Perturbative and non perturbative effects in the Standard Model and orbifolded ADS/CFT based theories

We study some perturbative and nonperturbative effects in the framework of the Standard Model of particle physics. In particular we consider the time dependence of the Higgs vacuum expectation value given by the dynamics of the StandardModel and study the non-adiabatic production of both bosons and fermions, which is intrinsically non-perturbative. In theHartree approximation, we analyze the general expressions that describe the dissipative dynamics due to the backreaction of the produced particles. Then, we solve numerically some relevant cases for the Standard Model phenomenology in the regime of relatively small oscillations of the Higgs vacuum expectation value (vev). As perturbative effects, we consider the leading logarithmic resummation in small Bjorken x QCD, concentrating ourselves on the Nc dependence of the Green functions associated to reggeized gluons. Here the eigenvalues of the BKP kernel for states of more than three reggeized gluons are unknown in general, contrary to the large Nc limit (planar limit) case where the problem becomes integrable. In this contest we consider a 4-gluon kernel for a finite number of colors and define some simple toy models for the configuration space dynamics, which are directly solvable with group theoretical methods. In particular we study the depencence of the spectrum of thesemodelswith respect to the number of colors andmake comparisons with the planar limit case. In the final part we move on the study of theories beyond the Standard Model, considering models built on AdS5 S5/Γ orbifold compactifications of the type IIB superstring, where Γ is the abelian group Zn. We present an appealing three family N = 0 SUSY model with n = 7 for the order of the orbifolding group. This result in a modified Pati–Salam Model which reduced to the StandardModel after symmetry breaking and has interesting phenomenological consequences for LHC.

Determinazione di amminotioli mediante elettroforesi capillare

The aminothiols are critical cellular components that play numerous and important roles in metabolism as key extracellular reducing agents, critical substrates for proteins synthesis and detoxificants of free radicals and peroxides. Because altered thiols levels in body fluids are linked to specific pathological conditions, their measurement is thus considered very important. One method to determine these compounds is the capillary electrophoresis, a technique that involves the separation of charged molecules on the basis of their movement under the influence of an applied electric field. The instrument used in this work is equipped with an amperometric detector recording the current of the thiols oxidized at the end of the capillary at a BDD electrode. The aim of this work is to find a valid method for the separations of the aminothiols analyzed, in terms of capillary coating and experimental conditions. In order to find an alternative and less expensive electrode than BDD and to increase sensitivity for the detection of the thiols, a modified electrode consisting in a carbon paste electrode containing Cobalt-phthalocyanine has been studied. In this electrode Cobalt-phthalocyanine works as electrocatalyst to enhance the oxidation reaction, meanwhile the graphite acts as conductive mean. This kind of electrode shows great sensibility and low detection limits for the thiols that have a free thiolic group, but it is not sensible to disulfides. The analysis of human plasma point out that the best method found for the capillary electrophoresis is not useful for the detection of aminothiols in a healthy person, because the very low concentrations in which they are present.

Konstruktion symmetrischer Designs

In dieser Arbeit werden zehn neue symmetrische (176,50,14) Designs und ein neues symmetrisches (144,66,30) Design durch Vorgabe von nichtauflösbaren Automorphismengruppen konstruiert. Im Jahre 1969 entdeckte G. Higman ein symmetrisches (176,50,14) Design, dessen volle Automorphismengruppe die sporadische einfache Gruppe HS der Ordnung 44.352.000 ist. Hier wurden nun Designs gesucht, die eine Untergruppe von HS zulassen. Folgende Untergruppen wurden betrachtet: die transitive und die intransitive Erweiterung einer elementarabelschen Gruppe der Ordnung 16 durch Alt(5), AGL(3,2), das direkte Produkt einer zyklischen Gruppe der Ordnung 5 mit Alt(5) und PSL(2,11). Die transitive Erweiterung von E(16) durch Alt(5) lieferte zwei neue Designs mit Automorphismengruppen der Ordnungen 960 bzw. 11.520; letzteres konnte auch mit der transitiven Erweiterung erhalten werden. Die Gruppe PSL(2,11) operiert auf den Punkten des Higman-Designs in drei Bahnen; sucht man nach symmetrischen (176,50,14) Designs, auf denen diese Gruppe in zwei Bahnen operiert, so erhält man acht neue Designs. Die übrigen Gruppen lieferten keine neuen Designs. Schließlich konnte ein neues symmetrisches (144,66,30) Design unter Verwendung der sporadischen Mathieu-Gruppe M(12) konstruiert werden. Dies war zu diesem Zeitpunkt außer dem Higman-Design das einzige bekannte symmetrische Design, dessen volle Automorphismengruppe im Wesentlichen eine sporadische einfache Gruppe ist.

