Latest Results The latest content available from Springer Latest Results <h3 class="a-plus-plus">Abstract</h3> <p class="a-plus-plus">As shown by McMullen in 1983, the coefficients of the Ehrhart polynomial of a lattice polytope can be written as a weighted sum of facial volumes. The weights in such a local formula depend only on the outer normal cones of faces, but are far from being unique. In this paper, we develop an infinite class of such local formulas. These are based on choices of fundamental domains in sublattices and obtained by polyhedral volume computations. We hereby also give a kind of geometric interpretation for the Ehrhart coefficients. Since our construction gives us a great variety of possible local formulas, these can, for instance, be chosen to fit well with a given polyhedral symmetry group. In contrast to other constructions of local formulas, ours does not rely on triangulations of rational cones into simplicial or even unimodular ones.</p> 2020-03-01 10.1007/s13366-019-00457-8 Latest Results <h3 class="a-plus-plus">Abstract</h3> <p class="a-plus-plus">In this paper we provide an algorithm, similar to the simplex algorithm, which determines a rational cp-factorization of a given matrix, whenever the matrix allows such a factorization. This algorithm can be used to show that every integral completely positive <span class="a-plus-plus inline-equation id-i-eq1"><span class="a-plus-plus equation-source format-t-e-x">\(2 \times 2\)</span></span> matrix has an integral cp-factorization. </p> 2020-01-28 10.1007/s10107-020-01467-4 Latest Results <h3 class="a-plus-plus">Abstract</h3> <p class="a-plus-plus">In this paper we use topological tools to investigate the structure of the algebraic <em class="a-plus-plus">K</em>-groups <span class="a-plus-plus inline-equation id-i-eq4"> <span class="a-plus-plus equation-source format-t-e-x">\(K_4(R)\)</span> </span> for <span class="a-plus-plus inline-equation id-i-eq5"> <span class="a-plus-plus equation-source format-t-e-x">\(R=Z[i]\)</span> </span> and <span class="a-plus-plus inline-equation id-i-eq6"> <span class="a-plus-plus equation-source format-t-e-x">\(R=Z[\rho ]\)</span> </span> where <span class="a-plus-plus inline-equation id-i-eq7"> <span class="a-plus-plus equation-source format-t-e-x">\(i := \sqrt{-1}\)</span> </span> and <span class="a-plus-plus inline-equation id-i-eq8"> <span class="a-plus-plus equation-source format-t-e-x">\(\rho := (1+\sqrt{-3})/2\)</span> </span>. We exploit the close connection between homology groups of <span class="a-plus-plus inline-equation id-i-eq9"> <span class="a-plus-plus equation-source format-t-e-x">\(\mathrm {GL}_n(R)\)</span> </span> for <span class="a-plus-plus inline-equation id-i-eq10"> <span class="a-plus-plus equation-source format-t-e-x">\(n\le 5\)</span> </span> and those of related classifying spaces, then compute the former using Voronoi’s reduction theory of positive definite quadratic and Hermitian forms to produce a very large finite cell complex on which <span class="a-plus-plus inline-equation id-i-eq11"> <span class="a-plus-plus equation-source format-t-e-x">\(\mathrm {GL}_n(R)\)</span> </span> acts. Our main result is that <span class="a-plus-plus inline-equation id-i-eq12"> <span class="a-plus-plus equation-source format-t-e-x">\(K_{4} ({\mathbb {Z}}[i])\)</span> </span> and <span class="a-plus-plus inline-equation id-i-eq13"> <span class="a-plus-plus equation-source format-t-e-x">\(K_{4} ({\mathbb {Z}}[\rho ])\)</span> </span> have no <em class="a-plus-plus">p</em>-torsion for <span class="a-plus-plus inline-equation id-i-eq14"> <span class="a-plus-plus equation-source format-t-e-x">\(p\ge 5\)</span> </span>.</p> 2019-03-01 10.1007/s40062-018-0212-8 Latest Results <h3 class="a-plus-plus">Abstract</h3> <p class="a-plus-plus">Standard weighted scoring rules do not directly accommodate the possibility that some voters might have dichotomous preferences in three-candidate elections. The direct solution to this issue would be to require voters to arbitrarily break their indifference ties on candidates and report strict rankings. This option was previously found to be a poor alternative when voters have completely independent preferences. The introduction of a small degree of dependence among voters’ preferences has typically been found to make a significant reduction of the impact of such negative outcomes in earlier studies. However, we find that the forced ranking option continues to be a poor choice when dependence is introduced among voters’ preferences. This conclusion is reinforced by the fact that other voting options like Approval Voting and Extended Scoring Rules have been found to produce much better results. These observations are made as a result of using a significant advancement in techniques that obtain probability representations for such outcomes.