Large sets of <Emphasis Type="Italic">t</Emphasis>-designs from groups

This paper addresses questions related to the existence and construction of large sets of t-(v, k, λ) designs. It contains material from my talk in the Combinatorial Designs Conference in honor of Alex Rosa’s 70th birthday, which took place in beautiful Bratislava, in July, 2007. Naturally, only a small number of “highlight” topics could be included, and for the most part these involve the use of symmetry, that is, it is assumed that the particular designs or large sets of designs, are invariant under a prescribed group of automorphisms. I present almost no proofs, but give references so that the reader can find a much wider repertory of theorems and constructions in the literature. For completeness, I include the statement of a few recursive constructions. The latter are extremely important on their own right, and deserve extensive attention elsewhere. I hope the reader becomes interested in the intriguing open problems posed at the end of the paper and succeeds in solving some of them. (Communicated by Peter Horák)     

Hawking’s Local Rigidity Theorem Without Analyticity

We prove the existence of a Hawking Killing vector-field in a full neighborhood of a local, regular, bifurcate, non-expanding horizon embedded in a smooth vacuum Einstein manifold. The result extends a previous result of Friedrich, Rácz and Wald, see [FRW, Prop.B.1], which was limited to the domain of dependence of the bifurcate horizon. So far, the existence of a Killing vector-field in a full neighborhood has been proved only under the restrictive assumption of analyticity of the space-time. Using this result we provide the first unconditional proof that a stationary black-hole solution must possess an additional, rotational Killing field in an open neighborhood of the event horizon. This work is accompanied by a second paper, where we prove a uniqueness result for smooth stationary black-hole solutions which are close (in a very precise, geometric sense) to the Kerr family of solutions, for arbitrary 0 < a < m.     

Dark energy reconstruction based on the Padé approximation; an expansion around the $$\varLambda $$CDM

We study the dynamical properties of dark energy based on a large family of Padé parameterizations for which the dark energy density evolves as the ratio between two polynomials in the scale factor of the universe. Using the latest cosmological data we perform a standard likelihood analysis in order to place constraints on the main cosmological parameters of different Padé models. We find that the basic cosmological parameters, namely \(({\varOmega _{m0}},h,{\sigma _{8}})\) are practically the same for all Padé parametrizations explored here. Concerning the free parameters which are related to dark energy we show that the best-fit values indicate that the equation of state parameter at the present time is in the phantom regime (\(w<-1\)); however, we cannot exclude the possibility of \(w>-1\) at \(1\sigma \) level. Finally, for the current family of Padé parametrizations we test their ability, via AIC, BIC and Jeffreys’ scale, to deviate from \(\varLambda \)CDM cosmology. Among the current Padé parametrizations, the model which contains two dark energy parameters is the one for which a small but non-zero deviation from \(\varLambda \)CDM cosmology is slightly allowed by the AIC test. Moreover, based on Jeffreys’ scale we show that a deviation from \(\varLambda \)CDM cosmology is also allowed and thus the possibility of having a dynamical dark energy in the form of Padé parametrization cannot be excluded.     

Gravitational Influence on Human Living Systems and the Evolution of Species on Earth

Gravity constituted the only constant environmental parameter, during the evolutionary period of living matter on Earth. However, whether gravity has affected the evolution of species, and its impact is still ongoing. The topic has not been investigated in depth, as this would require frequent and long-term experimentations in space or an environment of altered gravity. In addition, each organism should be studied throughout numerous generations to determine the profound biological changes in evolution. Here, we review the significant abnormalities presented in the cardiovascular, immune, vestibular and musculoskeletal systems, due to altered gravity conditions. We also review the impact that gravity played in the anatomy of snakes and amphibians, during their evolution. Overall, it appears that gravity does not only curve the space–time continuum but the biological continuum, as well.     

