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We consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold M isometrically immersed into another Riemannian manifold . We first assume the pull back Weitzenböck operator of bounded from below, and obtain an extrinsic lower bound for the first eigenvalue of Hodge-Laplacian. As applications, we obtain some rigidity results. Second, when the pull back Weitzenböck operator of bounded from both sides, we give a lower bound of the first eigenvalue by the Ricci curvature of M and some extrinsic geometry. As a consequence, we prove a weak Ejiri type theorem, that is, if the Ricci curvature bounded from below pointwisely by a function of the norm square of the mean curvature vector, then M is a homology sphere. In the end, we give an example to show that all the eigenvalue estimates are optimal when is the space form. 相似文献
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Eugenio Giannelli Gunter Malle Carolina Vallejo Rodríguez 《Journal of Pure and Applied Algebra》2019,223(2):900-907
We characterise finite groups such that for an odd prime p all the irreducible characters in its principal p-block have odd degree. We show that this situation does not occur in non-abelian simple groups of order divisible by p unless and the group is . As a consequence we deduce that if or if is not a composition factor of a group G, then the condition above is equivalent to having odd order. 相似文献
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In this paper we study solitons invariant with respect to the flow generated by a complete parallel vector field in a ambient Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product and the parallel field is . Similarly to what happens in the Euclidean setting, we call them translating solitons. We see that a translating soliton in can be seen as a minimal submanifold for a weighted volume functional. Moreover we show that this kind of solitons appear in a natural way in the context of a monotonicity formula for the mean curvature flow in . When is rotationally invariant and its sectional curvature is non-positive, we are able to characterize all the rotationally invariant translating solitons. Furthermore, we use these families of new examples as barriers to deduce several non-existence results. 相似文献
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Michael Giudici S.P. Glasby Cai Heng Li Gabriel Verret 《Journal of Pure and Applied Algebra》2019,223(3):1217-1226
Let Γ be a finite G-vertex-transitive digraph. The in-local action of is the permutation group induced by a vertex-stabiliser on the set of in-neighbours of the corresponding vertex. The out-local action is defined analogously. Note that and may not be isomorphic. We thus consider the problem of determining which pairs are possible. We prove some general results, but pay special attention to the case when and are both quasiprimitive. (Recall that a permutation group is quasiprimitive if each of its nontrivial normal subgroups is transitive.) Along the way, we prove a structural result about pairs of finite quasiprimitive groups of the same degree, one being (abstractly) isomorphic to a proper quotient of the other. 相似文献
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We consider the pseudo-Euclidean space , , with coordinates and metric , , where at least one is positive, and also tensors of the form , such that are differentiable functions of x. For such tensors, we use Lie point symmetries to find metrics that solve the Ricci curvature and the Einstein equations. We provide a large class of group-invariant solutions and examples of complete metrics defined globally in . As consequences, for certain functions , we show complete metrics , conformal to the pseudo-Euclidean metric g, whose scalar curvature is . 相似文献
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General dual curvature measures have recently been introduced by Lutwak, Yang and Zhang [24]. These new measures unify several other geometric measures of the Brunn–Minkowski theory and the dual Brunn–Minkowski theory. dual curvature measures arise from qth dual intrinsic volumes by means of Alexandrov-type variational formulas. Lutwak, Yang and Zhang [24] formulated the dual Minkowski problem, which concerns the characterization of dual curvature measures. In this paper, we solve the existence part of the dual Minkowski problem for and , and we also discuss the regularity of the solution. 相似文献
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Following his discovery that finite metric spaces have injective envelopes naturally admitting a polyhedral structure, Isbell, in his pioneering work on injective metric spaces, attempted a characterization of cellular complexes admitting the structure of an injective metric space. A bit later, Mai and Tang confirmed Isbell’s conjecture that a simplicial complex is injectively metrizable if and only if it is collapsible. Considerable advances in the understanding, classification and applications of injective envelopes have since been made by Dress, Huber, Sturmfels and collaborators, and most recently by Lang. Unfortunately a combination theory for injective polyhedra is still unavailable.Here we expose a connection to the duality theory of cubings –simply connected non-positively curved cubical complexes –which provides a more principled and accessible approach to Mai and Tang’s result, providing one with a powerful tool for systematic construction of locally-compact injective metric spaces:Main Theorem. Any complete pointed Gromov–Hausdorff limit of locally-finite piecewise- cubings is injective. □This result may be construed as a combination theorem for the simplest injective polytopes, -parallelopipeds, where the condition for retaining injectivity is the combinatorial non-positive curvature condition on the complex. Thus it represents a first step towards a more comprehensive combination theory for injective spaces.In addition to setting the earlier work on injectively metrizable complexes within its proper context of non-positively curved geometry, this paper is meant to provide the reader with a systematic review of the results – otherwise scattered throughout the geometric group theory literature – on the duality theory and the geometry of cubings, which make this connection possible. 相似文献
