Structure of complex tori with the automorphisms of maximal degree

Let G be a connected reductive complex affine algebraic group and K ⊂ G a maximal compact subgroup. Let M be a compact complex torus equipped with a flat Kähler structure and (EG, θ) a polystable Higgs G-bundle on M. Take any C∞ reduction of structure group EK ⊂ EG to the subgroup K that solves the Yang–Mills equation for (EG, θ). We prove that the principal G-bundle EG is polystable and the above reduction EK solves the Einstein–Hermitian equation for EG. We also prove that for a semistable (respectively, polystable) Higgs G-bundle (EG, θ) on a compact connected Calabi–Yau manifold, the underlying principal G-bundle EG is semistable (respectively, polystable).

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Non-defectivity of Grassmannian bundles over a curve

International Journal of Mathematics ◽

10.1142/s0129167x16400024 ◽

2016 ◽

Vol 27 (07) ◽

pp. 1640002 ◽

Cited By ~ 1

Author(s):

Insong Choe ◽

George H. Hitching

Keyword(s):

Vector Bundles ◽

Manuscripta Math ◽

Maximal Degree ◽

Geometric Proof ◽

Secant Varieties ◽

Grassmann Bundle ◽

Positive Rank

Let [Formula: see text] be the Grassmann bundle of two-planes associated to a general bundle [Formula: see text] over a curve [Formula: see text]. We prove that an embedding of [Formula: see text] by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the isotropic Segre invariant for maximal isotropic sub-bundles of orthogonal bundles over [Formula: see text], analogous to those given for vector bundles and symplectic bundles in [I. Choe and G. H. Hitching, Secant varieties and Hirschowitz bound on vector bundles over a curve, Manuscripta Math. 133 (2010) 465–477, I. Choe and G. H. Hitching, Lagrangian sub-bundles of symplectic vector bundles over a curve, Math. Proc. Cambridge Phil. Soc. 153 (2012) 193–214]. From the non-defectivity, we also deduce an interesting feature of a general orthogonal bundle of even rank over [Formula: see text], contrasting with the classical and symplectic cases: a general maximal isotropic sub-bundle of maximal degree intersects at least one other such sub-bundle in positive rank.

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Clique coverings of graphs V: maximal-clique partitions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700005414 ◽

1982 ◽

Vol 25 (3) ◽

pp. 337-356 ◽

Cited By ~ 5

Author(s):

N.J. Pullman ◽

H. Shank ◽

W.D. Wallis

Keyword(s):

Maximal Clique ◽

Maximal Degree ◽

Open Problems ◽

Partition Number

A maximal-clique partition of a graph G is a way of covering G with maximal complete subgraphs, such that every edge belongs to exactly one of the subgraphs. If G has a maximal-clique partition, the maximal-clique partition number of G is the smallest cardinality of such partitions. In this paper the existence of maximal-clique partitions is discussed – for example, we explicitly describe all graphs with maximal degree at most four which have maximal-clique partitions - and discuss the maximal-clique partition number and its relationship to other clique covering and partition numbers. The number of different maximal-clique partitions of a given graph is also discussed. Several open problems are presented.

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On Maximal Degree of Domination for Decision Making

Proceedings of the International Conference on Data Engineering 2015 (DaEng-2015) - Lecture Notes in Electrical Engineering ◽

10.1007/978-981-13-1799-6_15 ◽

2019 ◽

pp. 129-135

Author(s):

Suhirman ◽

Jasni Mohamad Zain

Keyword(s):

Decision Making ◽

Maximal Degree

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Relative logarithmic cohomology and Nambu structures of maximal degree

Journal de Mathématiques Pures et Appliquées ◽

10.1016/j.matpur.2020.07.005 ◽

2020 ◽

Vol 144 ◽

pp. 250-268

Author(s):

Konstantinos Kourliouros

Keyword(s):

Maximal Degree

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Improved Resource State for Verifiable Blind Quantum Computation

Entropy ◽

10.3390/e22090996 ◽

2020 ◽

Vol 22 (9) ◽

pp. 996

Author(s):

Qingshan Xu ◽

Xiaoqing Tan ◽

Rui Huang

Keyword(s):

Quantum Computing ◽

Quantum Computation ◽

Fault Tolerant ◽

Maximal Degree ◽

Recent Advances ◽

Quantum Computations ◽

Blind Quantum Computation ◽

Computation Graph ◽

Resource State

Recent advances in theoretical and experimental quantum computing raise the problem of verifying the outcome of these quantum computations. The recent verification protocols using blind quantum computing are fruitful for addressing this problem. Unfortunately, all known schemes have relatively high overhead. Here we present a novel construction for the resource state of verifiable blind quantum computation. This approach achieves a better verifiability of 0.866 in the case of classical output. In addition, the number of required qubits is 2N+4cN, where N and c are the number of vertices and the maximal degree in the original computation graph, respectively. In other words, our overhead is less linear in the size of the computational scale. Finally, we utilize the method of repetition and fault-tolerant code to optimise the verifiability.

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