PARTITIONING THE BLOCKS OF A STEINER TRIPLE SYSTEM INTO PARTIAL PARALLEL CLASSES

Does there exist a Steiner Triple System on v points, whose blocks can be partitioned into partial parallel classes of size m, where m ≤ [v⁄3], m | b and b is the number of blocks of the STS(v)? We give the answer for 9 ≤ v ≤ 43. We also show that whenever 2|b, v ≡ 3 (mod 6) we can find an STS(v) whose blocks can be partitioned into partial parallel classes of size 2, and whenever 4|b , v ≡ 3 (mod 6), there exists an STS(v) whose blocks can be partitioned into partial parallel classes of size 4.

Doubling and Tripling Constructions for Defining Sets in Steiner Triple Systems

A minimal defining set of a Steiner triple system on a points (STS(v)) is a partial Steiner triple system contained in only this STS(v), and such that any of its proper subsets is contained in at least two distinct STS(v)s. We consider the standard doubling and tripling constructions for STS(2v + 1) and STS(3v) from STS(v) and show how minimal defining sets of an STS(v) gives rise to minimal defining sets in the larger systems. We use this to construct some new families of defining sets. For example, for Steiner triple systems on, 3" points; we construct minimal defining sets of volumes varying by as much as 7(n-/-).

A conjecture on small embeddings of partial Steiner triple systems

A well-known, and unresolved, conjecture states that every partial Steiner triple system of order u can be embedded in a Steiner triple system of order v for all v equivalent to 1 or 3 (mod 6), v greater than or equal to 2u + 1. However, some partial Steiner triple systems of order u can be embedded in Steiner triple systems of order v < 2u + 1. A more general conjecture that considers these small embeddings is presented and verified for some cases. (C) 2002 Wiley Periodicals, Inc.

Large sets of large sets of Steiner triple systems of order 9

We describe a direct method of partitioning the 840 Steiner triple systems of order 9 into 120 large sets. The method produces partitions in which all of the large sets are isomorphic and we apply the method to each of the two non-isomorphic large sets of STS(9).

Existence and embeddings of partial Steiner triple systems of order ten with cubic leaves

Denote the set of 21 non-isomorphic cubic graphs of order 10 by L. We first determine precisely which L is an element of L occur as the leave of a partial Steiner triple system, thus settling the existence problem for partial Steiner triple systems of order 10 with cubic leaves. Then we settle the embedding problem for partial Steiner triple systems with leaves L is an element of L. This second result is obtained as a corollary of a more general result which gives, for each integer v greater than or equal to 10 and each L is an element of L, necessary and sufficient conditions for the existence of a partial Steiner triple system of order v with leave consisting of the complement of L and v - 10 isolated vertices. (C) 2004 Elsevier B.V. All rights reserved.

Embeddings of partial Steiner triple systems

Any partial Steiner triple system of order u can be embedded in a Steiner triple system of order v if v equivalent to 1, 3 (mod 6) and v greater than or equal to 3u - 2. (C) 2004 Elsevier Inc. All rights reserved.

Steiner triple systems with two disjoint subsystems

It is shown that there exists a triangle decomposition of the graph obtained from the complete graph of order v by removing the edges of two vertex disjoint complete subgraphs of orders u and w if and only if u, w, and v are odd, ((v)(2)) - ((u)(2)) - ((w)(2)) equivalent to 0 (mod 3), and v >= w + u + max {u, w}. Such decompositions are equivalent to group divisible designs with block size 3, one group of size u, one group of size w, and v - u - w groups of size 1. This result settles the existence problem for Steiner triple systems having two disjoint specified subsystems, thereby generalizing the well-known theorem of Doyen and Wilson on the existence of Steiner triple systems with a single specified subsystem. (c) 2005 Wiley Periodicals, Inc.

The spectrum of minimal defining sets of some Steiner systems

For a design D, define spec(D) = {\M\ \ M is a minimal defining set of D} to be the spectrum of minimal defining sets of D. In this note we give bounds on the size of an element in spec(D) when D is a Steiner system. We also show that the spectrum of minimal defining sets of the Steiner triple system given by the points and lines of PG(3,2) equals {16,17,18,19,20,21,22}, and point out some open questions concerning the Steiner triple systems associated with PG(n, 2) in general. (C) 2002 Elsevier Science B.V. All rights reserved.