N-Vinylamidderivate zur Synthese neuartiger Kern-Schale-Partikel und Hydrogele

N-Vinylamidderivate sind eine toxikologische unbedenkliche Monomerklasse. Mit diesen Monomeren wurden verschiedene technische Anwendungsgebiete im Bereich der Kern-Schale-Partikel und der fließfähigen und vernetzten Hydrogele untersucht. Kern-Schale-Partikel Für die Synthese von Kern-Schale-Partikeln wurden die N-Vinylamidderivate als Schalenpolymere auf kommerziellen Poly(styrol-stat.-butadien)-Kernpartikeln aufpolymerisiert. Mit Hilfe verschiedener Untersuchungsmethoden (DLS, SEM, FFF, Ultrazentrifuge) wurde die Kern-Schale-Strukturbildung und die Effizienz der Pfropfungsreaktion untersucht und eine erfolgreiche Synthese der Kern-Schale-Partikel belegt. Durch die gezielte Modifizierung des Schalenpolymers wurde ein kationisches, organisches Mikropartikelsystem entwickelt, charakterisiert und auf die Eignung als „Duales Flockungsmittel“ untersucht. Diese Versuche belegten die Eignung der modifizierten Kern-Schale-Partikel als „Duales Flockungsmittel“ und bieten eine Alternative zu kommerziell verwendeten Retentionsmitteln. Außerdem wurden die filmbildenden Eigenschaften der Poly(N﷓vinylformamid)-Kern-Schale-Dispersionen untersucht. Nach der Verfilmung der Dispersionen wurden transparente und harte Filme erhalten. Die Auswirkungen auf die mechanischen Eigenschaften der Filme wurden durch die Variation verschiedener Parameter eingehend studiert. Auf der Basis dieser Partikel wurden selbstvernetzende Dispersionssysteme entwickelt. Das P﷓(VFA)-Schalenpolymer wurde teilweise hydrolysiert und die generierten freien Aminogruppen des Poly(N-vinylamins) durch eine Michael-Addition mit einem divinylfunktionalisierten Acrylat (Tetraethylenglykoldiacrylat) vernetzt. Untersuchungen zur mechanischen Beständigkeit der Filme zeigten bei geringen Vernetzungsgraden eine deutliche Optimierung der maximalen Zugbelastungen. Die Untersuchungen belegten, dass die Verwendung des selbstvernetzenden Dispersionssystems als Dispersion für eine Polymerbeschichtung möglich ist. Hydrogele Die Synthese von fließfähigen und quervernetzten Hydrogelen erfolgte auf der Basis verschiedener N﷓Vinylamide. Mit Hilfe geeigneter Vernetzer wurden feste Hydrogelplatten synthetisiert und für die Auftrennung von DNA-Sequenzen mit Hilfe der Gelelektrophorese verwendet. Scharfe und gute Auftrennung der verschiedenen „DNA-Ladder Standards“ wurden durch die Variation des Vernetzeranteils, der Polymerzusammensetzung, der angelegten Spannung und der Verweilzeit in der Gelelektrophoresekammer mit P﷓(MNVA)-Hydrogelplatten erreicht. Fließfähige und quervernetzte Elektrolytgele auf Poly-(N-vinylamid)-Basis wurden in wartungsfreien pH﷓Elektroden eingesetzt. Die Eignung dieser Hydrogele wurden in Bezug auf die Anwendung eingehend charakterisiert. Elektroden befüllt mit Poly(N-vinylamid)-Gelen wurden in Dauerbelastungsexperimenten, direkt mit kommerziellen pH﷓Elektroden verglichen. Es konnte gezeigt werden, dass die fließfähigen und quervernetzten Poly-(N-vinylamid)-Elektrolytgele in ihren Messeigenschaften gleichwertige bzw. bessere Potentialstabilitäten aufweisen als kommerzielle Referenzelektroden. Die Hydrogele auf Basis von Poly(N-vinylamidderivaten) boten für die beiden getesteten Anwendungen eine toxikologisch unbedenkliche Alternative zu Poly(acrylamid)-Gelen. In dieser Arbeit konnten die durchgeführten Untersuchungen belegen, dass N﷓Vinylamide eine attraktive Monomerklasse ist, die erfolgreich in vielen technischen Anwendungen einsetzbar ist.

