Towards a New Family of Photoluminescent Organozinc 8‐Hydroxyquinolinates with a High Propensity to Form Noncovalent Porous Materials

We report on investigations of reactions of tBu₂Zn with 8‐hydroxyquinoline (q‐H) and the influence of water on the composition and structure of the final product. A new synthetic approach to photoluminescent zinc complexes with quinolinate ligands was developed that allowed the isolation of a series of structurally diverse and novel alkylzinc 8‐hydroxyquinolate complexes: the trinuclear alkylzinc aggregate [tBuZn(q)]₃ ( 1₃ ), the pentanuclear oxo cluster [(tBu)₃Zn₅(μ₄‐O)(q)₅] ( 2 ), and the tetranuclear hydroxo cluster [Zn(q)₂]₂[tBuZn(OH)]₂ ( 3 ). All compounds were characterized in solution by ¹H NMR, IR, UV/Vis, and photoluminescence (PL) spectroscopy, and in the solid state by X‐ray diffraction, TGA, and PL studies. Density functional theory calculations were also carried out for these new Zn^II complexes to rationalize their luminescence behavior. A detailed analysis of the supramolecular structures of 2 and 3 shows that the unique shape of the corresponding single molecules leads to the formation of extended 3D networks with 1D open channels. Varying the stoichiometry, shape, and supramolecular structure of the resulting complexes leads to changes in their spectroscopic properties. The close‐packed crystal structure of 1₃ shows a redshifted emission maximum in comparison to the porous crystal structure of 2 and the THF‐solvated structure of 3 .     

Relative Entropy in Determining Degressive Proportional Allocations

The principle of degressively proportional apportionment of goods, being a compromise between equality and proportionality, facilitates the application of many different allocation rules. Agents with smaller entitlements are more interested in an allocation that is as close to equality as possible, while those with greater entitlements prefer an allocation as close to proportionality as possible. Using relative entropy to quantify the inequity of allocation, this paper indicates an allocation that neutralizes these two contradictory approaches by symmetrizing the inequities perceived by the smallest and largest agents participating in the apportionment. First, based on some selected properties, the set of potential allocation rules was reduced to those generated by power functions. Then, the existence of the power function whose exponent is determined so as to generate the allocation that symmetrizes the relative entropy with respect to equal and proportional allocations was shown. As a result, all agents of the apportionment are more inclined to accept the proposed allocation regardless of the size of their entitlements. The exponent found in this way shows the significant relationship between the problem under study and the well-known Theil indices of inequality. The problem may also be seen from this viewpoint.     

A New Notion of Transitivity for Groups and Sets of Permutations

Let = {1, 2, ..., n} where n 2. The shape of an ordered setpartition P = (P₁, ..., P_k) of is the integer partition =(₁, ..., _k) defined by _i = |P_i|. Let G be a group of permutationsacting on . For a fixed partition of n, we say that G is -transitiveif G has only one orbit when acting on partitions P of shape. A corresponding definition can also be given when G is justa set. For example, if = (n – t, 1, ..., 1), then a -transitivegroup is the same as a t-transitive permutation group, and if = (n – t, t), then we recover the t-homogeneous permutationgroups. We use the character theory of the symmetric group S_n to establishsome structural results regarding -transitive groups and sets.In particular, we are able to generalize a celebrated resultof Livingstone and Wagner [Math. Z. 90 (1965) 393–403]about t-homogeneous groups. We survey the relevant examplescoming from groups. While it is known that a finite group ofpermutations can be at most 5-transitive unless it containsthe alternating group, we show that it is possible to constructa nontrivial t-transitive set of permutations for each positiveinteger t. We also show how these ideas lead to a combinatorialbasis for the Bose–Mesner algebra of the association schemeof the symmetric group and a design system attached to thisassociation scheme.     

Symmetric functions in noncommuting variables

Consider the algebra of formal power series in countably many noncommuting variables over the rationals. The subalgebra of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, complete homogeneous, and Schur symmetric functions as well as investigating their properties.

     

Characteristic and Ehrhart Polynomials

Let A be a subspace arrangement and let (A,t) be the characteristic polynomial of its intersection lattice L( A). We show that if the subspaces in A are taken from , where is the type B Weyl arrangement, then (A,t) counts a certain set of lattice points. One can use this result to study the partial factorization of (A,t) over the integers and the coefficients of its expansion in various bases for the polynomial ring R[t]. Next we prove that the characteristic polynomial of any Weyl hyperplane arrangement can be expressed in terms of an Ehrhart quasi-polynomial for its affine Weyl chamber. Note that our first result deals with all subspace arrangements embedded in while the second deals with all finite Weyl groups but only their hyperplane arrangements.     

Minimum Query Set for Decision Tree Construction

A new two-stage method for the construction of a decision tree is developed. The first stage is based on the definition of a minimum query set, which is the smallest set of attribute-value pairs for which any two objects can be distinguished. To obtain this set, an appropriate linear programming model is proposed. The queries from this set are building blocks of the second stage in which we try to find an optimal decision tree using a genetic algorithm. In a series of experiments, we show that for some databases, our approach should be considered as an alternative method to classical ones (CART, C4.5) and other heuristic approaches in terms of classification quality.     

Study of heating effect and acoustic properties of dextran stabilized magnetic fluid

This paper presents acoustic properties of water-based biocompatible fluids in which magnetite particles (Fe₃O₄) were coated with two layers of surfactants: sodium oleate and dextran. The attenuation coefficient of ultrasonic wave measurements shows good structural stability of the fluid under the influence of a magnetic field. Hyperthermic tests proved that the magnetic fluid is suitable for therapeutic use as an agent which can release thermal energy (hyperthermia).     

An analog of Schensted's algorithm for Shifted Young Tableaux

Schensted [Canad. J. Math. 13 (1961)] constructed an algorithm giving a bijective correspondence between permutations and pairs of Young Tableaux. The author develops an analogous algorithm relating permutations and triples consisting of two Shifted Tableaux and a set. Various properties of this algorithm are also examined.     

A note on Abel polynomials and rooted labeled forests

A special case of the Abel polynomials counts rooted labeled forests. This interpretation is used to obtain a combinatorial proof of the formula expressing xⁿ as a sum of these polynomials.