Exact semidefinite programming bounds for packing problems

In this paper we give an algorithm to round the floating point output of a semidefinite programming solver to a solution over the rationals or a quadratic extension of the rationals. We apply this to get sharp bounds for packing problems, and we use these sharp bounds to prove that certain optimal packing configurations are unique up to rotations. In particular, we show that the configuration coming from the $\mathsf{E}_8$ root lattice is the unique optimal code with minimal angular distance $\pi/3$ on the hemisphere in $\mathbb R^8$, and we prove that the three-point bound for the $(3, 8, \vartheta)$-spherical code, where $\vartheta$ is such that $\cos \vartheta = (2\sqrt{2}-1)/7$, is sharp by rounding to $\mathbb Q[\sqrt{2}]$. We also use our machinery to compute sharp upper bounds on the number of spheres that can be packed into a larger sphere.Comment: 24 page

Geometric Packings of Non-Spherical Shapes

The focus of this thesis lies on geometric packings of non-spherical shapes in three-dimensional Euclidean space. Since computing the optimal packing density is difficult, we investigate lower and upper bounds for the optimal value. For this, we consider two special kinds of geometric packings: translative packings and lattice packings. We study upper bounds for the optimal packing density of translative packings. These are packings in which just translations and no rotations of the solids are allowed. Cohn and Elkies determined a linear program for the computation of such upper bounds that is defined by infinitely many inequalities optimizing over an infinite dimensional set. We relax this problem to a semidefinite problem with finitely many constraints, since this kind of problem is efficiently solvable in general. In our computation we consider three-dimensional convex bodies with tetrahedral or icosahedral symmetry. To obtain a program that is not too large for current solvers, we use invariant theory of finite pseudo-reflection groups to simplify the constraints. Since we solve this program by using numerical computations, the solutions might be slightly infeasible. Therefore, we verify the obtained solutions to ensure that they can be made feasible for the Cohn-Elkies program. With this approach we find new upper bounds for three-dimensional superballs, which are unit balls for the l^3_p norm, for p ∈ (1, ∞) \ {2} . Furthermore, using our approach, we improve Zong’s recent upper bound for the translative packing density of tetrahedra from 0.3840 . . . to 0.3683... , which is very close to the best known lower bound of 0.3673... The last part of this thesis deals with lattice packings of superballs. Lattice packing sare translative packings in which the centers of the solids form a lattice. Thus, any lattice packing density is in particular a lower bound for the optimal translative packing density. Using a theorem of Minkowski, we compute locally optimal lattice packings for superballs. We obtain lattice packings for p ∈ [1, 8] whose density is at least as high as the density of the currently best lattice packings provided by Jiao, Stillinger, and Torquato. For p ∈ (1, 2)\[log 2 3, 1.6], we even improve these lattice packings. The upper bounds for p ∈ [3, 8], as well as the numerical results for the upper bounds for p ∈ [1, log 2 , 3], are remarkably close to the lower bounds we obtain by these lattice packings

Das Problem der Kugelpackung

Wie würdest du Tennisbälle oder Orangen stapeln? Oder allgemeiner formuliert: Wie dicht lassen sich identische 3-dimensionale Objekte überschneidungsfrei anordnen? Das Problem, welches auch Anwendungen in der digitalen Kommunikation hat, hört sich einfach an, ist jedoch für Kugeln in höheren Dimensionen noch immer ungelöst. Sogar die Berechnung guter Näherungslösungen ist für die meisten Dimensionen schwierig

New upper bounds for the density of translative packings of three-dimensional convex bodies with tetrahedral symmetry

In this paper we determine new upper bounds for the maximal density of translative packings of superballs in three dimensions (unit balls for the $l_3^p$-norm) and of Platonic and Archimedean solids having tetrahedral symmetry. These bounds give strong indications that some of the lattice packings of superballs found in 2009 by Jiao, Stillinger, and Torquato are indeed optimal among all translative packings. We improve Zong's recent upper bound for the maximal density of translative packings of regular tetrahedra from $0.3840\ldots$ to $0.3745\ldots$, getting closer to the best known lower bound of $0.3673\ldots$. We apply the linear programming bound of Cohn and Elkies which originally was designed for the classical problem of packings of round spheres. The proofs of our new upper bounds are computational and rigorous. Our main technical contribution is the use of invariant theory of pseudo-reflection groups in polynomial optimization

