Orthogonal decompositions of Lie algebras over finite commutative rings

Let [Formula: see text] be a finite commutative ring with identity. In this paper, we give a necessary condition for the existence of an orthogonal decomposition of the special linear Lie algebra over [Formula: see text]. Additionally, we study orthogonal decompositions of the symplectic Lie algebra and the special orthogonal Lie algebra over [Formula: see text].

On the Abuaf-Ueda Flop via Non-Commutative Crepant Resolutions

The Abuaf-Ueda flop is a 7-dimensional flop related to G₂ homogeneous spaces. The derived equivalence for this flop was first proved by Ueda using mutations of semi-orthogonal decompositions. In this article, we give an alternative proof for the derived equivalence using tilting bundles. Our proof also shows the existence of a non-commutative crepant resolution of the singularity appearing in the flopping contraction. We also give some results on moduli spaces of finite-length modules over this non-commutative crepant resolution.

Complex rotated CCA: a method to correlate lagged geophysical variables

<p>The nature of the climate system is very complex: a network of mutual interactions between ocean and atmosphere lead to a multitude of overlapping geophysical processes. As a consequence, the same process has often a signature on different climate variables but with spatial and temporal shifts. Orthogonal decompositions, such as Canonical Correlation Analysis (CCA), of geophysical data fields allow to filter out common dominant patterns between two different variables by maximizing cross-correlation. In general, however, CCA suffers from (i) the orthogonality constraint, which tends to produce unphysical patterns, and (ii) the use of direct correlations, which leads to signals that are merely shifted in time being considered as distinct patterns.</p><p>In this work, we propose an extension of CCA, complex rotated CCA (crCCA), to address both limitations. First, we generate complex signals by using the Hilbert transforms. To reduce the spatial leakage inherent in Hilbert transforms, we extend the time series using the Theta model, thus creating an anti-leakage buffer space. We then perform the orthogonal decomposition in complex space, allowing us to detect out-of-phase signals. Subsequent Varimax rotation removes the orthogonal constraints to allow more geophysically meaningful modes.</p><p>We applied crCCA to a pair of variables expected to be coupled: Pacific sea surface temperature and continental precipitation. We show that crCCA successfully captures the temporally and spatially complex modes of (i) seasonal cycle, (ii) canonical ENSO, and (iii) ENSO Modoki, in a compact manner that allows an easy geophysical interpretation. The proposed method has the potential to be useful especially, but not limited to, studies on the prediction of continental precipitation by other climate variables. An implementation of the method is readily available as a Python package.</p>

On Orthogonal Symmetric Chain Decompositions

The $n$-cube is the poset obtained by ordering all subsets of $\{1,\ldots,n\}$ by inclusion, and it can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$ chains, which is the minimum possible number. Two such decompositions of the $n$-cube are called orthogonal if any two chains of the decompositions share at most a single element. Shearer and Kleitman conjectured in 1979 that the $n$-cube has $\lfloor n/2\rfloor+1$ pairwise orthogonal decompositions into the minimum number of chains, and they constructed two such decompositions. Spink recently improved this by showing that the $n$-cube has three pairwise orthogonal chain decompositions for $n\geq 24$. In this paper, we construct four pairwise orthogonal chain decompositions of the $n$-cube for $n\geq 60$. We also construct five pairwise edge-disjoint symmetric chain decompositions of the $n$-cube for $n\geq 90$, where edge-disjointness is a slightly weaker notion than orthogonality, improving on a recent result by Gregor, Jäger, Mütze, Sawada, and Wille.