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Ts.Ch-D. Batueva O.V. Borodin M.A. Bykov A.O. Ivanova O.N. Kazak D.V. Nikiforov 《Discrete Mathematics》2017,340(11):2659-2664
The weight of an edge in a normal plane map (NPM) is the degree-sum of its end-vertices. An edge is of type if and . In 1940, Lebesgue proved that every NPM has an edge of one of the types , , or , where 7 and 6 are best possible. In 1955, Kotzig proved that every 3-connected planar graph has an edge with , which bound is sharp. Borodin (1989), answering Erd?s’ question, proved that every NPM has either a -edge, or -edge, or -edge.A vertex is simplicial if it is completely surrounded by 3-faces. In 2010, Ferencová and Madaras conjectured (in different terms) that every 3-polytope without simplicial 3-vertices has an edge with . Recently, we confirmed this conjecture by proving that every NPM has either a simplicial 3-vertex adjacent to a vertex of degree at most 10, or an edge of types , , or .By a -vertex we mean a -vertex incident with precisely triangular faces. The purpose of our paper is to prove that every NPM has an edge of one of the following types: , , , , , , or , where all bounds are best possible. In particular, this implies that the bounds in , , and can be attained only at NPMs having a simplicial 3-, 4-, or 5-vertex, respectively. 相似文献
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Fritz Gesztesy Lance L. Littlejohn Isaac Michael Richard Wellman 《Journal of Differential Equations》2018,264(4):2761-2801
In 1961, Birman proved a sequence of inequalities , for , valid for functions in . In particular, is the classical (integral) Hardy inequality and is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space of functions defined on . Moreover, implies ; as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite , these inequalities hold on the standard Sobolev space . Furthermore, in all cases, the Birman constants in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in (resp., ). We also show that these Birman constants are related to the norm of a generalized continuous Cesàro averaging operator whose spectral properties we determine in detail. 相似文献
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《Applied Mathematics Letters》2005,18(11):1256-1264
In this paper, we discuss the existence of positive periodic solutions to the nonlinear differential equation where is an -periodic continuous function with , is continuous and is also an -periodic function for each . Using the fixed point index theory in a cone, we get an essential existence result because of its involving the first positive eigenvalue of the linear equation with regard to the above equation. 相似文献
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Elena Angelini Francesco Galuppi Massimiliano Mella Giorgio Ottaviani 《Journal of Pure and Applied Algebra》2018,222(4):950-965
We prove that a general polynomial vector in three homogeneous variables of degrees has a unique Waring decomposition of rank 7. This is the first new case we are aware of, and likely the last one, after five examples known since the 19th century and the binary case. We prove that there are no identifiable cases among pairs in three homogeneous variables of degree , unless , and we give a lower bound on the number of decompositions. The new example was discovered with Numerical Algebraic Geometry, while its proof needs Nonabelian Apolarity. 相似文献
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Fixed point and coincidence results are presented for single-valued generalized -weakly contractive mappings on complete metric spaces , where is a lower semicontinuous function with and for all and is a function such that is nonempty and closed. Our results extend previous results given by Rhoades (2001) [1] and by Zhang and Song (2009) [2]. 相似文献
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