The Metric of a graph plays an essential role in the arrangement of different dimensional structures and finding their basis in various terms.The metric dimension of a graph is the selection of the minimum possible nu...The Metric of a graph plays an essential role in the arrangement of different dimensional structures and finding their basis in various terms.The metric dimension of a graph is the selection of the minimum possible number of vertices so that each vertex of the graph is distinctively defined by its vector of distances to the set of selected vertices.This set of selected vertices is known as the metric basis of a graph.In applied mathematics or computer science,the topic of metric basis is considered as locating number or locating set,and it has applications in robot navigation and finding a beacon set of a computer network.Due to the vast applications of this concept in computer science,optimization problems,and also in chemistry enormous research has been conducted.To extend this research to a four-dimensional structure,we studied the metric basis of the Klein bottle and proved that the Klein bottle has a constant metric dimension for the variation of all its parameters.Although the metric basis is variying in 3 and 4 values when the values of its parameter change,it remains constant and unchanged concerning its order or number of vertices.The methodology of determining the metric basis or locating set is based on the distances of a graph.Therefore,we proved the main theorems in distance forms.展开更多
In this paper, we study an infinite-dimensional Lie algebra Bq, called the q-analog Klein bottle Lie algebra. We show that Bq is a finitely generated simple Lie algebra with a unique (up to scalars) symmetric invari...In this paper, we study an infinite-dimensional Lie algebra Bq, called the q-analog Klein bottle Lie algebra. We show that Bq is a finitely generated simple Lie algebra with a unique (up to scalars) symmetric invariant bilinear form. The derivation algebra and the universal central extension of Bq are also determined.展开更多
The Bloch band theory and Brillouin zone(BZ)that characterize wave-like behaviors in periodic mediums are two cornerstones of contemporary physics,ranging from condensed matter to topological physics.Recent theoretica...The Bloch band theory and Brillouin zone(BZ)that characterize wave-like behaviors in periodic mediums are two cornerstones of contemporary physics,ranging from condensed matter to topological physics.Recent theoretical breakthrough revealed that,under the projective symmetry algebra enforced by artificial gauge fields,the usual two-dimensional(2D)BZ(orientable Brillouin two-torus)can be fundamentally modified to a non-orientable Brillouin Klein bottle with radically distinct manifold topology.However,the physical consequence of artificial gauge fields on the more general three-dimensional(3D)BZ(orientable Brillouin three-torus)was so far missing.Here,we theoretically discovered and experimentally observed that the fundamental domain and topology of the usual 3D BZ can be reduced to a non-orientable Brillouin Klein space or an orientable Brillouin half-turn space in a 3D acoustic crystal with artificial gauge fields.We experimentally identify peculiar 3D momentum-space non-symmorphic screw rotation and glide reflection symmetries in the measured band structures.Moreover,we experimentally demonstrate a novel stacked weak Klein bottle insulator featuring a nonzero Z2 topological invariant and self-collimated topological surface states at two opposite surfaces related by a nonlocal twist,radically distinct from all previous 3D topological insulators.Our discovery not only fundamentally modifies the fundamental domain and topology of 3D BZ,but also opens the door towards a wealth of previously overlooked momentum-space multidimensional manifold topologies and novel gaugesymmetry-enriched topological physics and robust acoustic wave manipulations beyond the existing paradigms.展开更多
文摘The Metric of a graph plays an essential role in the arrangement of different dimensional structures and finding their basis in various terms.The metric dimension of a graph is the selection of the minimum possible number of vertices so that each vertex of the graph is distinctively defined by its vector of distances to the set of selected vertices.This set of selected vertices is known as the metric basis of a graph.In applied mathematics or computer science,the topic of metric basis is considered as locating number or locating set,and it has applications in robot navigation and finding a beacon set of a computer network.Due to the vast applications of this concept in computer science,optimization problems,and also in chemistry enormous research has been conducted.To extend this research to a four-dimensional structure,we studied the metric basis of the Klein bottle and proved that the Klein bottle has a constant metric dimension for the variation of all its parameters.Although the metric basis is variying in 3 and 4 values when the values of its parameter change,it remains constant and unchanged concerning its order or number of vertices.The methodology of determining the metric basis or locating set is based on the distances of a graph.Therefore,we proved the main theorems in distance forms.
基金The first author was supported in part by the NSFC (10931006, 10871125) and the Innovation Program of Shanghai Municipal Education Commission (11ZZ18). The second author was supported by the NSFC (11326060). The third author was supported in part by the NSFC (11101285, 11026042, 11071068), the Shanghai Natural Science Foundation (11ZR1425900), the Innovation Program of Shanghai Municipal Education Commission (11YZ85), the Academic Discipline Project of Shanghai Normal University (DZL803) and ZJNSF (Y6100148).
文摘In this paper, we study an infinite-dimensional Lie algebra Bq, called the q-analog Klein bottle Lie algebra. We show that Bq is a finitely generated simple Lie algebra with a unique (up to scalars) symmetric invariant bilinear form. The derivation algebra and the universal central extension of Bq are also determined.
基金funding from the National Natural Science Foundation of China(62375118,6231101016,and 12104211)Shenzhen Science and Technology Innovation Commission(20220815111105001)+8 种基金SUSTech(Y01236148 and Y01236248)Zhengyou Liu acknowledges funding from the National Key R&D Program of China(2022YFA1404900 and 2018YFA0305800)the National Natural Science Foundation of China(11890701)the National Natural Science Foundation of China(12304484)Basic and Applied Basic Research Foundation of Guangdong Province(2414050002552)Shenzhen Science and Technology Innovation Commission(202308073000209)Perry Ping Shum acknowledges the National Natural Science Foundation of China(62220106006)Shenzhen Science and Technology Program(SGDX20211123114001001)Kexin Xiang acknowledges the Special Funds for the Cultivation of Guangdong College Students’Scientific and Technological Innovation(pdjh2023c21002).
文摘The Bloch band theory and Brillouin zone(BZ)that characterize wave-like behaviors in periodic mediums are two cornerstones of contemporary physics,ranging from condensed matter to topological physics.Recent theoretical breakthrough revealed that,under the projective symmetry algebra enforced by artificial gauge fields,the usual two-dimensional(2D)BZ(orientable Brillouin two-torus)can be fundamentally modified to a non-orientable Brillouin Klein bottle with radically distinct manifold topology.However,the physical consequence of artificial gauge fields on the more general three-dimensional(3D)BZ(orientable Brillouin three-torus)was so far missing.Here,we theoretically discovered and experimentally observed that the fundamental domain and topology of the usual 3D BZ can be reduced to a non-orientable Brillouin Klein space or an orientable Brillouin half-turn space in a 3D acoustic crystal with artificial gauge fields.We experimentally identify peculiar 3D momentum-space non-symmorphic screw rotation and glide reflection symmetries in the measured band structures.Moreover,we experimentally demonstrate a novel stacked weak Klein bottle insulator featuring a nonzero Z2 topological invariant and self-collimated topological surface states at two opposite surfaces related by a nonlocal twist,radically distinct from all previous 3D topological insulators.Our discovery not only fundamentally modifies the fundamental domain and topology of 3D BZ,but also opens the door towards a wealth of previously overlooked momentum-space multidimensional manifold topologies and novel gaugesymmetry-enriched topological physics and robust acoustic wave manipulations beyond the existing paradigms.