Binary Geometries Research Articles

Tuza’s Conjecture for Binary Geometries

Tuza’s Conjecture for Binary Geometries

Binary geometries, generalized particles and strings, and cluster algebras

The authors study the fundamental properties of scattering amplitudes of particles in any spacetime dimension. They introduce binary geometries, giving a completely rigid geometric realization of the combinatorics of generalized associahedra attached to any Dynkin diagram. Furthermore, they define open and closed ``cluster string integrals'', which provide a generalization of particle and string scattering amplitudes, and enjoy remarkable factorization properties at finite ${\ensuremath{\alpha}}^{\ensuremath{'}}$.

Experimental and Numerical Investigation of Flow Structures Around Different Binary Building Geometries

Experimental and Numerical Investigation of Flow Structures Around Different Binary Building Geometries

The first interferometric survey of massive YSOs in the K-band

Context. Circumstellar discs are essential for the formation of high mass stars, while multiplicity, and in particular binarity, appears to be an inevitable outcome, as the vast majority of massive stars (>8 M⊙) are found in binaries (up to 100%). Our understanding of the innermost regions of accretion discs around massive stars and the binarity of high-mass young stars is sparse because of the high spatial resolution and sensitivity required to trace these rare and distant objects. Aims. We aim to spatially resolve and constrain the sizes of the dust and ionised gas emission from the innermost regions of a sample of massive young stellar objects (MYSOs) for the first time, and to provide high-mass binary statistics for young stars at 2–300 au scales using direct interferometric measurements. Methods. We observed six MYSOs using long-baseline near-infrared K-band interferometry on the VLTI (GRAVITY, AMBER) in order to resolve and characterise the 2.2 μm hot dust emission originating from the inner rim of circumstellar discs around MYSOs, and the associated Brγ emission from ionised gas. We fitted simple geometrical models to the interferometric observables, and determined the inner radius of the dust emission. We placed MYSOs with K-band measurements in a size–luminosity diagram for the first time, and compared our findings to their low- and intermediate-mass counterparts (T Tauris and Herbig AeBes). We also compared the observed K-band sizes (i.e. inner rim radius) to the sublimation radius predicted by three different disc scenarios: a classical thick flattened structure with oblique heating in action, and direct heating from the protostar via an optically thin cavity with and without backwarming effects. Lastly, we applied binary geometries to trace close binarity among MYSOs. Results. The characteristic size of the 2.2 μm continuum emission towards this sample of MYSOs shows a large scatter at the given luminosity range. When the inner sizes of MYSOs are compared to those of lower mass Herbig AeBe and T Tauri stars, they appear to follow a universal trend in that the sizes scale with the square-root of the stellar luminosity. The Brγ emission originates from a similar or somewhat smaller and co-planar area compared to the 2.2 μm continuum emission. We discuss this new finding with respect to a disc-wind or jet origin. Finally, we report an MYSO binary fraction of 17–25% at milli-arcsecond separations (2–300 au). Conclusions. The size–luminosity diagram indicates that the inner regions of discs around young stars scale with luminosity independently of the stellar mass. The observed fraction of MYSO binaries in K-band is almost ‘flat’ for a wide range of separations (2–10 000 au). At the targeted scales (2–300 au), the MYSO binary fraction is lower than what was previously reported for the more evolved main sequence massive stars, which, if further confirmed, could implicate predictions from massive binary formation theories. Lastly, with this study, we can finally spatially resolve the crucial star–disc interface in a sample of MYSOs, showing that au-scale discs are prominent in high-mass star formation and are similar to their low-mass equivalents, while the ionised gas can be linked to disc wind and disc accretion models similar to Herbig AeBes.

The Smallest Matroids with no Large Independent Flat

We show that a simple rank-$r$ matroid with no $(t+1)$-element independent flat has at least as many elements as the matroid $M_{r,t}$ defined to be the direct sum of $t$ binary projective geometries whose ranks pairwise differ by at most $1$. We also show for $r \geqslant 2t$ that $M_{r,t}$ is the unique example for which equality holds.

Stringy canonical forms and binary geometries from associahedra, cyclohedra and generalized permutohedra

Stringy canonical forms are a class of integrals that provide α′-deformations of the canonical form of any polytopes. For generalized associahedra of finite-type cluster algebras, there exist completely rigid stringy integrals, whose configuration spaces are the so-called binary geometries, and for classical types are associated with (generalized) scattering of particles and strings. In this paper, we propose a large class of rigid stringy canonical forms for another class of polytopes, generalized permutohedra, which also include associahedra and cyclohedra as special cases (type An and Bn generalized associahedra). Remarkably, we find that the configuration spaces of such integrals are also binary geometries, which were suspected to exist for generalized associahedra only. For any generalized permutohedron that can be written as Minkowski sum of coordinate simplices, we show that its rigid stringy integral factorizes into products of lower integrals for massless poles at finite α′, and the configuration space is binary although the u equations take a more general form than those “perfect” ones for cluster cases. Moreover, we provide an infinite class of examples obtained by degenerations of type An and Bn integrals, which have perfect u equations as well. Our results provide yet another family of generalizations of the usual string integral and moduli space, whose physical interpretations remain to be explored.

Excluding the Cycle Geometries of the Kuratowski Graphs from Binary Geometries

We prove that a binary combinatorial geometry (or simple matroid) of rank n not containing M(K5) (respectively, M(K3,3)) as a minor contains at most 8n (respectively, 10n) points. Here, M(K5) is the cycle geometry of the complete graph K5 and M(K3,3) is the cycle geometry of the complete bipartite graph K3,3. These bounds are probably not tight. The proof uses the notion of a bond graph. Using these results, we obtain upper bounds on the critical exponents of these geometries.

A characterization of binary geometries by a double elimination axiom

This note gives a characterization of binary geometries by means of a double elimination axiom which is a strengthening of the usual circuit elimination axiom used in defining matroids in general.

Uniform binary geometries

In a geometric lattice every interval can be mapped isomorphically into an upper interval (containing 1) by a strong map. A natural question thus arises as to what extent certain assumptions on the “upper interval structure” determine the whole lattice. We consider conditions of the following sort: that above a certain levelm any two upper intervals of the same length be isomorphic. This property, called uniformity, is studied for binary geometries. The geometries satisfying the strongest uniformity condition (m = 1) are determined (except for one open case). As is to be expected the corresponding problem for lower intervals is easier and is solved completely.