Large Cross-free sets in Steiner triple systems

A {\em cross-free} set of size $m$ in a Steiner triple system $(V,{\cal{B}})$ is three pairwise disjoint $m$-element subsets $X_1,X_2,X_3\subset V$ such that no $B\in {\cal{B}}$ intersects all the three $X_i$-s. We conjecture that for every admissible $n$ there is an STS$(n)$ with a cross-free set of size $\lfloor{n-3\over 3}\rfloor$ which if true, is best possible. We prove this conjecture for the case $n=18k+3$, constructing an STS$(18k+3)$ containing a cross-free set of size $6k$. We note that some of the $3$-bichromatic STSs, constructed by Colbourn, Dinitz and Rosa, have cross-free sets of size close to $6k$ (but cannot have size exactly $6k$). The constructed STS$(18k+3)$ shows that equality is possible for $n=18k+3$ in the following result: in every $3$-coloring of the blocks of any Steiner triple system STS$(n)$ there is a monochromatic connected component of size at least $\lceil{2n\over 3}\rceil+1$ (we conjecture that equality holds for every admissible $n$). The analogue problem can be asked for $r$-colorings as well, if r-1 \equiv 1,3 \mbox{ (mod 6)} and $r-1$ is a prime power, we show that the answer is the same as in case of complete graphs: in every $r$-coloring of the blocks of any STS$(n)$, there is a monochromatic connected component with at least ${n\over r-1}$ points, and this is sharp for infinitely many $n$.Comment: Journal of Combinatorial Designs, 201

Continuum in the Excitation Spectrum of the S=1 Compound CsNiCl_3

Recent neutron scattering experiments on CsNiCl_3 reveal some features which are not well described by the nonlinear sigma model nor by numerical simulations on isolated S=1 spin chains. In particular, in real systems the intensity of the continuum of multiparticle excitations, at T=6K, is about 5 times greater than predicted. Also the gap is slightly higher and the correlation length is smaller. We propose a theoretical scenario where the interchain interaction is approximated by a staggered magnetic field, yielding to a correct prediction of the observed quantities.Comment: 4 pages, 2 figures (.eps), RevTe

On the Integral of the Absolute Value of the Pinned Wiener Process

Let W~=W~t,0≤t≤1, be the pinned Wiener process and let ξ = ∫10|W~|. We show that the Laplace transform of ξ,ϕ(s)=Ee−ξs satisfies ∫∞0 e−us ϕ(2√s3/2)s−1/2 ds = −√π Ai(u)/Ai′(u) where Ai is Airy\u27s function. Using (∗), we find a simple recurrence for the moments, Eξn (which seem to be difficult to calculate by direct or by other techniques) namely Eξn = en√π(36√2)−n/Γ(3n+1/2) where e0 = 1,gk = Γ(3k+1/2)/Γ(k+1/2) and for n ≥ 1, en=gn+∑nk=1en−k(n k) (6k+1)/(6k−1) gk

Groups whose locally maximal product-free sets are complete

Let G be a finite group and S a subset of G. Then S is product-free if S ∩ SS = ∅, and complete if G∗ ⊆ S ∪ SS. A product-free set is locally maximal if it is not contained in a strictly larger product-free set. If S is product-free and complete then S is locally maximal, but the converse does not necessarily hold. Street and Whitehead [J. Combin. Theory Ser. A 17 (1974), 219–226] defined a group G as filled if every locally maximal product-free set S in G is complete (the term comes from their use of the phrase ‘S fills G’ to mean S is complete). They classified all abelian filled groups, and conjectured that the finite dihedral group of order 2n is not filled when n = 6k +1 (k ≥ 1). The conjecture was disproved by two of the current authors [C.S. Anabanti and S.B. Hart, Australas. J. Combin. 63 (3) (2015), 385–398], where we also classified the filled groups of odd order. In this paper we classify filled dihedral groups, filled nilpotent groups and filled groups of order 2n p where p is an odd prime. We use these results to determine all filled groups of order up to 2000