Derivate des Ionentransporters Valinomycin

In der vorliegenden Arbeit wurden Derivate des Ionentransporters Valinomycin synthetisiert und als Sensorelemente in Modellmembransysteme inkorporiert. Als Modellmembranen wurden festkörpergestützte Lipiddoppelschichten (tBLMs, tethered bilayer lipid membranes) verwendet. rnValinomycin transportiert selektiv Kalium-Ionen durch Membranen, was durch einen Rückgang des Widerstandes über elektrochemische Messmethoden nachgewiesen werden kann. Es ist ein zyklisches Dodecadepsipeptid, das aus zwei verschiedenen Aminosäuren (L- und D-Valin) und  Hydroxysäuren (L Milchsäure und D Hydroxyisovaleriansäure) besteht. In dieser Arbeit wurde ein L Valin durch ein L-Lysin ausgetauscht, um eine freie Aminogruppe zum Anbinden verschiedenster Liganden zu erhalten. rnDie Synthese erfolgte in Lösung über einen linearen Decadepsipeptid-Precursor, hierbei wurde hauptsächlich mit Benzyl- und Boc-Schutzgruppen gearbeitet. An den Precursor wurden dann unterschiedlich modifizierte Lysin-Didepside gebunden und das daraus erhaltene lineare Dodecadepsipeptid zyklisiert.rnInsgesamt wurden sechs verschiedene Derivate synthetisiert, deren Ionentransportfähigkeit mit Hilfe eines angebundenen Liganden blockiert wurde. Diese Blockade kann entweder mechanisch durch Festhalten des Ionencarriers an der Oberfläche der Membran oder chemisch durch Einbringen einer Ladung erfolgen, da geladene Moleküle eine Membran nicht überwinden können. rnAcetyl-Lysin-Valinomycin wurde als Testsystem hergestellt, um zu zeigen, dass die Synthese einen funktionsfähigen Ionencarrier ergeben hatte. Im nächsten Schritt wurde Lysin-Valinomycin mit freier Aminogruppe synthetisiert, um es als pH-Sensor zu nutzen und damit zu überprüfen, ob das chemische Einbringen einer Ladung möglich ist. Es konnte ein pH abhängiger Kalium-Transport nachgewiesen werden, die Blockade der Ionentransportfähigkeit über eine eingebrachte Ladung ist somit möglich. rnAuf dem gleichen Konzept beruht Ferrocen-Valinomycin. Wird der Ferrocen-Ligand oxidiert, liegt eine positive Ladung vor und der Ionencarrier kann die Membran nicht mehr überwinden. Eine Reduktion macht diesen Prozess reversibel. Ferrocen-Valinomycin konnte innerhalb einer tBLM chemisch oxidiert und reduziert werden, dieses System kann somit als chemischer Redox-Sensor eingesetzt werden.rnEine mechanische Blockade liegt dem Biotin- und dem Sulfonamid-Valinomycin zugrunde. Dabei soll die Zugabe von Streptavidin bzw. BCA II (bovine Carboanhydrase) den Ionentransport durch die Membran stoppen. Beide Valinomycin-Derivate zeigten aber keine Ionentransportfähigkeit, eine Inkorporation in tBLMs konnte jedoch über SPR gezeigt werden. rnDie Synthese eines fluoreszenz-markierten (FITC) Valinomycins ergab zwar auch keinen transportfähigen Ionencarrier, aber mit diesem Derivat konnte der Diffusionskoeffizient von Valinomycin in sBLMs mit Hilfe von Fluorescence Recovery after photobleaching (FRAP) bestimmt werden.rn