</p> 2017-11-01 10.1007/s11135-016-0446-7 Latest Results <h3 class="a-plus-plus">Abstract</h3> <p class="a-plus-plus">Given a permutation group acting on coordinates of <span class="a-plus-plus inline-equation id-i-eq1"> <span class="a-plus-plus equation-source format-t-e-x">\({{\mathbb {R}}}^n\)</span> </span>, we consider lattice-free polytopes that are the convex hull of an orbit of one integral vector. The vertices of such polytopes are called <em class="a-plus-plus">core points</em> and they play a key role in a recent approach to exploit symmetry in integer convex optimization problems. Here, naturally the question arises, for which groups the number of core points is finite up to translations by vectors fixed by the group. In this paper we consider transitive permutation groups and prove this type of finiteness for the <span class="a-plus-plus inline-equation id-i-eq2"> <span class="a-plus-plus equation-source format-t-e-x">\(2\)</span> </span>-homogeneous ones. We provide tools for practical computations of core points and obtain a complete list of representatives for all <span class="a-plus-plus inline-equation id-i-eq3"> <span class="a-plus-plus equation-source format-t-e-x">\(2\)</span> </span>-homogeneous groups up to degree twelve. For transitive groups that are not <span class="a-plus-plus inline-equation id-i-eq4"> <span class="a-plus-plus equation-source format-t-e-x">\(2\)</span> </span>-homogeneous we conjecture that there exist infinitely many core points up to translations by the all-ones-vector. We prove our conjecture for two large classes of groups: For imprimitive groups and groups that have an irrational invariant subspace.</p> 2015-01-01 10.1007/s00454-014-9638-x Latest Results <h3 class="a-plus-plus">Abstract</h3> <p class="a-plus-plus">A large amount of literature in social choice theory deals with quantifying the probability of certain election outcomes. One way of computing the probability of a specific voting situation under the Impartial Anonymous Culture assumption is via counting integral points in polyhedra. Here, Ehrhart theory can help, but unfortunately the dimension and complexity of the involved polyhedra grows rapidly with the number of candidates. However, if we exploit available polyhedral symmetries, some computations become possible that previously were infeasible. We show this in three well known examples: Condorcet’s paradox, Condorcet efficiency of plurality voting and in Plurality voting vs Plurality Runoff.</p> 2013-04-01 10.1007/s00355-012-0667-1 Latest Results <h3 class="a-plus-plus">Abstract</h3> <p class="a-plus-plus">In this note we give a short overview on symmetry exploiting techniques in three different branches of polyhedral computations: The representation conversion problem, integer linear programming and lattice point counting. We describe some of the future challenges and sketch some directions of potential developments.</p> 2013-01-01 10.1007/978-3-319-00200-2_15 Latest Results <h3 class="a-plus-plus">Abstract</h3> <p class="a-plus-plus">The contact polytope of a lattice is the convex hull of its shortest vectors. In this paper we classify the facets of the contact polytope of the Leech lattice up to symmetry. There are 1,197,362,269,604,214,277,200 many facets in 232 orbits.</p> 2010-12-01 10.1007/s00454-010-9266-z Latest Results <h3 class="a-plus-plus">Abstract</h3> <p class="a-plus-plus">We report on the recently developed C++ tools <span class="a-plus-plus literal">PermLib</span> and <span class="a-plus-plus literal">SymPol</span> that are designed to support high performance work with symmetric polyhedra. The callable library <span class="a-plus-plus literal">PermLib</span> provides basic support for permutation group algorithms and data structures. It can in particular be used for the development of optimization algorithms that combine methods from polyhedral combinatorics and computational group theory. The software <span class="a-plus-plus literal">SymPol</span> is such an application helping to detect polyhedral symmetries and to analyze faces of polyhedra up to symmetries. It in particular provides successfully used decomposition methods for polyhedral representation conversions up to symmetries.