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A graph $X$ is defined inductively to be $(a_0,\dots,a_{n-1})$-regular if $X$ is $a_0$-regular and for every vertex $v$ of $X$, the sphere of radius $1$ around $v$ is an $(a_1,\dots,a_{n-1})$-regular graph. Such a graph $X$ is said to be highly regular (HR) of level $n$ if $a_{n-1}\neq 0$. Chapman, Linial and Peled studied HR-graphs of level 2 and provided several methods to construct families of graphs which are expanders "globally and locally". They ask whether such HR-graphs of level 3 exist. In this paper we show how the theory of Coxeter groups, and abstract regular polytopes and their generalisations, can lead to such graphs. Given a Coxeter system $(W,S)$ and a subset $M$ of $S$, we construct highly regular quotients of the 1-skeleton of the associated Wythoffian polytope $\mathcal{P}_{W,M}$, which form an infinite family of expander graphs when $(W,S)$ is indefinite and $\mathcal{P}_{W,M}$ has finite vertex links. The regularity of the graphs in this family can be deduced from the Coxeter diagram of $(W,S)$. The expansion stems from applying superapproximation to the congruence subgroups of the linear group $W$. This machinery gives a rich collection of families of HR-graphs, with various interesting properties, and in particular answers affirmatively the question asked by Chapman, Linial and Peled., Comment: 22 pages, 2 tables. Dedicated to the memory of John Conway and Ernest Vinberg

A polytope $${{ \mathsf {P}}}$$ in some euclidean space is called quasi-regular if each facet $${{ \mathsf {F}}}$$ of $${{ \mathsf {P}}}$$ is regular and the symmetry group $${\mathbf {G}}({{ \mathsf {F}}})$$ of $${{ \mathsf {F}}}$$ is a subgroup of the symmetry group $${\mathbf {G}}({{ \mathsf {P}}})$$ of $${{ \mathsf {P}}}$$ . Further, $${{ \mathsf {P}}}$$ is of full rank if its rank and dimension are the same. In this paper, the quasi-regular polytopes of full rank that are not regular are classified. Similarly, an apeirotope of full rank sits in a space of one fewer dimension; the discrete quasi-regular apeirotopes that are not regular are also classified here. One curiosity of the classification is the difference between even and odd dimensions, in that certain families are present in $${\mathbb {E}}^d$$ if d is even, but are absent if d is odd.

We show that for all integers m⩾ 2 ,and all integers k⩾ 2 ,the orthogonal groups O±(2m,F2k) act on abstract regular polytopes of rank 2m, and the symplectic groups Sp(2m,F2k) act on abstract regular polytopes of rank 2 m+ 1., SCOPUS: ar.j, info:eu-repo/semantics/published

Recently the classification of all possible faithful transitive permutation representations of the group of symmetries of a regular toroidal map was accomplished. In this paper we complete this investigation on a surface of genus 1 considering the group of a regular toroidal hypermap of type $(3,3,3)$ that is a subgroup of index $2$ of the group of symmetries of a toroidal map of type $\{6,3\}$., Comment: 8 pages, 2 figures

Neural networks are widely used as a model for classification in a large variety of tasks. Typically, a learnable transformation (i.e. the classifier) is placed at the end of such models returning a value for each class used for classification. This transformation plays an important role in determining how the generated features change during the learning process. In this work, we argue that this transformation not only can be fixed (i.e. set as non-trainable) with no loss of accuracy and with a reduction in memory usage, but it can also be used to learn stationary and maximally separated embeddings. We show that the stationarity of the embedding and its maximal separated representation can be theoretically justified by setting the weights of the fixed classifier to values taken from the coordinate vertices of the three regular polytopes available in $\mathbb{R}^d$, namely: the $d$-Simplex, the $d$-Cube and the $d$-Orthoplex. These regular polytopes have the maximal amount of symmetry that can be exploited to generate stationary features angularly centered around their corresponding fixed weights. Our approach improves and broadens the concept of a fixed classifier, recently proposed in \cite{hoffer2018fix}, to a larger class of fixed classifier models. Experimental results confirm the theoretical analysis, the generalization capability, the faster convergence and the improved performance of the proposed method. Code will be publicly available., Comment: arXiv admin note: substantial text overlap with arXiv:1902.10441

We show that every non-degenerate regular polytope can be used to construct a thin, residually-connected, chamber-transitive incidence geometry, i.e. a regular hypertope. These hypertopes are related to the semi-regular polyotopes with a tail-triangle Coxeter diagram constructed by Monson and Schulte. We discuss several interesting examples derived when this construction is applied to generalised cubes. In particular, we produce an example of a rank 5 finite locally spherical proper hypertope of hyperbolic type. No such examples were previously known.