STABILITY AND FOLDING OF MUTANT FORMS OF ISO-1 CYTOCHROME-C FROM SACCHAROMYCES CEREVISIAE

Partially functional forms of iso-1-cytochrome c from Saccharomyces cerevisiae were obtained by replacements of the evolutionarily conserved proline 71 with valine, isoleucine and threonine (Ernst et.al.,1985). Pro-71 lies at the juncture of two short helical regions and is believed to be important for proper local polypeptide chain folding within the iso-1-cytochrome c structure.^ To study folding in the absence of intermolecular disulfide dimer formation the free sulfhydryl group of Cys-102 was modified in both wild type and mutant proteins with an alkylating reagent, methyl methanethiosulfonate. Spectral analysis of the wild type and mutant proteins shows that the native-like functional (or partially functional) folded structure of cytochrome c is retained in the chemically modified derivatives. The replacement of Pro-71 with valine, isoleucine or threonine reduces the intensity of the 696 nm absorbance band which is an indicator of the Met-80 ligation to the heme. Thermal stability and guanidine hydrochloride unfolding studies of the mutant proteins shows a destabilization of the protein as a result of mutation. The degree of destabilization depends on the chemical nature of the substituent amino acid in the mutant protiens.^ Kinetics of folding/unfolding reactions of the proteins were monitored by fluorescence changes using stopped flow mixing to obtain guanidine hydrochloride concentration jumps ending below, within, and above the transition zone. The replacement of Pro-71 alters the rate on one of the fastest phases, $\tau\sb3$, while the two other phases, $\tau\sb1$ & $\tau\sb2$, remain the same.^ Slow refolding kinetic studies indicate that replacement of Pro-71 does not completely eliminate the absorbance or fluorescence detected slow phases leading to the conclusion that Pro-71 is not involved in the generation of the slow phases in the folding kinetics of iso-1-cytochrome c.^ The alkaline conformational change involving the disappearance of the 696 nm absorbance band occurs with increasing pH in the alkaline pH region (Davis et al., 1974). The apparent pK of this conformational change in mutant proteins is shifted as much as two pH units compared to wild type. The equilibrium and kinetic data of alkaline transition for the wild type follows a simple mechanism proposed by Davis et al., (1974) for horse heart cytochrome c. A more complex mechanism is proposed for the behavior of the mutant proteins. ^

Mapping of contact sites in complex formation between transducin and light-activated rhodopsin by covalent crosslinking: Use of a photoactivatable reagent

Interaction of light-activated rhodopsin with transducin (T) is the first event in visual signal transduction. We use covalent crosslinking approaches to map the contact sites in interaction between the two proteins. Here we use a photoactivatable reagent, N-[(2-pyridyldithio)-ethyl], 4-azido salicylamide. The reagent is attached to the SH group of cytoplasmic monocysteine rhodopsin mutants by a disulfide-exchange reaction with the pyridylthio group, and the derivatized rhodopsin then is complexed with T by illumination at λ >495 nm. Subsequent irradiation of the complex at λ310 nm generates covalent crosslinks between the two proteins. Crosslinking was demonstrated between T and a number of single cysteine rhodopsin mutants. However, sites of crosslinks were investigated in detail only between T and the rhodopsin mutant S240C (cytoplasmic loop V-VI). Crosslinking occurred predominantly with Tα. For identification of the sites of crosslinks in Tα, the strategy used involved: (i) derivatization of all of the free cysteines in the crosslinked proteins with N-ethylmaleimide; (ii) reduction of the disulfide bond linking the two proteins and isolation of all of the Tα species carrying the crosslinked moiety with a free SH group; (iii) adduct formation of the latter with the N-maleimide moiety of the reagent, maleimido-butyryl-biocytin, containing a biotinyl group; (iv) trypsin degradation of the resulting Tα derivatives and isolation of Tα peptides carrying maleimido-butyryl-biocytin by avidin-agarose chromatography; and (v) identification of the isolated peptides by matrix-assisted laser desorption/ionization time-of-flight mass spectrometry. We found that crosslinking occurred mainly to two C-terminal peptides in Tα containing the amino acid sequences 310–313 and 342–345.

