A reduction of the spectrum problem for odd sun systems and the prime case

A k‐cycle with a pendant edge attached to each vertex is called a k‐sun. The existence problem for k‐sun decompositions of K_v, with k odd, has been solved only when k = 3 or 5. By adapting a method used by Hoffmann, Lindner, and Rodger to reduce the spectrum problem for odd cycle systems of the complete graph, we show that if there is a $k$‐sun system of $K_v$ ( $k$ odd) whenever $v$ lies in the range $2k<v<6k$ and satisfies the obvious necessary conditions, then such a system exists for every admissible $v\ge 6k$. Furthermore, we give a complete solution whenever k is an odd prime.     

Strong difference families over arbitrary graphs

The concept of a strong difference family formally introduced in Buratti [J Combin Designs 7 (1999), 406–425] with the aim of getting group divisible designs with an automorphism group acting regularly on the points, is here extended for getting, more generally, sharply‐vertex‐transitive Γ‐decompositions of a complete multipartite graph for several kinds of graphs Γ. We show, for instance, that if Γ has e edges, then it is often possible to get a sharply‐vertex‐transitive Γ‐decomposition of K_{m × e} for any integer m whose prime factors are not smaller than the chromatic number of Γ. This is proved to be true whenever Γ admits an α‐labeling and, also, when Γ is an odd cycle or the Petersen graph or the prism T₅ or the wheel W₆. We also show that sometimes strong difference families lead to regular Γ‐decompositions of a complete graph. We construct, for instance, a regular cube‐decomposition of K_16m for any integer m whose prime factors are all congruent to 1 modulo 6. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 443–461, 2008     

1‐rotational k‐factorizations of the complete graph and new solutions to the Oberwolfach problem

We consider k‐factorizations of the complete graph that are 1‐rotational under an assigned group G, namely that admit G as an automorphism group acting sharply transitively on all but one vertex. After proving that the k‐factors of such a factorization are pairwise isomorphic, we focus our attention to the special case of k = 2, a case in which we prove that the involutions of G necessarily form a unique conjugacy class. We completely characterize, in particular, the 2‐factorizations that are 1‐rotational under a dihedral group. Finally, we get infinite new classes of previously unknown solutions to the Oberwolfach problem via some direct and recursive constructions. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 87–100, 2008     

Some progress on the existence of 1-rotational Steiner triple systems

A Steiner triple system of order v (briefly STS(v)) is 1-rotational under G if it admits G as an automorphism group acting sharply transitively on all but one point. The spectrum of values of v for which there exists a 1-rotational STS(v) under a cyclic, an abelian, or a dicyclic group, has been established in Phelps and Rosa (Discrete Math 33:57–66, 1981), Buratti (J Combin Des 9:215–226, 2001) and Mishima (Discrete Math 308:2617–2619, 2008), respectively. Nevertheless, the spectrum of values of v for which there exists a 1-rotational STS(v) under an arbitrary group has not been completely determined yet. This paper is a considerable step forward to the solution of this problem. In fact, we leave as uncertain cases only those for which we have v =  (p ³−p)n +  1 ≡ 1 (mod 96) with p a prime, n \not o 0{n \not\equiv 0} (mod 4), and the odd part of (p ³ − p)n that is square-free and without prime factors congruent to 1 (mod 6).     

On Resolvable Difference Families

A Steiner 2-design is said to be G-invariantly resolvable if admits an automorphism group G and a resolution invariant under G. Introducing and studying resolvable difference families, we characterize the class of G-invariantly resolvable Steiner 2-designs arising from relative difference families over G. Such designs have been already studied by Genma, Jimbo, and Mishima [13] in the case in which G is cyclic. Developping their results, we prove that any (p, k, 1)-DF (p prime) whose base blocks exactly cover p–1/k(k–1) distinct cosets of the k-th roots of unity (mod p), leads to a Ckp-invariantly resolvable cyclic (kp,k,1)-BBD. This induced us to propose several constructions for DF's having this property. In such a way we prove, in particular, the existence of a C5p-invariantly resolvable cyclic (5p, 5, 1)-BBD for each prime p = 20n + 1 < 1.000.     

Eshelby Tensor as a Tensor of Free Enthalpy

The balance equations of mass, momentum, energy and entropy at a phase boundary imply phase boundary conditions which determine the position of the boundary as a function of temperature. This is true when either the phase boundary is sharp or when it occurs through a transition zone, albeit the latter case seems to require strongly symmetric geometry.     

