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### The non-existence of right cyclic planar functions of degree pn for n ⩾ 2

Nobuo Nakagawa
1993 Journal of combinatorial theory. Series A
cyclic subgroup for a prime number p and n >/2.  ...  DEDICATED TO PROFESSOR TUYOSHI OYAMA ON HIS 60TH BIRTHDAY In this article we show that there exists no planar function of degree p~ from a finite group G into an abelian group H which contains a large  ...  Especially, there exists no right cyclic planar function of degree pn for n>~2.  ...

### Linear analysis of quandrature domains. II

Björn Gustafsson, Mihai Putinar
2000 Israel Journal of Mathematics
A few simple examples of the interplay between planar domains and pairs of matrices with a cyclic vector end the paper.  ...  The natural correspondence between bounded planar quadrature domains, in the terminology of Aharonov-Shapiro, and certain square matrices with a distinguished cyclic vector is further exploited.  ...  There exists a linear transformation A of C d with a cyclic vector ~ so that P(A) polynomials Qk(z) of degree k (exactly), 0 <_ k < d, with the property d-1 (26) Q(z,-2) --IP(z)[ 2 -IQk(z)l 2. k=O Isr.  ...

### Uniform Random Planar Graphs with Degree Constraints [article]

Chris Dowden
2013 arXiv   pre-print
,n with exactly m(n) edges (this can be thought of as a uniform random planar graph with a constraint on the average degree).  ...  In Part II, we investigate the same three topics for a uniform random planar graph with constraints on the maximum and minimum degrees.  ...  n→∞ Proof We shall suppose, hoping for a contradiction, that there exists a function m(n) such that lim supn→∞ P[Pn,m will contain any non-acyclic components of order ≤ k] = 1.  ...

### Whispering Bloch modes

B. Maling, R. V. Craster
2016 Proceedings of the Royal Society A
We investigate eigenvalue problems for the planar Helmholtz equation in open systems with a high order of rotational symmetry.  ...  , with confinement instead resulting from the geometry of a series of inclusions arranged in a ring.  ...  We are grateful to Alexander Movchan and Stewart Haslinger for useful conversations about multipole techniques and to the anonymous referees for comments that led to several improvements to the article  ...

### Cyclic Multiplicative-Additive Proof Nets of Linear Logic with an Application to Language Parsing [chapter]

Vito Michele Abrusci, Roberto Maieli
2016 Lecture Notes in Computer Science
In its turn, the correctness criterion for CyMLL PNs can be considered as the non-commutative counterpart of the famous Danos-Regnier (DR) criterion for PNs of the pure multiplicative fragment (MLL) of  ...  In particular, it presents a syntax for PNs of the cyclic multiplicative and additive fragment of linear logic (CyMALL).  ...  What complicate matters from a computational point of view is the fact that, dislike the multiplicative case, we could have chosen different additive proof nets for capturing the polymorphic lexical behavior  ...

### Phong Tessellation and PN Polygons for Polygonal Models [article]

Gerben J. Hettinga, Jiri Kosinka
2017 Eurographics State of the Art Reports
In addition, a generalisation of the associated quadratic normal field is given as well as a simple algorithm for evaluating the polygonal extensions for a polygon with vertex normals on the GPU.  ...  We extend Phong tessellation and point normal (PN) triangles from the original triangular setting to arbitrary polygons by use of generalised barycentric coordinates and S-patches.  ...  Acknowledgements This paper is in part based on the first author's MSc thesis at the University of Groningen, and more details can be found therein.  ...

### Combinatorial Dyson–Schwinger equations in noncommutative field theory

2013 Journal of Noncommutative Geometry
We give here the Hopf algebra structure describing the noncommutative renormalization of a recently introduced translation-invariant model on Moyal space.  ...  We define Hochschild one-cocyles B_+^γ which allows us to write down the combinatorial Dyson-Schwinger equations for noncommutative quantum field theory.  ...  Rivasseau for discussions. D. K. thanks K. Yeats for discussions. A. T. was partially suported by a Univ.  ...

### Shortest Vertex-Disjoint Two-Face Paths in Planar Graphs [article]

Eric Colin De Verdière
2008 arXiv   pre-print
Let G be a directed planar graph of complexity n, each arc having a nonnegative length. Let s and t be two distinct faces of G; let s_1,...,s_k be vertices incident with s; let t_1,...  ...  We give an algorithm to compute k pairwise vertex-disjoint paths connecting the pairs (s_i,t_i) in G, with minimal total length, in O(kn n) time.  ...  Acknowledgements We would like to thank Dion Gijswijt and Günter Rote for stimulating discussions.  ...