Stability regions around the components of the triple system 2001 SN263

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

The aster mission: Exploring for the first time a triple system asteroid

2001 SN263 is a triple system asteroid. Although it was discovery in 2001, in 2008 astronomical observation carried out by Arecibo observatory revealed that it is actually a system with three bodies orbiting each other. The main central body is an irregular object with a diameter about 2.8 km, while the other two are small objects with less than 1 km across. This system presents an orbital eccentricity of 0.47, with perihelion of 1.04 and aphelion of 1.99, which means that it can be considered as a Near Earth Object. This interesting system was chosen as the target for the Aster mission - first Brazilian space exploration undertaking. A small spacecraft with 150 kg of total mass, 30 kg of payload with 110 W available for the instruments, is scheduled to be launched in 2015, and in 2018 it will approach and will be put in orbit of the triple system. This spacecraft will use electric propulsion and in its payload it will carry image camera, laser rangefinder, infrared spectrometer, mass spectrometer, and experiments to be performed in its way to the asteroid. This mission represents a great challenge for the Brazilian space program. It is being structured to allow the full engagement of the Brazilian universities and technological companies in all the necessary developments to be carried out. In this paper, we present some aspects of this mission, including the transfer trajectories to be used, and details of buss and payload subsystems that are being developed and will be used. Copyright ©2010 by the International Astronautical Federation. All rights reserved.

Stable retrograde orbits around the triple system 2001 SN263

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

The magnetically-active, low-mass, triple system WDS 19312+3607 (Research Note)

Aims. We investigated in detail the system WDS 19312+3607, whose primary is an active M4.5Ve star previously inferred to be young (τ ~ 300–500 Ma) based on its high X-ray luminosity. Methods. We collected intermediate- and low-resolution optical spectra taken with 2 m-class telescopes, photometric data from the B to 8 μm bands, and data for eleven astrometric epochs with a time baseline of over 56 years for the two components in the system, G 125–15 and G 125–14. Results. We derived the M4.5V spectral types of both stars, confirmed their common proper motion, estimated their heliocentric distance and projected physical separation, determined their Galactocentric space velocities, and deduced a most-probable age of older than 600 Ma. We discovered that the primary, G 125–15, is an inflated, double-lined, spectroscopic binary with a short period of photometric variability of 1.6 d, which we associated with orbital synchronisation. The observed X-ray and Hα emissions, photometric variability, and abnormal radius and effective temperature of G 125–15 AB are indicative of strong magnetic activity, possibly because of the rapid rotation. In addition, the estimated projected physical separation between G 125–15 AB and G 125–14 of about 1200 AU ensures that WDS 19312+3607 is one of the widest systems with intermediate M-type primaries. Conclusions. G 125–15 AB is a nearby (d ≈ 26 pc), bright (J ≈ 9.6 mag), active spectroscopic binary with a single proper-motion companion of the same spectral type at a wide separation. They are thus ideal targets for specific follow-ups to investigate wide and close multiplicity or stellar expansion and surface cooling because of the lower convective efficiency.

Orbital and Physical Properties of the σ Ori Aa, Ab, B Triple System

We provide a complete characterization of the astrophysical properties of the σ Ori Aa, Ab, B hierarchical triple system and an improved set of orbital parameters for the highly eccentric σ Ori Aa, Ab spectroscopic binary. We compiled a spectroscopic data set comprising 90 high-resolution spectra covering a total time span of 1963 days. We applied the Lehman-Filhés method for a detailed orbital analysis of the radial velocity curves and performed a combined quantitative spectroscopic analysis of the σ Ori Aa, Ab, B system by means of the stellar atmosphere code FASTWIND. We used our own plus other available information on photometry and distance to the system for measuring the radii, luminosities, and spectroscopic masses of the three components. We also inferred evolutionary masses and stellar ages using the Bayesian code BONNSAI. The orbital analysis of the new radial velocity curves led to a very accurate orbital solution of the σ Ori Aa, Ab pair. We provided indirect arguments indicating that σ Ori B is a fast-rotating early B dwarf. The FASTWIND+BONNSAI analysis showed that the Aa, Ab pair contains the hottest and most massive components of the triple system while σ Ori B is a bit cooler and less massive. The derived stellar ages of the inner pair are intriguingly younger than the one widely accepted for the σ Orionis cluster, at 3 ± 1 Ma. The outcome of this study will be of key importance for a precise determination of the distance to the σ Orionis cluster, the interpretation of the strong X-ray emission detected for σ Ori Aa, Ab, B, and the investigation of the formation and evolution of multiple massive stellar systems and substellar objects.

Generalizations of Cole's Systems

There are four resolvable Steiner triple systems on fifteen elements. Some generalizations of these systems are presented here.

Some generalization of Desargues and Veronese configurations

2000 Mathematics Subject Classification: 51E14, 51E30.