Quadratic-time, linear-space algorithms for generating orthogonal polygons with a given number of vertices

Programa de Financiamento Plurianual, Fundação para a Ciéncia e TecnologiaPrograma POSIPrograma POCTI, FCTFondo Europeo de Desarrollo Regiona

Thermal States in Conformal QFT. II

We continue the analysis of the set of locally normal KMS states w.r.t. the translation group for a local conformal net A of von Neumann algebras on the real line. In the first part we have proved the uniqueness of KMS state on every completely rational net. In this second part, we exhibit several (non-rational) conformal nets which admit continuously many primary KMS states. We give a complete classification of the KMS states on the U(1)-current net and on the Virasoro net Vir_1 with the central charge c=1, whilst for the Virasoro net Vir_c with c>1 we exhibit a (possibly incomplete) list of continuously many primary KMS states. To this end, we provide a variation of the Araki-Haag-Kastler-Takesaki theorem within the locally normal system framework: if there is an inclusion of split nets A in B and A is the fixed point of B w.r.t. a compact gauge group, then any locally normal, primary KMS state on A extends to a locally normal, primary state on B, KMS w.r.t. a perturbed translation. Concerning the non-local case, we show that the free Fermi model admits a unique KMS state.Comment: 36 pages, no figure. Dedicated to Rudolf Haag on the occasion of his 90th birthday. The final version is available under Open Access. This paper contains corrections to the Araki-Haag-Kaster-Takesaki theorem (and to a proof of the same theorem in the book by Bratteli-Robinson). v3: a reference correcte

MATHEMATICAL NOTES

In [1] Guggenheimer states that a Jordan polygon has two principal vertices that are exposed points of its convex hull, and he refers to Meisters\u27s paper [3]. Such a statement cannot be found in Meisters\u27s paper, and in fact it is false. The polygon illustrated in the figure below provides a counterexample. By a Jordan polygon P = V1 . . . VN is meant a simple closed polygonal plane curve with N sides Vl V2, V2 V3, . . . , VN-1, VN, VN V1, joining the N vertices V1, . . . ,VN. In [3] any consecutive vertices Vi-1, Vi, and Vi+1 of a Jordan polygon P are said to form an ear (regarded as the region enclosed by the triangle Vi-1 Vi Vi-1) at the vertex Vi if the open chord joining Vi-1 andVi+1 lies entirely inside the polygon P. Two such ears are called nonoverlapping if the interiors of their triangular regions are disjoint. The following Two-Ears Theorem was proved in [3]. TWO-EARS THEOREM. Except for triangles, every Jordan polygon has at least two nonoverlapping ears

POLYGONS HAVE EARS

We refer to a simple closed polygonal plane curve with a finite number of sides as a Jordan polygon. We assume the truth of the famous Jordan Curve Theorem only for Jordan polygons. (For elementary proofs see Appendix 2 of Chapter V of [4] or Appendix B1 of [5].) Three consecutive vertices V1, V2, V3 of a Jordan polygon P = V1V2V3V4... VnV1 (n \u3e 4) are said to form an ear (regarded as the region enclosed by the triangle V1V2V3) at the vertex V2 if the (open) chord joining V1 and V3 lies entirely inside the polygon P. We say that two ears are non-overlapping if their interior regions are disjoint; otherwise they are overlapping. If we remove or cut off an ear V1V2V3 (by drawing the chord V1V3) from the Jordan polygon P, then there remains the Jordan polygon P\u27 = V1V3V4 ... VnV1 which has one less vertex than P. The property of Jordan polygons expressed by the following theorem seems to provide a particularly simple and conceptual bridge from the Jordan Curve Theorem for Polygons to the Triangulation Theorem for Jordan Polygons; at least simpler perhaps than that given in Appendix B2 of [5]