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《Discrete Mathematics》2020,343(1):111641
A graph is called -induced-saturated if does not contain an induced copy of , but removing any edge from creates an induced copy of and adding any edge of to creates an induced copy of . Martin and Smith studied a related problem, and proved that there does not exist a -induced-saturated graph, where is the path on 4 vertices. Axenovich and Csikós gave examples of families of graphs for which -induced-saturated graph exists, and asked if there exists a -induced-saturated graph when . Our aim in this short note is to show that there exists a -induced-saturated graph. 相似文献
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In this paper, the sufficient condition in terms of the RIC and ROC for the stable and robust recovery of signals in both noiseless and noisy settings was established via weighted minimization when there is partial prior information on support of signals. An improved performance guarantee has been derived. We can obtain a less restricted sufficient condition for signal reconstruction and a tighter recovery error bound under some conditions via weighted minimization. When prior support estimate is at least 50% accurate, the sufficient condition is weaker than the analogous condition by standard minimization method, meanwhile the reconstruction error upper bound is provably to be smaller under additional conditions. Furthermore, the sufficient condition is also proved sharp. 相似文献
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For a random walk on we study the asymptotic behaviour of the associated centre of mass process . For lattice distributions we give conditions for a local limit theorem to hold. We prove that if the increments of the walk have zero mean and finite second moment, is recurrent if and transient if . In the transient case we show that has a diffusive rate of escape. These results extend work of Grill, who considered simple symmetric random walk. We also give a class of random walks with symmetric heavy-tailed increments for which is transient in . 相似文献
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Aleksander Vesel 《Discrete Mathematics》2019,342(4):1139-1146
The generalized Fibonacci cube is the graph obtained from the -cube by removing all vertices that contain a given binary string as a substring. If is an induced subgraph of , then the cube-complement of is the graph induced by the vertices of which are not in . In particular, the cube-complement of a generalized Fibonacci cube is the subgraph of induced by the set of all vertices that contain as a substring. The questions whether a cube-complement of a generalized Fibonacci cube is (i) connected, (ii) an isometric subgraph of a hypercube or (iii) a median graph are studied. Questions (ii) and (iii) are completely solved, i.e. the sets of binary strings that allow a graph of this class to be an isometric subgraph of a hypercube or a median graph are given. The cube-complement of a daisy cube is also studied. 相似文献
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Trying to interpret B. Zilber's project on model theory of quantum mechanics we study a way of building limit models from finite-dimensional approximations. Our point of view is that of metric model theory, and we develop a method of taking ultraproducts of unbounded operators. We first calculate the Feynman propagator for the free particle as defined by physicists as an inner product of the eigenvector of the position operator with eigenvalue and , where is the time evolution operator. However, due to a discretising effect, the eigenvector method does not work as expected, and straightforward calculations give the wrong value. We look at this phenomenon, and then complement this by showing how to instead correctly calculate the kernel of the time evolution operator (for both the free particle and the harmonic oscillator) in the limit model. We believe that our method of calculating these is new. 相似文献
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Lionel Nguyen Van Thé 《Expositiones Mathematicae》2019,37(2):192-199
Say that a graph is representable in if there is a map from its vertex set into the Euclidean space such that iff and are both edges or both non-edges in . The purpose of this note is to present the proof of the following result, due to Einhorn and Schoenberg in Einhorn and Schoenberg (1966): if finite is neither complete nor independent, then it is representable in . A similar result also holds in the case of finite complete edge-colored graphs. 相似文献
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Saeed Nasseh Sean Sather-Wagstaff Ryo Takahashi Keller VandeBogert 《Journal of Pure and Applied Algebra》2019,223(3):1272-1287
We construct a local Cohen–Macaulay ring R with a prime ideal such that R satisfies the uniform Auslander condition (UAC), but the localization does not satisfy Auslander's condition (AC). Given any positive integer n, we also construct a local Cohen–Macaulay ring R with a prime ideal such that R has exactly two non-isomorphic semidualizing modules, but the localization has non-isomorphic semidualizing modules. Each of these examples is constructed as a fiber product of two local rings over their common residue field. Additionally, we characterize the non-trivial Cohen–Macaulay fiber products of finite Cohen–Macaulay type. 相似文献