A Remark on Concircularly Flat Totally Real Minimal Submanifolds in CP^n

Let M be a concircularly fiat totally real minimal submanifold in CP4. The infimum Vm of the volume V （M） of M is obtained, also the necessary and sufficient conditions of ＂V（M）=Vm＂ is given.     

A generalized Hirzebruch Riemann–Roch theorem

This short Note proves a generalization of the Hirzebruch Riemann–Roch theorem equivalent to the Cardy condition. This is done using an earlier result that explicitly describes what the Mukai pairing on Hochschild homology descends to in Hodge cohomology via the Hochschild–Kostant–Rosenberg map twisted by the root Todd genus. To cite this article: A.C. Ramadoss, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Harder–Narasimhan theory for linear codes (with an appendix on Riemann–Roch theory)

In this text we develop some aspects of Harder–Narasimhan theory, slopes, semistability and canonical filtration, in the setting of combinatorial lattices. Of noticeable importance is the Harder–Narasimhan structure associated to a Galois connection between two lattices. It applies, in particular, to matroids.We then specialize this to linear codes. This could be done from at least three different approaches: using the sphere-packing analogy, or the geometric view, or the Galois connection construction just introduced. A remarkable fact is that these all lead to the same notion of semistability and canonical filtration. Relations to previous propositions toward a classification of codes, and to Wei's generalized Hamming weight hierarchy, are also discussed.Last, we study the two important questions of the preservation of semistability (or more generally the behavior of slopes) under duality, and under tensor product. The former essentially follows from Wei's duality theorem for higher weights—and its matroid version—which we revisit in an appendix, developing analogues of the Riemann–Roch, Serre duality, Clifford, and gap and gonality sequence theorems. Likewise the latter is closely related to the bound on higher weights of a tensor product, conjectured by Wei and Yang, and proved by Schaathun in the geometric language, which we reformulate directly in terms of codes. From this material we then derive semistability of tensor product.     

A Remark on Steinness

In this paper, the authors prove that if Mn is a complete noncompact Kahler manifold with a pole p, and its holomorphic bisectional curvature is asymptotically non-negative to p, then it is a Stein manifold.     

Equivariant Riemann–Roch theorems for curves over perfect fields

We prove an equivariant Riemann–Roch formula for divisors on algebraic curves over perfect fields. By reduction to the known case of curves over algebraically closed fields, we first show a preliminary formula with coefficients in . We then prove and shed some further light on a divisibility result that yields a formula with integral coefficients. Moreover, we give variants of the main theorem for equivariant locally free sheaves of higher rank.     

Riemann–Roch theory for weighted graphs and tropical curves

We define a divisor theory for graphs and tropical curves endowed with a weight function on the vertices; we prove that the Riemann–Roch theorem holds in both cases. We extend Baker’s Specialization Lemma to weighted graphs.     

A Remark on Inverse Problems for Sturm�CLiouville Operators on Graphs

We compare some formulations of the inverse spectral problems for differential operators on graphs with the classical inverse problems on an interval.     

A Remark on Kleiman–Piene's Question for Gauss Maps

We discuss some algebraic properties of the monoid generated by (left) translations in left distributive structures.This furnishes methods for enriching the original structure with a compatible associative product.     

A D’Alembert Formula for Flat Surfaces in the 3-Sphere

We find all the flat surfaces in the unit 3-sphere $\mathbb{S}^{3}We find all the flat surfaces in the unit 3-sphere that pass through a given regular curve of with a prescribed tangent plane distribution along this curve. The formula that solves this problem may be seen as a geometric analogue of the classical D’Alembert formula that solves the Cauchy problem for the homogeneous wave equation. We also provide several applications of this geometric D’Alembert formula, including a classification of the flat M?bius strips of  .      

The Riemann–Roch theorem for the Dynnikov–Novikov discrete complex analysis

We prove an analog of the Riemann–Roch Theorem for the Dynnikov–Novikov discrete complex analysis.     

A Remark on the Singularities of Solutions for Three-dimensional Nonlinear Wave Equations

Synopsis As a remark on [1], it is pointed out that C~2 solutions of u_(tt)-sum from to n=1 to 3(u_(x_4x_4))=u_t~2 with compact supported initial data must vanish everywhere in R_+~4. A remark on Fritz John＇s paper [1] is given in this paper. In[2],a remarkable result is obtained that the Cauchy problem for nonlinear wave euation with small initial data always has global solution if the space dimension≥6.For the on-dimensional wave equation,the situation is just the     

Equivariant Riemann–Roch for G ‐Quasi‐Projective Varieties‐I

Let G denote a complex linear algebraic group. In this paper we establish several forms of equivariant Riemann–Roch valid for the category of Gquasiprojective complex varieties. The main application is to the operation of convolution that appears in the construction of modules over convolution algebras, for example the Hecke algebras associated to a complex reductive group as well as in the representation theory of quantum groups. The paper concludes with a discussion of some of these applications. The results in this paper hold mostly at the level of Grothendieck groups. The second part of this paper will discuss the extension to higher Ktheory in detail.     

A Remark on Finite Horizon Markov Decision processes

本文是利用压缩算子求解的折扣有限水平Markov决策过程逼近非折扣情形的一点注记。这里涉及的状态集与活动集均为可数集。     

Para-CR Structures of Codimension 2 on Tangent Bundles in Riemann–Finsler Geometry

We determine a 2-codimensional para-CR structure on the slit tangent bundle T0 M of a Finsler manifold（M,F） by imposing a condition regarding the almost paracomplex structure P associated to F when restricted to the structural distribution of a framed para-f-structure.This condition is satisfied when（M,F） is of scalar flag curvature（particularly constant） or if the Riemannian manifold（M,g） is of constant curvature.     

A Remark on Partly Dissipative Reaction Diffusion Systems on IR~n

The long time behavior of the solutions of some partly dissipative reaction diffusion systems is studied.We prove the existence of a compact(L~2×L~2-H~1×L~2)attractor for a partly dissipative reaction diffusion system in R~n.This improves a previous result obtained by A.Rodrigues-Bernal and B.Wang concerning the existence of a compact(L~2×L~2-L~2×L~2)attractor for the same system.     

A Remark on З-Permutability of Finite Groups

Let З be a complete set of Sylow subgroups of a finite group G, that is, З contains exactly one and only one Sylow p-subgroup of G for each prime p. A subgroup of a finite group G is said to be З-permutable if it permutes with every member of З. Recently, using the Classification of Finite Simple Groups, Heliel, Li and Li proved tile following result： If the cyclic subgroups of prime order or order 4 iif p = 2） of every member of З are З-permutable subgroups in G, then G is supersolvable. In this paper, we give an elementary proof of this theorem and generalize it in terms of formation.     

A Remark on a Theorem of Szegö

Let be a polynomial with complex coefficients and define, for , where ||P|| is the euclidean norm of the polynomial P. By a theorem of Szegö where is the Mahler measure of F. Recently, J. Dégot proved an effective version of this result. In this paper we sharpen Dégot's result, under the additional hypotheses that F is a square-free polynomial with integer coefficients and without reciprocal factors.     

A Remark on the Hörmander Inequality

A sufficient condition is given in order to ensure the Hörmander inequality for a (formally) self-adjoint transversally elliptic pseudodifferential operator P in the case of the symplectic form changing rank on the characteristic manifold.     

A Remark on Harmonic Maps Between Pseudo-Riemannian Spheres

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