Development, validation and data quality assurance of screening methods: a case study

Despite the growing importance of qualitative screening tests in routine laboratories involved in the EU official control, their validation is not as deeply explained in Commission Decision 2002/657/EC as the validation of quantitative confirmatory methods. At the same time, the issue of quality assurance of screening assays defining internal quality control (IQC) procedures as required by accreditation bodies is undoubtedly less developed in this analytical field. As an example the present study describes the development, the validation and the IQC implemented for a commercial enzyme linked immunosorbent assay (ELISA) able to detect 17-α-19-nortestosterone (α-NT) and 17-β-19-nortestosterone (β-NT) isomers in bullock urine. In order to select a suitable sample treatment, two SPE purification protocols were preliminary compared. The chosen method was therefore fully validated determining the mandatory parameters required by Commission Decision 2002/657/EC: specificity, detection capability and robustness. An in-depth discussion was carried out illustrating the possible validation approaches and their implications especially in the assessment of the key performance characteristic: detection capability. Finally, the control charts implemented for continuous method verification during analyses of real samples were reported.     

Some Results on 1‐Rotational Hamiltonian Cycle Systems

A Hamiltonian cycle system of (briefly, a HCS(v)) is 1‐rotational under a (necessarily binary) group G if it admits G as an automorphism group acting sharply transitively on all but one vertex. We first prove that for any there exists a 3‐perfect 1‐rotational HCS. This allows to get the existence of another infinite class of 3‐perfect (but not Hamiltonian) cycle decompositions of the complete graph. Then we prove that the full automorphism group of a 1‐rotational HCS under G is G itself unless the HCS is the 2‐transitive one. This allows us to give a partial answer to the problem of determining which abstract groups are the full automorphism group of a HCS. Finally, we revisit and simplify by means of a careful group theoretic discussion a formula by Bailey, Ollis, and Preece on the number of inequivalent 1‐rotational HCSs under G. This leads us to a formula counting all 1‐rotational HCSs up to isomorphism. Though this formula heavily depends on some parameters that are hard to compute, it allows us to say that, for any , there are at least nonisomorphic 1‐rotational (and hence symmetric) HCS().     

Assay of urinary alpha-fluoro-beta-alanine by gas chromatography-mass spectrometry for the biological monitoring of occupational exposure to 5-fluorouracil in oncology nurses and pharmacy technicians

The validation of an analytical method for the measurement of the unnatural amino acid alpha-fluoro-beta-alanine (AFBA), the main metabolite of the antineoplastic drug 5-fluorouracil (5FU), in urine for the biological monitoring of the exposure of hospital workers to the drug when preparing the therapeutical doses and administering to cancer patients is described. The method employed a two-step extractive derivatization of the analyte from urine to the N-trifluoroacety-n-butyl ester derivative and detection by selected-ion monitoring gas chromatography-mass spectrometry of structurally specific fragments. The limit of detection was 20 ng/mL with quantification accuracy better than +/-20% and precision (CV%) better than +/-20% in the range 0.020-10 microg/mL. Norleucine was used as the internal standard and the sample-to-sample analysis time was less than 15 min. The validated method has been applied to the biological monitoring of some hospital workers potentially exposed to 5FU and to matched control subjects. On a total number of 65 analyzed urine samples from control and exposed subjects, only three, obtained from exposed subjects, were found to be positive, with values of 20, 30 and 1150 ng/mL, respectively.     

Further progress on difference families with block size 4 or 5

A strong indication about the existence of a (7p, 4, 1) difference family with p ≡ 7 (mod 12) a prime has been given in [11]. Here, developing some ideas of that paper, we give, much more generally, a strong indication about the existence of a cyclic (pq, 4, 1) difference family whenever p and q are primes congruent to 7 (mod 12) and of a cyclic (pq, 5, 1) difference family whenever p and q are primes congruent to 11 (mod 20). Indeed we give an algorithm for their construction that seems to be always successful and we have checked it works whenever both primes p and q do not exceed 1,000. All our (pq, 4, 1) and (pq, 5, 1) difference families have the nice property of admitting a multiplier of order 3 or 5, respectively, that fixes almost all base blocks. As an intermediate result we also find an optimal (p, 5, 1) optical orthogonal code for every prime p ≡ 11 (mod 20) not exceeding 10,000.