

A graph is called weakly triangulated if it contains no chordless cycle on five or more vertices (also called hole) and no complement of such a cycle (also called antihole). Equivalently, we can define weakly triangulated graphs as antihole-free graphs whose induced cycles are isomorphic either to C₃ or to C₄. The perfection of weakly triangulated graphs was proved by Hayward [Hayward, J Combin Theory B. 39 (1985), 200–208] and generated intense studies to efficiently solve, for these graphs, the classical NP-complete problems that become polynomial on perfect graphs. If we replace, in the definition above, the C₄ by an arbitrary C_p (p even, at least equal to 6), we obtain new classes of graphs whose perfection is shown in this article. In fact, we prove a more general result: for any even integer p ≥ 6, the graphs whose cycles are isomorphic either to C₃ or to one of C_p, C_p+2, …, C_{2p 6} are perfect. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 73–79, 1999     

On the binary codes with parameters of doubly-shortened 1-perfect codes

We show that any binary (n = 2 ^k − 3, 2 ^n−k , 3) code C ₁ is a cell of an equitable partition (perfect coloring) (C ₁, C ₂, C ₃, C ₄) of the n-cube with the quotient matrix ((0, 1, n−1, 0)(1, 0, n−1, 0)(1, 1, n−4, 2)(0, 0, n−1, 1)). Now the possibility to lengthen the code C ₁ to a 1-perfect code of length n + 2 is equivalent to the possibility to split the cell C ₄ into two distance-3 codes or, equivalently, to the biparticity of the graph of distances 1 and 2 of C ₄. In any case, C ₁ is uniquely embedable in a twofold 1-perfect code of length n + 2 with some structural restrictions, where by a twofold 1-perfect code we mean that any vertex of the space is within radius 1 from exactly two codewords. By one example, we briefly discuss 2 − (n, 3, 2) multidesigns with similar restrictions. We also show a connection of the problem with the problem of completing latin hypercuboids of order 4 to latin hypercubes.     

Triangular actions onC 3

Free algebraic actions of a connected algebraic groupG onC ³ which can be triangularized are shown to be trivial, that isC ³ is equivariantly isomorphic toGxC ^3–dimG . This result follows directly from the case of the additive groupG=G _a and is shown to hold for quasi-algebraic actions as well. Connections with the classification of homogeneous affine varieties are discussed.Partially supported by NSF grant DMS 8420315     

A binary perfect code of length 15 and codimension 0

We construct three new binary perfect codesC ₁,C ₂ andC ₃ of length 15. We show that dim(ker(C _i))=i fori=1, 2 and 3. It follows that the codimension ofC ₁ equals 0.     

Cantor exchange systems and renormalization

We prove a one-to-one correspondence between (i) C¹⁺ conjugacy classes of C^1+H Cantor exchange systems that are C^1+H fixed points of renormalization and (ii) C¹⁺ conjugacy classes of C^1+H diffeomorphisms f with a codimension 1 hyperbolic attractor Λ that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Λ. However, we prove that there is no C^1+α Cantor exchange system, with bounded geometry, that is a C^1+α fixed point of renormalization with regularity α greater than the Hausdorff dimension of its invariant Cantor set.     

Rigidity of critical circle mappings I

We prove that two C ³ critical circle maps with the same rotation number in a special set ? are C ^1+α conjugate for some α>0 provided their successive renormalizations converge together at an exponential rate in the C ⁰ sense. The set ? has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C ^∞ critical circle maps with the same rotation number that are not C ^1+β conjugate for any β>0. The class of rotation numbers for which such examples exist contains Diophantine numbers. Received November 1, 1998 / final version received July 7, 1999     

Two shape preserving lagrangeC 2-interpolants

Summary We present a LagrangeC ²-interpolant to scattered convex data which preserves convexity. We also present a LagrangeC ²-interpolant to uniformly spaced monotone data sites which preserves monotonicity. In both cases no further conditions are required on the data values. These interpolants are explicitely described and local. Error isO(h ³) when the function to be interpolated isC ³.     

Finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves

We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations over curves over perfect fields. For example, if k is finitely generated over ℚ and X → C is a quadric fibration of odd relative dimension at least 11, then CH ⁱ (X) is finitely generated for i ≤ 4.     

Volumes of certain small geodesic balls and almost-Hermitian geometry

Let D be the characteristic connection of an almost-Hermitian manifold, V _D ^m (r) the volume of a small geodesic ball for the connection D and C C _D ¹ the first non-trivial term of the Taylor expansion of V _D ^m (r). NK-manifolds are characterized in terms of C C _D ¹ and a family of Hermitian manifolds for which _M C C _D ¹ dvol is a spectral invariant is given and one proves that C C _D ¹ and the spectrum of the complex Laplacian, together, determine the class in which a compact Hermitian manifold lines.     

Finite sets on curves and surfaces

A complete proof is given for Schnirelmann’s theorem on the existence of a square inC ² Jordan curves. The following theorems are then proved, using the same method: 1. On every hypersurface inR ⁿ,C ³-diffeomorphic toS ⁿ⁻¹, there exist 2n points which are the vertices of a regular 2 ⁿ -cellC _n. 2. Every planeC′ Jordan curve can beC′ approximated by a curve on which there are 2N distinct points which are the vertices of a centrally symmetric 2N-gon (anglesπ not excluded). 3. On every planeC ² curve there exist 5 distinct points which are the vertices of an axially symmetric pentagon with given base anglesa, π/2≦a<π. (The angle at the vertex on the axis of symmetry might beπ). Research supported by Grant AF-AFOSR-664-64, Air Force Office of Scientific Research.     

A Nonlinear Version of Noether's Type Theorem

Abstract

Let L be a line bundle on a smooth curve C, which defines a birational morphism onto Φ(C) ? P ^r . We prove that, under suitable assumptions on L, which are satisfied by Castelnuovo's curves, a generic section in H ⁰(C, L ²) can be written as α² + β² + γ², with α, β, γ ∈ H ⁰(C, L). If there are no quadrics of rank 3 containing Φ(C), this is true for any section. For canonical curves, this gives a non linear version of Noether's Theorem.     

On Gevrey solvability and regularity

In this paper we study global C^∞ and Gevrey solvability for a class of sublaplacian defined on the torus T ³. We also prove Gevrey regularity for a class of solutions of certain operators that are globally C^∞ hypoelliptic in the N ‐dimensional torus (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Evolution Equations for Special Lagrangian 3-Folds in C3

This is the third in a series of papers constructing explicit examples of special Lagrangian submanifolds in C ^m . The previous paper (Math. Ann. 320 (2001), 757–797), defined the idea of evolution data, which includes an (m – 1)-submanifold P in R ⁿ , and constructed a family of special Lagrangian m-folds N in C ^m , which are swept out by the image of P under a 1-parameter family of affine maps _t : R ⁿ C ^m , satisfying a first-order o.d.e. in t. In this paper we use the same idea to construct special Lagrangian 3-folds in C³. We find a one-to-one correspondence between sets of evolution data with m = 3 and homogeneous symplectic 2-manifolds P. This enables us to write down several interesting sets of evolution data, and so to construct corresponding families of special Lagrangian 3-folds in C³.Our main results are a number of new families of special Lagrangian 3-foldsin C³, which we write very explicitly in parametric form. Generically these are nonsingular as immersed 3-submanifolds, and diffeomorphic to R³ or ¹× R². Some of the 3-folds are singular, and we describe their singularities, which we believe are of a new kind.We hope these 3-folds will be helpful in understanding singularities ofcompact special Lagrangian 3-folds in Calabi–Yau 3-folds. This will beimportant in resolving the SYZ conjecture in Mirror Symmetry.