This paper studies the following question: Given a surface Σ and an integer n, what is the maximum number of cliques in an n-vertex graph embeddable in Σ? We characterise the extremal graphs for this question, and prove that the answer is between 8(n− ω)+ 2 and 8n + 3 2 2 + o(2), where ω is the maximum integer such that the complete graph Kω embeds in Σ. For the surfaces S0, S1, S2, N1, N2, N3 ...

The interplay between between gauge-field winding numbers, θ-vacua, and the Dirac operator spectrum in finite-volume gauge theories is reconsidered. To assess the weight of each topological sector, we compare the mass-dependent chiral condensate in gauge field sectors of fixed topological index with the answer obtained by summing over the topological charge. Also the microscopic Dirac operator ...

The self-interference pattern of a Laguerre Gaussian beam using a Fresnel bi-prism is shown to be very different from what could be expected from a usual laser beam. Actually it resembles the interference pattern that could be obtained using a double slit experiment. The interferences are shifted and the topological charge and its sign can be readily determined considering the shift order of th...

We define the crossing number, a natural invariant in topological graph theory, and survey general results on the crossing number in the Euclidean plane. We also discuss progress on conjectures for two particular cases: complete bipartite graphs and complete graphs. Finally, we show some applications of the crossing number to problems in incidence geometry and VLSI chip design.

We have recently presented evidence that in configurations dominating the regularized pure-glue QCD path integral, the topological charge density constructed from overlap Dirac operator organizes into an ordered space-time structure. It was pointed out that, among other properties, this structure exhibits two important features: it is lowdimensional and geometrically global, i.e. consisting of ...

We give a unifying description of all inequivalent vector bundles over the 2dimensional sphere S2 by constructing suitable global projectors p via equivariant maps. Each projector determines the projective module of finite type of sections of the corresponding complex rank 1 vector bundle over S2. The canonical connection ∇ = p ◦ d is used to compute the topological charges. Transposed projecto...

We define a new kind of crossing number which generalizes both the bipartite crossing number and the outerplanar crossing number. We calculate exact values of this crossing number for many complete bipartite graphs and also give a lower bound. 1 Preliminaries The bipartite crossing number of a bipartite graph G was defined by Watkins in [W] to be the minimum number of crossings over all biparti...

in this paper we find sufficient conditions on primitive inverse topological semigroup s under which: the inversion inv : (h(s)) (h(s)) is continuous; we show that every topologically periodic countable compact primitive inverse topological semigroups with closed h-classes is topologically isomorphic to an orthogonal sum p i2= bi (gi) of topological brandt extensions bi (gi) of countably compac...

in this paper, we have dened and studied a generalized form of topological vectorspaces called s-topological vector spaces. s-topological vector spaces are dened by using semi-open sets and semi-continuity in the sense of levine. along with other results, it is provedthat every s-topological vector space is generalized homogeneous space. every open subspaceof an s-topological vector space is ...