Nora Virus Persistent Infections Are Not Affected by the RNAi Machinery

Drosophila melanogaster is widely used to decipher the innate immune system in response to various pathogens. The innate immune response towards persistent virus infections is among the least studied in this model system. We recently discovered a picorna-like virus, the Nora virus which gives rise to persistent and essentially symptom-free infections in Drosophila melanogaster. Here, we have used this virus to study the interaction with its host and with some of the known Drosophila antiviral immune pathways. First, we find a striking variability in the course of the infection, even between flies of the same inbred stock. Some flies are able to clear the Nora virus but not others. This phenomenon seems to be threshold-dependent; flies with a high-titer infection establish stable persistent infections, whereas flies with a lower level of infection are able to clear the virus. Surprisingly, we find that both the clearance of low-level Nora virus infections and the stability of persistent infections are unaffected by mutations in the RNAi pathways. Nora virus infections are also unaffected by mutations in the Toll and Jak-Stat pathways. In these respects, the Nora virus differs from other studied Drosophila RNA viruses

The triterpene echinocystic acid and its 3-O-glucoside derivative are revealed as potent and selective glucocorticoid receptor agonists

Glucocorticoids are steroid hormones widely used to control many inflammatory conditions. These effects are primarily attributed to glucocorticoid receptor transrepressional activities but with concomitant receptor transactivation associated with considerable side effects. Accordingly, there is an immediate need for selective glucocorticoid receptor agonists able to dissociate transactivation from transrepression. Triterpenoids have structural similarities with glucocorticoids and exhibit anti-inflammatory and apoptotic activities via mechanisms that are not well-defined. In this study, we examined whether echinocystic acid and its 3-O-glucoside derivative act, at least in part, through the regulation of glucocorticoid receptor and whether they can constitute selective receptor activators. We showed that echinocystic acid and its glucoside induced glucocorticoid receptor nuclear translocation by 75% and 55%. They suppressed the nuclear factor-kappa beta transcriptional activity by 20% and 70%, respectively, whereas they have no glucocorticoid receptor transactivation capability and stimulatory effect on the expression of the phosphoenolopyruvate carboxykinase target gene in HeLa cells. Interestingly, their suppressive effect is diminished in glucocorticoid receptor low level COS-7 cells, verifying the receptor involvement in this process. Induced fit docking calculations predicted favorable binding in the ligand binding domain and structural characteristics which can be considered consistent with the experimental observations. Further, glucocorticoids exert apoptotic activities; we have demonstrated here that the echinocystic acids in combination with the synthetic glucocorticoid, dexamethasone, induce apoptosis. Taken together, our results indicate that echinocystic acids are potent glucocorticoid receptor regulators with selective transrepressional activities (dissociated from transactivation), highlighting the potential of echinocystic acid derivatives as more promising treatments for inflammatory conditions

Parametric investigation of flow-sound interaction mechanism of circular cylinders in cross-flow

Flow-excited acoustic resonance in heat exchangers has been an ongoing issue for the past century. The main challenge in this issue, is in the actual prediction of the resonance occurrence. This is due to the complexity of the flow-sound interaction mechanism that takes place between the packed cylinders. Most of the research lately has therefore shifted focus to simpler geometries that resemble the same mechanisms of flow-sound interaction found in actual heat-exchangers. The research presented hereafter summarizes an extensive experimental parametric work performed on multiple simple configurations such as single, tandem and side-by-side cylinders in cross-flow. The main objective of the research is to identify the critical parameters that should be included in the damping criteria to reliably predict the occurrence of acoustic resonance in tube bundles. Special attention is given to the geometrical characteristics of the duct (i.e. cross-sectional area) and how they affect the acoustic resonance. To achieve this; more than one hundred experiments have been performed in three different wind-tunnels of different cross-sectional areas. The research is motivated by the fact that most of the criteria developed to date, fail to predict the destructive phenomena of acoustic resonance in 30-40% of the cases