Dinuclear Lanthanide(III) Complexes from the Use of Methyl 2-Pyridyl Ketoxime: Synthetic,Structural, and Physical Studies

The first use of methyl 2-pyridyl ketoxime (mepaoH) in homometallic lanthanide(III) [Ln(III)] chemistry is described. The 1:2 reactions of Ln(NO₃)₃·nH₂O (Ln = Nd, Eu, Gd, Tb, Dy; n = 5, 6) and mepaoH in MeCN have provided access to complexes [Ln₂(O₂CMe)₄(NO₃)₂(mepaoH)₂] (Ln = Nd, 1; Ln = Eu, 2; Ln = Gd, 3; Ln = Tb, 4; Ln = Dy, 5); the acetato ligands derive from the Ln^III—mediated hydrolysis of MeCN. The 1:1 and 1:2 reactions between Dy(O₂CMe)₃·4H₂O and mepaoH in MeOH/MeCN led to the all-acetato complex [Dy₂(O₂CMe)₆(mepaoH)₂] (6). Treatment of 6 with one equivalent of HNO₃ gave 5. The structures of 1, 5, and 6 were solved by single-crystal X-ray crystallography. Elemental analyses and IR spectroscopy provide strong evidence that 2–4 display similar structural characteristics with 1 and 5. The structures of 1–5 consist of dinuclear molecules in which the two Ln^III centers are bridged by two bidentate bridging (η¹:η¹:μ₂) and two chelating-bridging (η¹:η²:μ₂) acetate groups. The Ln^III atoms are each chelated by a N,N’-bidentate mepaoH ligand and a near-symmetrical bidentate nitrato group. The molecular structure of 6 is similar to that of 5, the main difference being the presence of two chelating acetato groups in the former instead of the two chelating nitrato groups in the latter. The geometry of the 9-coordinate Ln^III centers in 1, 5 and 6 can be best described as a muffin-type (MFF-9). The 3D lattices of the isomorphous 1 and 5 are built through H-bonding, π⋯π stacking and C-H⋯π interactions, while the 3D architecture of 6 is stabilized by H bonds. The IR spectra of the complexes are discussed in terms of the coordination modes of the organic and inorganic ligands involved. The Eu(III) complex 2 displays a red, metal-ion centered emission in the solid state; the Tb^III atom in solid 4 emits light in the same region with the ligand. Magnetic susceptibility studies in the 2.0–300 K range reveal weak antiferromagnetic intramolecular Gd^III…Gd^III exchange interactions in 3; the J value is −0.09(1) cm⁻¹ based on the spin Hamiltonian Ĥ = −J(Ŝ_Gd1·Ŝ_Gd2).     

Ferromagnetic coupling in a 1D coordination polymer containing a symmetric [Cu(mu1,1-N3)2Cu(mu1,1-N3)2Cu]2+ core and based on an organic ligand obtained from the solid state

Addition of rctt-tetrakis(2-pyridyl)cyclobutane (2,2'-tpcb) in a Cu(II)/N(3)- solution afforded the 1D coordination polymer [Cu(3)(N(3))(6)(2,2'-tpcb)(DMF)(2)](n) (1). The ligand 2,2'-tpcb serves as a tetradentate bis-chelating ligand by linking linear [(DMF)Cu(mu(1,1)-N(3))(2)Cu(N(3))(2)(mu(1,1)-N(3))(2)Cu(DMF)] trinuclear units to produce a zigzag chain. Within each centrosymmetric trinuclear unit there exist two irregularly asymmetric end-on double azido-bridged [Cu(mu(1,1)-N(3))(2)Cu](2+) cores, while one of the largest Cu-Nazide-Cu angles is observed. Magnetic susceptibility data, measured from 2 to 300 K, show bulk moderate ferromagnetic coupling within the magnetically isolated trinuclear units. These data were fitted to the appropriate equation derived from the Hamiltonian H = -J(1)(S(A1)S(B) + S(A2)S(B)) - J(2)S(A1)S(A2), giving the parameters J1 = +70(3) cm(-1), J2 = -3(2) cm(-1), g = 2.12(1), with an intertrimer interaction parameter theta = -0.74(2) K. The coupling constants were correlated with the structural parameters. Density functional calculations reproduce very well the experimental J values and show that ferromagnetism for this complex is mainly due to the topology of the magnetic orbitals and the different coordination spheres of two neighboring Cu(II) atoms, resulting in a small overlap of the orbitals possessing the unpaired electrons.     

Two frustrated, bitetrahedral single-molecule magnets

Two unusual mixed-valent {MnIII6MnII} bitetrahedra display frustrated magnetic exchange, leading to S = 13/2 +/- 1 and 11/2 +/- 1 ground states and slow magnetization relaxation.     

A rare ferromagnetic manganese(III) 'cube'

A {Mn(III)4} distorted cube has been synthesized that possesses an S = 8 ground state and single-molecule magnetism behaviour.