### Shortest vertex-disjoint two-face paths in planar graphs

Éric Colin De Verdière, Alexander Schrijver
2011 ACM Transactions on Algorithms
Let G be a directed planar graph of complexity n, each arc having a nonnegative length.  ...  We give an algorithm to compute k pairwise vertex-disjoint paths connecting the pairs (si, ti) in G, with minimal total length, in O(kn log n) time.  ...  Acknowledgements We would like to thank Dion Gijswijt and Günter Rote for stimulating discussions.  ...

### A census of hyperbolic platonic manifolds and augmented knotted trivalent graphs [article]

Matthias Goerner
2017 arXiv   pre-print
For the octahedral case, we also identify which manifolds are complements of an augmented knotted trivalent graph and give the corresponding link.  ...  Generalizing an earlier publication by the author and others where this was done in case of the hyperbolic ideal tetrahedron, we give a census of hyperbolic Platonic manifolds and all of their Platonic  ...  The author gratefully thanks Stavros Garoufalidis for many helpful conversations. The author was supported in part by a National Science Foundation grant DMS-11-07452.  ...

### Preservation of bifurcations of Hamiltonian boundary value problems under discretisation [article]

Robert I McLachlan, Christian Offen
2018 arXiv   pre-print
We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators.  ...  Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their bifurcations  ...  Acknowledgements We thank Peter Donelan, Bernd Krauskopf, and Hinke Osinga for useful discussions. This research was supported by the Marsden Fund of the Royal Society Te Apārangi.  ...

### Size-Independent Energy Transfer in Biomimetic Nanoring Complexes

Patrick Parkinson, Nuntaporn Kamonsutthipaijit, Harry L. Anderson, Laura M. Herz
2016 ACS Nano
Binding of the antenna unit creates a local strain in the nanoring, and this strain was measured as a function of the size of the nanoring, by UV−vis-NIR titration, providing information on the conformational  ...  Strikingly, the energy-transfer rate is independent of nanoring size, indicating the existence of strain-localized acceptor states, spread over about six porphyrin units, arising from the noncovalent antenna-nanoring  ...  ACKNOWLEDGMENTS We thank the European Research Council (grant 320969) and the EPSRC for generous financial support and the EPSRC UK Mass Spectrometry Facility at Swansea University for mass spectra.  ...

### On the norms of the random walks on planar graphs

Andrzej Żuk
1997 Annales de l'Institut Fourier
-For all n, we have the following property : ' The sphere Sn is a cyclic graph and X U 5i U ... U Sn is a planar graph.  ...  -Let X be a planar graph such that N(v) > k > 7 for each vertex v € X. Then 1)N(e)=N+a(e). 2) For v ^ e, N^(v) < 2, Nc(v) < 2 and either (a) N-{v) =N-a{v) =1 or (b)JV-0;)=7v-^)=2.  ...

### Ising model and the positive orthogonal Grassmannian [article]

Pavel Galashin, Pavlo Pylyavskyy
2019 arXiv   pre-print
We completely describe by inequalities the set of boundary correlation matrices of planar Ising networks embedded in a disk. Specifically, we build on a recent result of M.  ...  Under our correspondence, the Kramers--Wannier high/low temperature duality transforms into the cyclic symmetry of the Grassmannian, and using this cyclic symmetry, we prove that the spaces under consideration  ...  Thus the cyclic shift operation on OG ≥0 (n, 2n) yields an automorphism of X n which has order 2n for n > 2 and order n for n = 1, 2. For n = 2, it sends σ 12 to 1−σ 12 1+σ 12 .  ...

### Exponential forms and path integrals for complex numbers in n dimensions [article]

Silviu Olariu (National Institute of Physics and Nuclear Engineering, Tandem Laboratory, Magurele, Bucharest, Romania)
2000 arXiv   pre-print
Azimuthal angles, which are cyclic variables, appear in these forms at the exponent, and this leads to the concept of residue for path integrals of n-complex functions.  ...  The exponential function of an n-complex number is expanded in terms of functions called in this paper cosexponential functions, which are generalizations to n dimensions of the circular and hyperbolic  ...  The logarithm exists as a planar n-complex function with real components for all values of x 0 , ..., x n−1 for which ρ = 0.  ...
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