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Every set of natural numbers determines a generating function convergent for $q in (-1,1)$ whose behavior as $q rightarrow 1^-$ determines a germ. These germs admit a natural partial ordering that can be used to compare sets of natural numbers in a manner that generalizes both cardinality of finite sets and density of infinite sets. For any finite set $D$ of positive integers, call a set $S$ $D$-avoiding if no two elements of $S$ differ by an element of $D$. We study the problem of determining, for fixed $D$, all $D$-avoiding sets that are maximal in the germ order. In many cases, we can show that there is exactly one such set. We apply this to the study of one-dimensional packing problems.
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Let $pi_1=(d_1^{(1)}, ldots,d_n^{(1)})$ and $pi_2=(d_1^{(2)},ldots,d_n^{(2)})$ be graphic sequences. We say they emph{pack} if there exist edge-disjoint realizations $G_1$ and $G_2$ of $pi_1$ and $pi_2$, respectively, on vertex set ${v_1,dots,v_n}$ such that for $jin{1,2}$, $d_{G_j}(v_i)=d_i^{(j)}$ for all $iin{1,ldots,n}$. In this case, we say that $(G_1,G_2)$ is a $(pi_1,pi_2)$-textit{packing}. A clear necessary condition for graphic sequences $pi_1$ and $pi_2$ to pack is that $pi_1+pi_2$, their componentwise sum, is also graphic. It is known, however, that this condition is not sufficient, and furthermore that the general problem of determining if two sequences pack is $NP$- complete. S.~Kundu proved in 1973 that if $pi_2$ is almost regular, that is each element is from ${k-1, k}$, then $pi_1$ and $pi_2$ pack if and only if $pi_1+pi_2$ is graphic. In this paper we will consider graphic sequences $pi$ with the property that $pi+mathbf{1}$ is graphic. By Kundus theorem, the sequences $pi$ and $mathbf{1}$ pack, and there exist edge-disjoint realizations $G$ and $mathcal{I}$, where $mathcal{I}$ is a 1-factor. We call such a $(pi,mathbf{1})$ packing a {em Kundu realization}. Assume that $pi$ is a graphic sequence, in which each term is at most $n/24$, that packs with $mathbf{1}$. This paper contains two results. On one hand, any two Kundu realizations of the degree sequence $pi+mathbf{1}$ can be transformed into each other through a sequence of other Kundu realizations by swap operations. On the other hand, the same conditions ensure that any particular 1-factor can be part of a Kundu realization of $pi+mathbf{1}$.
Based on results from the physics and mathematics literature which suggest a series of clearly defined conjectures, we formulate three simple scenarios for the fate of hard sphere crystallization in high dimension: (A) crystallization is impeded and the glass phase constitutes the densest packing, (B) crystallization from the liquid is possible, but takes place much beyond the dynamical glass transition and is thus dynamically implausible, or (C) crystallization is possible and takes place before (or just after) dynamical arrest, thus making it plausibly accessible from the liquid state. In order to assess the underlying conjectures and thus obtain insight into which scenario is most likely to be realized, we investigate the densest sphere packings in dimension $d=3$-$10$ using cell-cluster expansions as well as numerical simulations. These resulting estimates of the crystal entropy near close-packing tend to support scenario C. We additionally confirm that the crystal equation of state is dominated by the free volume expansion and that a meaningful polynomial correction can be formulated.
In this paper we introduce and study the class of d-ball packings arising from edge-scribable polytopes. We are able to generalize Apollonian disk packings and the well-known Descartes theorem in different settings and in higher dimensions. After introducing the notion of Lorentzian curvature of a polytope we present an analogue of the Descartes theorem for all regular polytopes in any dimension. The latter yields to nice curvature relations which we use to construct integral Apollonian packings based on the Platonic solids. We show that there are integral Apollonian packings based on the tetrahedra, cube and dodecahedra containing the sequences of perfect squares. We also study the duality, unicity under Mobius transformations as well as generalizations of the Apollonian groups. We show that these groups are hyperbolic Coxeter groups admitting an explicit matrix representation. An unexpected invariant, that we call Mobius spectra, associated to Mobius unique polytopes is also discussed.
We introduce and analyze an exactly soluble one-dimensional Ising model with long range interactions which exhibits a mixed order transition (MOT), namely a phase transition in which the order parameter is discontinuous as in first order transitions while the correlation length diverges as in second order transitions. Such transitions are known to appear in a diverse classes of models which are seemingly unrelated. The model we present serves as a link between two classes of models which exhibit MOT in one dimension, namely, spin models with a coupling constant which decays as the inverse distance squared and models of depinning transitions, thus making a step towards a unifying framework.
A classical tool in the study of real closed fields are the fields $K((G))$ of generalised power series (i.e., formal sums with well-ordered support) with coefficients in a field $K$ of characteristic 0 and exponents in an ordered abelian group $G$. A fundamental result of Berarducci ensures the existence of irreducible series in the subring $K((G^{leq 0}))$ of $K((G))$ consisting of the generalised power series with non-positive exponents. It is an open question whether the factorisations of a series in such subring have common refinements, and whether the factorisation becomes unique after taking the quotient by the ideal generated by the non-constant monomials. In this paper, we provide a new class of irreducibles and prove some further cases of uniqueness of the factorisation.
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