Structural Analysis and Molecular Model of a Self-Incompatibility RNase from Wild Tomato1

Self-incompatibility RNases (S-RNases) are an allelic series of style glycoproteins associated with rejection of self-pollen in solanaceous plants. The nucleotide sequences of S-RNase alleles from several genera have been determined, but the structure of the gene products has only been described for those from Nicotiana alata. We report on the N-glycan structures and the disulfide bonding of the S3-RNase from wild tomato (Lycopersicon peruvianum) and use this and other information to construct a model of this molecule. The S3-RNase has a single N-glycosylation site (Asn-28) to which one of three N-glycans is attached. S3-RNase has seven Cys residues; six are involved in disulfide linkages (Cys-16-Cys-21, Cys-46-Cys-91, and Cys-166-Cys-177), and one has a free thiol group (Cys-150). The disulfide-bonding pattern is consistent with that observed in RNase Rh, a related RNase for which radiographic-crystallographic information is available. A molecular model of the S3-RNase shows that four of the most variable regions of the S-RNases are clustered on one surface of the molecule. This is discussed in the context of recent experiments that set out to determine the regions of the S-RNase important for recognition during the self-incompatibility response.

Reconsideration of the catalytic center and mechanism of mammalian paraoxonase/arylesterase.

For three decades, mammalian paraoxonase (A-esterase, aromatic esterase, arylesterase; PON, EC 3.1.8.1) has been thought to be a cysteine esterase demonstrating structural and mechanistic homologies with the serine esterases (cholinesterases and carboxyesterases). Human, mouse, and rabbit PONs each contain only three cysteine residues, and their positions within PON have been conserved. In purified human PON, residues Cys-41 and Cys-352 form an intramolecular disulfide bond and neither could function as an active-center cysteine. Highly purified, enzymatically active PON contains a single titratable sulfhydryl group. Thus, Cys-283 is the only probable candidate for an active-center cysteine. Through site-directed mutagenesis of the human cDNA, Cys-283 was replaced with either serine (C283S) or alanine (C283A). The expressed C283 (wild-type) enzyme was inactivated by para-hydroxymercuribenzoate, but the C283S and C283A mutant enzymes were not inactivated. C283A and C283S mutant enzymes retained both paraoxonase and arylesterase activities, and the Km values for paraoxon and phenyl acetate were similar to those of the wild type. Clearly, residue Cys-283 is free in active PON, but a free sulfhydryl group is not required for either paraoxonase or arylesterase activities. Consequently, it is necessary to examine other models for the active-site structure and catalytic mechanism of PON.

Topological structures of complex belief systems

The concepts of substantive beliefs and derived beliefs are defined, a set of substantive beliefs S like open set and the neighborhood of an element substantive belief. A semantic operation of conjunction is defined with a structure of an Abelian group. Mathematical structures exist such as poset beliefs and join-semilattttice beliefs. A metric space of beliefs and the distance of belief depending on the believer are defined. The concepts of closed and opened ball are defined. S′ is defined as subgroup of the metric space of beliefs Σ and S′ is a totally limited set. The term s is defined (substantive belief) in terms of closing of S′. It is deduced that Σ is paracompact due to Stone's Theorem. The pseudometric space of beliefs is defined to show how the metric of the nonbelieving subject has a topological space like a nonmaterial abstract ideal space formed in the mind of the believing subject, fulfilling the conditions of Kuratowski axioms of closure. To establish patterns of materialization of beliefs we are going to consider that these have defined mathematical structures. This will allow us to understand better cultural processes of text, architecture, norms, and education that are forms or the materialization of an ideology. This materialization is the conversion by means of certain mathematical correspondences, of an abstract set whose elements are beliefs or ideas, in an impure set whose elements are material or energetic. Text is a materialization of ideology.

Fatty acids as haptens: exploring the limits of antigenicity

Fatty acids (FAs) are relatively small, hydrophobic and highly mobile molecular structures with vital biological functions and a ubiquitous distribution. Surprisingly, however, they can be rendered immunogenic. We have synthesised a novel immunogen in which dicarboxylic linoleic acid was conjugated to a carrier protein. Dicarboxylic fatty acids (DCA) differ from their normal counterparts only by their possession of a carboxyl group at each end of the molecule. When conjugated to proteins as haptens, they are, therefore, presented to the immune system with a free carboxyl group at the distal end, instead of a methyl group. Polyclonal IgG antibodies raised in response to this unique immunogen could bind not only conjugated hapten with high affinity, but also the equivalent free FA in mono and dicarboxylic form. Similar conjugates constructed from normal FAs produced much weaker antibody responses and could scarcely be considered antigenic at all. The cross-reactivities of the anti-DCA antibodies with FA variants differing in the number, position and configuration of their double bonds showed that the antibody paratope (binding site) was structured to accommodate the hapten in a way that depended on the precise shape of the acyl chain. We suggest that FAs become much more effective as B-cell epitopes when presented with their hydrophilic carboxyl group exposed on the surface of immunogenic conjugates. This type of epitope is determined by the particular double bond pattern of the unsaturated acyl chain, as well as the polar head group. (C) 2003 Elsevier Ltd. All rights reserved.

