The outer-distance of nodes in random trees

The outer-distance of a nodeu in a rooted treeT _n is the height of the subtree determined byu and all nodesv such thatu is on the path joiningv and the root ofT. We show that the expected outer-distance of nodes of treesT _n in certain families is asymptotic toB logn where the constantB depends on .     

On the theorems of Drake and Lenz

A generalization and an improvement of the results of Drake and Lenz on the constructions of projective Hjelmslev planes are obtained. Using this, some new series of invariant pairs including 4 new pairs (t, 2),t 1000, for projective Hjelmslev planes are obtained.     

Graph factorization,general triple systems,and cyclic triple systems

In this self-contained exposition, results are developed concerning one-factorizations of complete graphs, and incidence matrices are used to turn these factorization results into embedding theorems on Steiner triple systems. The result is a constructive graphical proof that a Steiner triple system exists for any order congruent to 1 or 3 modulo 6. A pairing construction is then introduced to show that one can also obtain triple systems which are cyclically generated.     

An algorithmic approach to Hermite-Birkhoff-interpolation

Summary This paper deals with an algorithmic approach to the Hermite-Birkhoff-(HB)interpolation problem. More precisely, we will show that Newton's classical formula for interpolation by algebraic polynomials naturally extends to HB-interpolation. Hence almost all reasons which make Newton's method superior to just solving the system of linear equations associated with the interpolation problem may be repeated. Let us emphasize just two: Newton's formula being a biorthogonal expansion has a well known permanence property when the system of interpolation conditions grows. From Newton's formula by an elementary argument due to Cauchy an important representation of the interpolation error can be derived. All of the above extends to HB-interpolation with respect to canonical complete ebyev-systems and naturally associated differential operators [7]. A numerical example is given.     

A variable metric algorithm for unconstrained minimization without evaluation of derivatives

Summary This paper presents a modification of the BFGS-method for unconstrained minimization that avoids computation of derivatives. The gradients are approximated by the aid of differences of function values. These approximations are calculated in such a way that a complete convergence proof can be given. The presented algorithm is implementable, no exact line search is required. It is shown that, if the objective function is convex and some usually required conditions hold, the algorithm converges to a solution. If the Hessian matrix of the objective function is positive definite and satisfies a Lipschitz-condition in a neighbourhood of the solution, then the rate of convergence is superlinear.     

Some bounds for partially balanced block designs

Bounds on eigenvalues of theC-matrix for a partially balanced block (PBB) design are given together with some bounds on the number of blocks. Furthermore, a certain equiblock-sized PBB design is characterized. These results contain, as special cases, the known results for variance-balanced block designs and so on.     

Cellularity in polyhedra

A compact subset X of a polyhedron P is cellular in P if there is a pseudoisotropy of P shrinking precisely X to a point. A proper surjection between polyhedra f:P→Q is cellular if each point inverse of f is cellular in P. It is shown that if f:P→Q is a cellular map and either P or Q is a generalized n-manifold, n≠4, then f is approximable by homeomorphisms. Also, if P or Q is an n-manifold with boundary, n≠4, 5, then a cellular map f:P→Q is approximable by homeomorphisms. A cellularity criterion for a special class of cell-like sets in polyhedra is established.     

Continuum models of two-dimensional random droplets

We study the statistical mechanical properties of a two-dimensional assembly of free particles coupled to a mechanical reservoir. The particles-reservoir interaction is modelised by an Hamiltonian depending on the convex hull of the particles only. We concentrate on models whose energy is the sum of an area-term, a perimeter term and possibly a term preventing the particles occupying the interior of the convex hull. The range of coupling constants insuring a thermodynamic behaviour, as well as the associated free energy per particle are exactly determined.Work partially supported by the Swiss National Foundation for Scientific Research.     

Synthesis and Sm/Sm doping effects on photoluminescence properties of Sr4Al14O25

Complete and partial samarium reduction was achieved under strong reducing atmosphere by solid-state and combustion synthesis of Sr_3.96Sm_0.04Al₁₄O₂₅. Dependence of different fluxing agents on the formation of various strontium aluminates was examined. The samples were investigated by X-ray powder diffraction, temperature dependent luminescence decay and photoluminescence measurements. Excitation with UV radiation resulted in sharp and well resolved emission lines of samarium ions. Distinct temperature behavior for Sm²⁺ and Sm³⁺ were detected in the range of 100-500 K. Estimated emission thermal quenching values (TQ_1/2) for divalent samarium were approximately 270 K while for trivalent state around 660 K. Measured luminescence decay values of Sm²⁺ are substantially lower than for Sm³⁺,≈1.7 and ≈2.7 ms, respectively. The spectral feature of Sm²⁺ emission spectrum indicates that dopant occupies low symmetry site in Sr₄Al₁₄O₂₅ compound.