</p> 2010-01-01 10.1007/978-3-642-15582-6_48 Latest Results <h3 class="a-plus-plus">Summary</h3> <p class="a-plus-plus">Let &lt;InlineEquation ID=IE&quot;1&quot;&gt;&lt;EquationSource Format=&quot;TEX&quot;&gt;&lt;![CDATA[&lt;InlineEquation ID=IE&quot;2&quot;&gt;&lt;EquationSource Format=&quot;TEX&quot;&gt;&lt;![CDATA[&lt;InlineEquation ID=IE&quot;3&quot;&gt;&lt;EquationSource Format=&quot;TEX&quot;&gt;&lt;![CDATA[&lt;InlineEquation ID=IE&quot;4&quot;&gt;&lt;EquationSource Format=&quot;TEX&quot;&gt;&lt;![CDATA[&lt;InlineEquation ID=IE&quot;5&quot;&gt;&lt;EquationSource Format=&quot;TEX&quot;&gt;&lt;![CDATA[&lt;InlineEquation ID=IE&quot;6&quot;&gt;&lt;EquationSource Format=&quot;TEX&quot;&gt;&lt;![CDATA[&lt;InlineEquation ID=IE&quot;7&quot;&gt;&lt;EquationSource Format=&quot;TEX&quot;&gt;&lt;![CDATA[&lt;InlineEquation ID=IE&quot;8&quot;&gt;&lt;EquationSource Format=&quot;TEX&quot;&gt;&lt;![CDATA[&lt;InlineEquation ID=IE&quot;9&quot;&gt;&lt;EquationSource Format=&quot;TEX&quot;&gt;&lt;![CDATA[&lt;InlineEquation ID=IE&quot;10&quot;&gt;&lt;EquationSource Format=&quot;TEX&quot;&gt;&lt;![CDATA[&lt;InlineEquation ID=IE&quot;11&quot;&gt;&lt;EquationSource Format=&quot;TEX&quot;&gt;&lt;![CDATA[$]]&gt;&lt;/EquationSource&gt;&lt;/InlineEquation&gt;]]&gt;&lt;/EquationSource&gt;&lt;/InlineEquation&gt;]]&gt;&lt;/EquationSource&gt;&lt;/InlineEquation&gt;]]&gt;&lt;/EquationSource&gt;&lt;/InlineEquation&gt;]]&gt;&lt;/EquationSource&gt;&lt;/InlineEquation&gt;]]&gt;&lt;/EquationSource&gt;&lt;/InlineEquation&gt;]]&gt;&lt;/EquationSource&gt;&lt;/InlineEquation&gt;]]&gt;&lt;/EquationSource&gt;&lt;/InlineEquation&gt;]]&gt;&lt;/EquationSource&gt;&lt;/InlineEquation&gt;]]&gt;&lt;/EquationSource&gt;&lt;/InlineEquation&gt;]]&gt;&lt;/EquationSource&gt;&lt;/InlineEquation&gt;X$ be a discrete subset of Euclidean $d$-space. We allow subsequently continuous movements of single elements, whenever the minimum distance to other elements does not decrease. We discuss the question, if it is possible to move all elements of $X$ in this way, for example after removing a finite subset $Y$ from $X$. Although it is not possible in general, we show the existence of such finite subsets $Y$ for many discrete sets $X$, including all lattices. We define the \textit{instability degree} of $X$ as the minimum cardinality of such a subset $Y$ and show that the maximum instability degree among lattices is attained by perfect lattices. Moreover, we discuss the $3$-dimensional case in detail.</p> 2006-09-01 10.1007/s10998-006-0015-2 Latest Results <h3 class="a-plus-plus">Abstract.</h3> <p class="a-plus-plus">We consider finite lattice ball packings with respect to parametric density and show that densest packings are attained in critical lattices if the number of translates and the density parameter are sufficiently large. A corresponding result is not valid for general centrally symmetric convex bodies.</p> 2005-01-01 10.1007/s00605-004-0262-3 Latest Results <h3 class="a-plus-plus">Abstract.</h3> <p class="a-plus-plus">We show that the shapes of convex bodies containing <em class="a-plus-plus">m</em> translates of a convex body <em class="a-plus-plus">K</em>, so that their Minkowskian surface area is minimum, tends for growing <em class="a-plus-plus">m</em> to a convex body <em class="a-plus-plus">L</em>.</p> 2004-10-01 10.1007/s00013-004-4617-0 Latest Results <h3 class="a-plus-plus">Abstract</h3> <p class="a-plus-plus">We consider finite packings of equal spheres in Euclidean 3–space E<sup class="a-plus-plus">3</sup>. The convex hull of the sphere centers is the packing polytope. In the first part of the paper we prove a tight inequality between the surface area of the packing polytope and the number of sphere centers on its boundary, and investigate in particular the equality cases. The inequality follows from a more general inequality for cell complexes on packing polytopes.</p> 2003-01-01 10.1007/978-3-642-55566-4_35 Latest Results <h3 class="a-plus-plus">Abstract</h3> <p class="a-plus-plus">For a centrally symmetric convex <span class="a-plus-plus inline-equation id-i-e1"> <span class="a-plus-plus equation-source format-t-e-x"> \(K \subset E^2 \) </span> </span> and a covering lattice <em class="a-plus-plus">L</em> for <em class="a-plus-plus">K</em>, a lattice polygon <em class="a-plus-plus">P</em> is called a covering polygon, if <span class="a-plus-plus inline-equation id-i-e2"> <span class="a-plus-plus equation-source format-t-e-x"> \(P \subset (L \cap P) + K\) </span> </span>. We prove that <em class="a-plus-plus">P</em> is a covering polygon, if and only if its boundary bd(<em class="a-plus-plus">P</em>) is covered by (L ∩ P) + K. Further we show that this characterization is false for non-symmetric planar convex bodies and in Euclidean <em class="a-plus-plus">d</em>–space, <em class="a-plus-plus">d</em> ≥ 3, even for the unit ball <em class="a-plus-plus">K = B</em> <sup class="a-plus-plus">d</sup>.</p> 2002-09-01 10.1023/A:1022310416382 Latest Results 2002-06-01 10.1007/BF03024730