Simplicial complexes are gaining increasing scientific attention as they are generalized network structures that can represent the many-body interactions existing in complex systems raging from the brain to high-order social networks. Simplicial complexes are formed by simplicies, such as nodes, links, triangles and so on. Cell complexes further extend these generalized network structures as they are formed by regular polytopes such as squares, pentagons etc. Pseudo-fractal simplicial and cell complexes are a major example of generalized network structures and they can be obtained by gluing $2$-dimensional $m$-polygons ($m=2$ triangles, $m=4$ squares, $m=5$ pentagons, etc.) along their links according to a simple iterative rule. Here we investigate the interplay between the topology of pseudo-fractal simplicial and cell complexes and their dynamics by characterizing the critical properties of link percolation defined on these structures. By using the renormalization group we show that the pseudo-fractal simplicial and cell complexes have a continuous percolation threshold at $p_c=0$. When the pseudo-fractal structure is formed by polygons of the same size $m$, the transition is characterized by an exponential suppression of the order parameter $P_{\infty}$ that depends on the number of sides $m$ of the polygons forming the pseudo-fractal cell complex, i.e., $P_{\infty}\propto p\exp(-\alpha/p^{m-2})$. Here these results are also generalized to random pseudo-fractal cell-complexes formed by polygons of different number of sides $m$., Comment: (11 pages,4 figures)

In this paper, we prove that for any positive integers n , s , t {n,s,t} such that n ≥ 10 {n\geq 10} , s , t ≥ 2 {s,t\geq 2} and n - 1 ≥ s + t {n-1\geq s+t} , there exists a regular polytope with Schläfli type { 2 s , 2 t } {\{2^{s},2^{t}\}} and its automorphism group is of order 2 n {2^{n}} . Furthermore, we classify regular polytopes with automorphism groups of order 2 n {2^{n}} and Schläfli types { 4 , 2 n - 3 } , { 4 , 2 n - 4 } {\{4,2^{n-3}\},\{4,2^{n-4}\}} and { 4 , 2 n - 5 } {\{4,2^{n-5}\}} , therefore giving a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, Period. Math. Hungar. 53 2006, 1–2, 231–255].

Submitted by Maria Fernandes (maria.elisa@ua.pt) on 2018-07-20T10:11:16Z No. of bitstreams: 1 2018FL.pdf: 383925 bytes, checksum: 1ca8b5c60c6bba02e54c0edd7a77d15a (MD5) Approved for entry into archive by Rita Gonçalves (ritaisabel@ua.pt) on 2018-07-24T10:54:43Z (GMT) No. of bitstreams: 1 2018FL.pdf: 383925 bytes, checksum: 1ca8b5c60c6bba02e54c0edd7a77d15a (MD5) Made available in DSpace on 2018-07-24T10:54:43Z (GMT). No. of bitstreams: 1 2018FL.pdf: 383925 bytes, checksum: 1ca8b5c60c6bba02e54c0edd7a77d15a (MD5) Previous issue date: 2018-08-15 This research was supported by a Marsden grant (UOA1218) of the Royal Society of New Zealand and by the Portuguese funds through the CIDMA – Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), through CIDMA – Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013. The authors also thank an anonymous referee for useful comments on a preliminary version of this paper. published

In this paper we exhibit the first infinite family of abstract 4-polytopes whose connection groups are not string C-groups. In addition we present some new results on methods of representing the connection group of a polytope in terms of its automorphism group. We also analyze the connection groups of all of the pyramids over the finite regular 3-tori.

Abstract polytopes are combinatorial structures with certain properties drawn from the study of geometric structures, like the Platonic solids, and of maps on surfaces. Of particular interest are the polytopes with maximal possible symmetry (subject to certain natural constraints). Symmetry can be measured by the effect of automorphisms on the ‘flags’ of the polytope, which are maximal chains of elements of increasing rank (dimension). An abstract polytope of rank n is said to be chiral if its automorphism group has precisely two orbits on the flags, such that two flags that differ in one element always lie in different orbits. Examples of chiral polytopes have been difficult to find and construct. In this paper, we introduce a new covering method that allows the construction of some infinite families of chiral polytopes, with each member of a family having the same rank as the original, but with the size of the members of the family growing linearly with one (or more) of the parameters making up its ‘type’ (Schlafli symbol). In particular, we use this method to construct several new infinite families of chiral polytopes of ranks 3, 4, 5 and 6.