BaV3O8: A possible Majumdar-Ghosh system with S=1/2

BaV3O8 contains both magnetic V4+(S=1/2) ions and non-magnetic V5+(S=0) ions. The V4+ ions are arranged in a coupled Majumdar-Ghosh chain like network. Our magnetic susceptibility chi(T) data fit well with the Curie-Weiss formula in the temperature range of 80-300K and it yields a Curie constant C=0.39cm3K/mole-V4+ and an antiferromagnetic Weiss temperature theta=-26K. The chi(T) curve shows a broad maximum at T~25K indicative of short-range order (SRO) and an anomaly corresponding to long-range order (LRO) at TN~6K. The value of the frustration index (f=mod[theta/TN]~5) suggests that the system is moderately frustrated. Above the LRO temperature the experimental magnetic susceptibility data match well with the coupled Majumdar-Ghosh chain model with the ratio of the nnn (next-nearest neighbor) to nn (nearest neighbor) magnetic coupling alpha=2 and Jnnn/kB=40K. In a mean-field approach when considering the inter-chain interactions, we obtain the total inter-chain coupling to be about 16K. The LRO anomaly at TN is also observe in the specific heat Cp(T) data and is not sensitive to an applied magnetic field up to 90kOe. A 51V NMR signal corresponding to the non-magnetic vanadium was observed. Anomalies at 6K were observed in the variation with temperature of the 51V NMR linewidth and in the spin-lattice relaxation rate 1/T1, indicating that they are sensitive to the LRO onset and fluctuations at the magnetic V sites. The existence of two components (one short and another long) is observed in the spin-spin relaxation rate 1/T2 data in the vicinity of TN. The shorter component seems to be intimately connected with the magnetically ordered state. We suggest that both magnetically ordered and non-long range ordered (non-LRO) regions coexist in this compound below the long range ordering temperature.Comment: Accepted in Phys. Rev.

Observation of a Transient Magnetization Plateau in a Quantum Antiferromagnet on the Kagome Lattice

The magnetization process of an S=1/2 antiferromagnet on the kagome lattice, [Cu_3(titmb)_2(OCOCH_3)_6]H_2O {titmb= 1,3,5-tris(imidazol-1-ylmethyl)-2,4,6 trimethylbenzene} has been measured at very low temperatures in both pulsed and steady fields. We have found a new dynamical behavior in the magnetization process: a plateau at one third of the saturation magnetization appears in the pulsed field experiments for intermediate sweep rates of the magnetic field and disappears in the steady field experiments. A theoretical analysis using exact diagonalization yields J_1=-19K and J_2=6K, for the nearest neighbor and second nearest neighbor interactions, respectively. This set of exchange parameters explains the very low saturation field and the absence of the plateau in the thermodynamic equilibrium as well as the two-peak feature in the magnetic heat capacity. Supported by numerical results we argue that a dynamical order by disorder phenomenon could explain the transient appearance of the 1/3 plateau in pulsed field experiments.Comment: 7 pages, 5 figure

Two-photon annihilation in the pair formation cascades in pulsar polar caps

The importance of the photon-photon pair production process ($\gamma+ \gamma^{\prime}\to e^{+}+e^{-}$) to form pair production cascades in pulsar polar caps is investigated within the framework of the Ruderman-Sutherland vacuum gap model. It is found that this process is unimportant if the polar caps are not hot enough, but will play a non-negligible role in the pair formation cascades when the polar cap temperatures are in excess of the critical temperatures, $T_{cri}$, which are around $4\times 10^6K$ when $P=0.1$s and will slowly increase with increasing periods. Compared with the $\gamma-B$ process, it is found that the two-photon annihilation process may ignite a central spark near the magnetic pole, where $\gamma-B$ sparks can not be formed due to the local weak curvatures. This central spark is large if the gap is dominated by the ``resonant ICS mode''. The possible connection of these central sparks with the observed pulsar ``core'' emission components is discussed.Comment: 7 pages, 3 Postscript figures, LaTex, accepted for publication in Astronomy and Astrophysic