Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

The classic Cayley identity states that

det(∂)(detX)^s=s(s+1)?(s+n−1)(detX)^s−1$\det (\partial ){(\det X)}^s=s(s+1)?(s+n-1){(\det X)}^{s-1}$

where X=(_xij)$X=(x_{ij})$ is an n×n$n\times n$ matrix of indeterminates and ∂=(∂/∂_xij)$\partial =(\partial /\partial x_{ij})$ is the corresponding matrix of partial derivatives. In this paper we present straightforward algebraic/combinatorial proofs of a variety of Cayley-type identities, both old and new. The most powerful of these proofs employ Grassmann algebra (= exterior algebra) and Grassmann–Berezin integration. Among the new identities proven here are a pair of “diagonal-parametrized” Cayley identities, a pair of “Laplacian-parametrized” Cayley identities, and the “product-parametrized” and “border-parametrized” rectangular Cayley identities.     

The generalized second law in the emergent universe

This Letter studies whether the generalized second law of thermodynamics is fulfilled in the transition from a generic initial Einstein static phase to the inflationary phase, with constant Hubble rate, and from the end of the latter to the conventional era of thermal radiation dominated expansion. As it turns out, the said law is satisfied provided the radiation component does not largely contribute to the total energy of the static phase.     

A suggested statistical test for measuring bivariate nonlinear dependence

We devise a new asymptotic statistical test to assess independence in bivariate continuous distributions. Our approach is based on the Cramér–von Mises test, in which the empirical process is viewed as the Kullback–Leibler divergence, that is, as the distance between the data under the independence hypothesis and the data empirically observed. We derive the theoretical characteristic function of the limit distribution of the test statistic and find the critical values through computer simulation. A Monte Carlo experiment is considered as assessing the validation and power performance of the test by assuming a bivariate nonlinear dependence structure with fat tails. Two extra examples, respectively, consider stationary and conditionally nonstationary series. Results confirm that our suggested test is consistent and powerful in the presence of bivariate nonlinear dependence even if the environment is non-Gaussian. Our case is illustrated with high-frequency data from stocks listed on the NYSE that recently experienced so-called mini-flash crashes.     

Convex domination in the composition and cartesian product of graphs

In this paper we characterize the convex dominating sets in the composition and Cartesian product of two connected graphs. The concepts of clique dominating set and clique domination number of a graph are defined. It is shown that the convex domination number of a composition G[H] of two non-complete connected graphs G and H is equal to the clique domination number of G. The convex domination number of the Cartesian product of two connected graphs is related to the convex domination numbers of the graphs involved.     

A methodology for the fabrication by FIB of needle-shape specimens around sub-surface features at the nanometre scale

A novel methodology for the preparation by focused ion beam (FIB) of needle-shape specimens in specific sites underneath the sample surface for their study by electron tomography (ET) is proposed. In particular, we demonstrate this methodology for the fabrication of needles containing InAs/InP quantum dots (QDs). The main challenge of this methodology is the location of specific QDs in the FIB equipment, as they are not visible with the secondary electrons detector. In order to overcome this difficulty, a series of marks visible both in conventional transmission electron microscopy and in the FIB are introduced before the preparation of the needles. The conditions for the fabrication by FIB of needles with optimized characteristics for their study by ET are also detailed.     

Ionically Self-Assembled, Multi-Luminophore One-Dimensional Micro- and Nanoscale Aggregates of Thiacarbocyanine GUMBOS

Groups of uniform materials based on organic salts (GUMBOS), derived from thiacarbocyanine (TC)-based dyes with increasing methyne chain lengths, were prepared through a single-step metathesis reaction between the iodide form of the TC dye and lithium bis(perfluoroethylsulfonyl)imide as the lipophilic anion source. Ionic self-assembly of these fluorescent hydrophobic GUMBOS resulted in aqueous dispersions of one-dimensional micro-and nano-scale molecular aggregates. Blended binary and ternary aggregates containing multiple TC GUMBOS were also prepared. These nanostructures exhibited a variety of aspect ratios, affording tunable F?rster resonance energy transfer (FRET) and aggregation-dependent spectroscopic properties.     

Entangled and sequential quantum protocols with dephasing

Sequences of commuting quantum operators can be parallelized using entanglement. This transformation is behind some optimal quantum metrology protocols and recent results on quantum circuit complexity. We show that dephasing quantum maps in arbitrary dimension can also be parallelized. This implies that for general dephasing noise the protocol with entanglement is not more fragile than the corresponding sequential protocol and, conversely, the sequential protocol is not less effective than the entangled one. We derive this result using tensor networks. Furthermore, we only use transformations strictly valid within string diagrams in dagger compact closed categories. Therefore, they apply verbatim to other theories, such as geometric quantization and topological quantum field theory. This clarifies and characterizes to some extent the role of entanglement in general quantum theories.     

On the velocity and dissipation signature of vortex tubes in isotropic turbulence

The properties of vortex tubes are extracted and analyzed from a DNS database at various Re_λ, with the objective to characterize the associated distributions of induced velocity and kinetic energy dissipation. The induced velocity exhibits an inverse power-law scaling in the far field, different from Burgers’ r⁻¹ scaling, supporting the interpretation that tubes are the remnants of vortex sheets after roll-up, and suggesting a possible link with the Kolmogorov k^−5/3 spectral scaling. The energy dissipation signature is characterized by a local maximum near the edge of the vortex core, and an absolute peak at its center, which can be tentatively explained appealing to the occurrence of a bi-axial configuration of the strain-rate tensor.