A Cl closing lemma for equivariant endomorphisms under actions of finite groups

A C^l closing lemma is proved for equivariant endomorphisms under actions of finite groups. Our result shows that, for such an endomorphism, a nonwandering orbit together with its symmetric conjugates can be closed up under a Cl-small equivariant perturbation Project partially supported by the National Natural Science Foundation of China (Grant No. 19531070).     

An inner product lemma

Given the operator product BA in which both A and B are symmetric positive‐definite operators, for which symmetric positive‐definite operators C is BA symmetric positive‐definite in the C inner product 〈x, y〉C? This question arises naturally in preconditioned iterative solution methods, and will be answered completely here. Copyright © 2004 John Wiley & Sons, Ltd.     

An algorithmic hypergraph regularity lemma

Szemerédi 's Regularity Lemma is a powerful tool in graph theory. It asserts that all large graphs admit bounded partitions of their edge sets, most classes of which consist of uniformly distributed edges. The original proof of this result was nonconstructive, and a constructive proof was later given by Alon, Duke, Lefmann, Rödl, and Yuster. Szemerédi's Regularity Lemma was extended to hypergraphs by various authors. Frankl and Rödl gave one such extension in the case of 3‐uniform hypergraphs, which was later extended to k‐uniform hypergraphs by Rödl and Skokan. W.T. Gowers gave another such extension, using a different concept of regularity than that of Frankl, Rödl, and Skokan. Here, we give a constructive proof of a regularity lemma for hypergraphs.     

On the sensitivity of sectional-Anosov flows

We study sectional-Anosov flows on compact 3-manifolds for which the maximal invariant and nonwandering sets coincide. We prove that every vector field close to one of these flows is sensitive with respect to initial conditions.     

An extension lemma and homogeneous programming

An extension lemma, which is equivalent to the generalized Gordan's theorem of the alternative, due to Fan, Glicksberg, and Hoffman, is applied to present a duality theory for a general class of homogeneous programs, with and without a constraint qualification of Slater type. In addition, an existence theorem for optimal solutions of homogeneous programs is given.The author thanks an anonymous referee for valuable suggestions about an earlier draft of this paper.     

An adjacency lemma for critical multigraphs

In edge colouring it is often useful to have information about the degree distribution of the neighbours of a given vertex. For example, the well-known Vizing's Adjacency Lemma provides a useful lower bound on the number of vertices of maximum degree adjacent to a given one in a critical graph. We consider an extension of this problem, where we seek information on vertices at distance two from a given vertex in a critical graph. We extend and, simultaneously, generalize to multigraphs two results proved, respectively, by Zhang [Every planar graph with maximum degree seven is Class 1, Graphs Combin. 16 (2000) 467-495] and Sanders and Zhao [Planar graphs of maximum degree seven are Class 1, J. Combin. Theory Ser. B 83 (2001) 201-212].     

An extension of the mountain pass lemma

We present a more general form of the mountain pass lemma. It asserts that a C¹ functional which satisfies the Palais–Smale condition admits a critical value when the connectedness of certain level sets changes. We also give an improved form of a theorem given in [A. Bahri, H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc. 267 (1) (1981) 1–32], which characterizes the existence of the critical value by means of contractibility properties of the level sets.     

A $${C^\infty}$$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces

We prove a \({C^\infty}\) closing lemma for Hamiltonian diffeomorphisms of closed surfaces. This is a consequence of a \({C^\infty}\) closing lemma for Reeb flows on closed contact three-manifolds, which was recently proved as an application of spectral invariants in embedded contact homology. A key new ingredient of this paper is an analysis of an area-preserving map near its fixed point, which is based on some classical results in Hamiltonian dynamics: existence of KAM invariant circles for elliptic fixed points, and convergence of the Birkhoff normal form for hyperbolic fixed points.     

The arithmetic fundamental lemma: An update

Science China Mathematics - This is an expository article on the recent progress on the arithmetic fundamental lemma conjecture, based largely on Zhang (2019). Beside stating the local conjecture,...     

An operator analogue of the Schwarz-Loewner lemma

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 48, pp. 71–74, 1987.     

An existence lemma for groups generated by 3-transpositions

Oblatum 29-VI-1991 & 28-I-1992     

An application of the neyman-pearson lemma to gaussian processes

Letξ _i (t),i=1,2,t ∈ [0, 1], be Gaussian zero mean processes with continous sample paths. Bounds for the probabilitiesβ _i =P{ξ _i -α _i ∈B},i=1,2, are given, where a_iε C0, 1., and B is a Borel subset of C[0, 1] Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 207, pp. 5–12, 1993. Translated by V. Sudakov.