Antitumour imidazotetrazines: probes for the major groove of DNA

The antitumour imidazotetrazinones are believed to act as prodrugs for the triazene series of alkylating agents, showing a marked pteference for the alkylation of the middle guanine residue in a run of three or more contiguous guanines. However, the. exact nature of the interactions of imidazotetrazinones within the micro~environment of DNA are; as yet unknown. In order to examine such interactions a three pronged approach involving molecular modelling, synthetic chemistry and biological analysis has been undertaken during the course of this project. . Molecular modelling studies have shown that for the 8-carboxamido substituted imidazotetrazinones antitumour activity is dependent upon the. presence of a free NH group which can be involved in the formation of both intramolecular and intermolecular hydrogen bonds, and the presence of a non-bulky substituent with a small negative potential . volume. Modelling studies involving the docking of .mitozolomide into the major groove of DNA in the region of a triguanine sequence has shown that a number of hydrogen bonding interactions are feasible. A series of 8-substituted carboxamide derivatives of mitozolomide have been synthesised via the 8-acid chloride and 8-carboxylic acid derivatives including a number of peptide analogues. The peptide derivatives were based upon the key structural features of the helix-turn-helix motif of DNA-binding proteins with a view to developing agents that are capable of binding to DNA with greater selectivity. An examination of the importance of intramolecular hydrogen bonding in influencing the antitumour activity:of :the imidazotetrazinones has led to the synthesis of the novel pyrimido[4',5' :4,3]pyrazolo[5,1-d]-1,2,3,5-tetrazine ring system. In general, in vitro cytotoxicity assays showed that the new derivatives were less active against the TLX5 lymphoma cell line. than the parent compound mitozolomide despite an increased potential for hydrogen bonding interactions. Due to the high reactivity of the: tetrazinone ring system it is difficult to study the interactions between the imidazotetrazinones and DNA. Consequently a number of structural analogues that are stable under physiological conditions have been. prepared based upon the 1,2,3 triazin-4(3H)-one ring system fused with both benzene and pyrazole rings. Although the 3-methylbenzotriazinones failed to antagonise the cytotoxic activity of temozolomide encouraging results with a 3-methylpyrazolotriazinone may suggest the existence of an imidazotetrazinone receptor site within DNA. The potential of guanine rich sequences to promote the alkylating selectivity of imidazotetrazinones by acting as a catalyst for ring cleavage and thereby generation of the alkylating agent was examined. Experiments involving the monitoring: of the rate of breakdown of mitozolomide incubated in the presence of synthetic oIigonucleotides did not reveal any catalytic effect resulting from the DNA. However, it was noted that the breakdown of mitozolomide was dependent upon the type of buffer used in the incubations and this may indeed mask any catalysis by the oligonucleotides.

Spectral synthesis and topologies on ideal spaces for Banach *-algebras

This paper continues the study of spectral synthesis and the topologies τ∞ and τr on the ideal space of a Banach algebra, concentrating on the class of Banach *-algebras, and in particular on L1-group algebras. It is shown that if a group G is a finite extension of an abelian group then τr is Hausdorff on the ideal space of L1(G) if and only if L1(G) has spectral synthesis, which in turn is equivalent to G being compact. The result is applied to nilpotent groups, [FD]−-groups, and Moore groups. An example is given of a non-compact, non-abelian group G for which L1(G) has spectral synthesis. It is also shown that if G is a non-discrete group then τr is not Hausdorff on the ideal lattice of the Fourier algebra A(G).

Packing of R^2 by Crosses

A cross in Rn is a cluster of unit cubes comprising a central one and 2n arms. In their monograph Algebra and Tiling, Stein and Szabó suggested that tilings of ℝn by crosses should be studied. The question of the existence of such a tiling has been answered by various authors for many special cases. In this paper we completely solve the problem for ℝ2. In fact we do not only characterize crosses for which there exists a tiling of ℝ2 but for each cross we determine its maximum packing density.