Submitted by Maria Fernandes (maria.elisa@ua.pt) on 2019-10-01T15:37:07Z No. of bitstreams: 1 JACDegreesMapsRev2.pdf: 379058 bytes, checksum: ff4da6eb4d35f76f34081923f4eadcfd (MD5) Approved for entry into archive by Rita Gonçalves (ritaisabel@ua.pt) on 2019-10-02T12:07:54Z (GMT) No. of bitstreams: 1 JACDegreesMapsRev2.pdf: 379058 bytes, checksum: ff4da6eb4d35f76f34081923f4eadcfd (MD5) Made available in DSpace on 2019-10-02T12:07:54Z (GMT). No. of bitstreams: 1 JACDegreesMapsRev2.pdf: 379058 bytes, checksum: ff4da6eb4d35f76f34081923f4eadcfd (MD5) Previous issue date: 2019-09-17 published

As the quantum analog of the classical one-time pad, the private quantum channel (PQC) plays a fundamental role in the construction of the maximally mixed state (from any input quantum state), which is very useful for studying secure quantum communications and quantum channel capacity problems. However, the undoubted existence of a relation between the geometric shape of regular polytopes and private quantum channels in the higher dimension has not yet been reported. Recently, it was shown that a one-to-one correspondence exists between single-qubit PQCs and three-dimensional regular polytopes (i.e., regular polyhedra). In this paper, we highlight these connections by exploiting two strategies known as a generalized Gell-Mann matrix and modified quantum Fourier transform. More precisely, we explore the explicit relationship between PQCs over a qutrit system (i.e., a three-level quantum state) and regular 4-polytopes. Finally, we attempt to devise a formula for such connections on higher dimensional cases.

In this paper we describe an infinite chiral 4 -polytope in the Euclidean 3 -space. This builds on previous work of Bracho, Hubard and the author, where a finite chiral 4 -polytope in the Euclidean 4 -space is constructed. These two polytopes show that there are finite and infinite chiral polytopes of full rank as defined by McMullen.

Symmetry can be widely found in natural phenomenon. Regular polygons and polyhedra are the most basic and important symmetrical structures in 2D and 3D Euclidean space. Four-dimensional regular polytopes (4-RPs) are the 4D analogs of regular polyhedra in three dimensions and regular polygons in two dimensions. After introducing the fundamental root systems of 4-RPs, this article presents three interesting methods to visualize 4-RPs using a fundamental region algorithm.

We are interested in enumerating the integer points in certain polytopes that are naturally associated with directed graphs. These polytopes generalize Stanley's order polytopes and also ( P , ω ) -partitions. A classical result states that the number of integer points in any given rational polytope can be expressed by a formula that is piecewise a quasipolynomial in certain parameters of the polytope, and, remarkably, the domains of validity of the involved quasipolynomials overlap. In the case of our special polytopes, the quasipolynomials are shown to be polynomials. We investigate the domains of validity of these polynomials and demonstrate how the overlaps can be used to explore the zero set of the polynomials. We have a closer look at the counting of Gelfand-Tsetlin patterns, which can be phrased as the counting of integer points in a polytope associated with a particular directed graph. We conjecture that the zeros that can be deduced by studying the overlaps essentially determine the enumeration formula in this case.

We study the problem of how to obtain an integer realization of a 3d polytope when an integer realization of its dual polytope is given. We focus on grid embeddings with small coordinates and develop novel techniques based on Colin de Verdi\`ere matrices and the Maxwell-Cremona lifting method. We show that every truncated 3d polytope with n vertices can be realized on a grid of size O(n^{9log(6)+1}). Moreover, for every simplicial 3d polytope with n vertices with maximal vertex degree {\Delta} and vertices placed on an L x L x L grid, a dual polytope can be realized on an integer grid of size O(n L^{3\Delta + 9}). This implies that for a class C of simplicial 3d polytopes with bounded vertex degree and polynomial size grid embedding, the dual polytopes of C can be realized on a polynomial size grid as well., Comment: Full version of the Graph Drawing 2013 conference version, 23 pages, 5 figures