Direct Proof Research Articles

Two results about the Sackin and Colless indices for phylogenetic trees and their shapes.

The Sackin and Colless indices are two widely-used metrics for measuring the balance of trees and for testing evolutionary models in phylogenetics. This short paper contributes two results about the Sackin and Colless indices of trees. One result is the asymptotic analysis of the expected Sackin and Colless indices of tree shapes (which are full binary rooted unlabelled trees) under the uniform model where tree shapes are sampled with equal probability. Another is a short direct proof of the closed formula for the expected Sackin index of phylogenetic trees (which are full binary rooted trees with leaves being labelled with taxa) under the uniform model.

The Medawar Prize Acceptance Speech 2022.

The Medawar Prize Acceptance Speech 2022.

Generalization of ETS-NOCV and ALMO-COVP Energy Decomposition Analysis to Connect Any Two Electronic States and Comparative Assessment.

Energy decomposition analysis (EDA) is a useful tool for obtaining chemically meaningful insights into molecular interactions. The extended transition-state method with natural orbitals for chemical valence (ETS-NOCV) and the absolutely localized molecular orbital-based method with complementary occupied-virtual pairs (ALMO-COVP) are two successful EDA schemes. Working within ground-state generalized Kohn-Sham density functional theory (DFT), we extend these methods to perform EDA between any two electronic states that can be connected by a unitary transformation of density matrices. A direct proof that the NOCV eigenvalues are symmetric pairs is given, and we also prove that the charge and energy difference defined by ALMO are invariant under certain orbital rotations, allowing us to define COVPs. We point out that ETS is actually a 1-point quadrature to obtain the effective Fock matrix, and though it is reasonably accurate, it can be systematically further improved by adding more quadrature points. We explain why the calculated amount of transferred charge measured by ALMO-COVP is typically much smaller than that of ETS-NOCV and explain why the ALMO-COVP values should be preferred. While the two schemes are independent, ETS-NOCV and ALMO-COVP in fact give a very similar chemical picture for a variety of chemical interactions, including H-H+, the transition structure for the Diels-Alder reaction between ethene and butadiene, and two hydrogen-bonded complexes, H2O···F- and H2O···HF.

Semantic Factorization and Descent

Let {mathbb {A}} be a 2-category with suitable opcomma objects and pushouts. We give a direct proof that, provided that the codensity monad of a morphism p exists and is preserved by a suitable morphism, the factorization given by the lax descent object of the two-dimensional cokernel diagram of p is up to isomorphism the same as the semantic factorization of p, either one existing if the other does. The result can be seen as a counterpart account to the celebrated Bénabou–Roubaud theorem. This leads in particular to a monadicity theorem, since it characterizes monadicity via descent. It should be noted that all the conditions on the codensity monad of p trivially hold whenever p has a left adjoint and, hence, in this case, we find monadicity to be a two-dimensional exact condition on p, namely, to be an effective faithful morphism of the 2-category {mathbb {A}}.

A direct proof of the Sharp Gårding inequality for symbols with limited smoothness

We give a proof of the (possibly optimal) Sharp Gårding inequality for system operators with symbol of limited smoothness directly from the original symmetrization arguments by Friedrichs and Kumano-Go. The fact that only a few derivatives of the regularized symbol are really important was already there.

ON THE NUMBER OF NEARLY SELF-CONJUGATE PARTITIONS

AbstractA partition $\lambda $ of n is said to be nearly self-conjugate if the Ferrers graph of $\lambda $ and its transpose have exactly $n-1$ cells in common. The generating function of the number of such partitions was first conjectured by Campbell and recently confirmed by Campbell and Chern (‘Nearly self-conjugate integer partitions’, submitted for publication). We present a simple and direct analytic proof and a combinatorial proof of an equivalent statement.

Direct proofs of intrinsic properties of prox-regular sets in Hilbert spaces

Abstract We provide new proofs of two intrinsic properties of prox-regular sets in Hilbert spaces.

Commutator method for averaging lemmas

We introduce a commutator method with multipliers to prove averaging lemmas, the regularizing effect for the velocity average of solutions for kinetic equations. This method requires only elementary techniques in Fourier analysis and shows a new range of assumptions that are sufficient for the velocity average to be in $L^2([0,T],H^{1/2}_x).$ This result not only shows an interesting connection between averaging lemmas and local smoothing property of dispersive equations, but also provide a direct proof for the regularizing effect for the measure-valued solutions of scalar conservation laws in space dimension one.

Decay Rate Estimates for the Wave Equation with Subcritical Semilinearities and Locally Distributed Nonlinear Dissipation

We study the stabilization and the wellposedness of solutions of the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable observability inequality. It is well known that observability inequalities play a critical role in characterizing the long time behaviour of solutions of evolution equations, which is the main goal of this study. In order to address this, we truncate the nonlinearities, and thereby construct approximate solutions for which it is possible to obtain a priori bounds and prove the essential observability inequality. The treatment of these approximate solutions is still a challenging task and requires the use of Strichartz estimates and some microlocal analysis tools such as microlocal defect measures. We include an appendix on the latter topic here to make the article self contained and supplement details to proofs of some of the theorems which can be already be found in the lecture notes of Burq and Gérard (http://www.math.u-psud.fr/~burq/articles/coursX.pdf, 2001). Once we establish essential observability properties for the approximate solutions, it is not difficult to prove that the solution of the original problem also possesses a similar feature via a delicate passage to limit. In the last part of the paper, we establish various decay rate estimates for different growth conditions on the nonlinear dissipative effect. We in particular generalize the known results on the subject to a considerably larger class of dissipative effects.

Establishment of a method to expose and measure pollution in excised porcine skin with electron paramagnetic resonance spectroscopy

Health problems associated with the amount of air pollutants are increasing worldwide. Pollution damages not only the lungs; it also has an impact on skin health and is co-responsible for the development of skin diseases.Anti-pollution products are on the rise in the cosmetic market but so far, there is no established method to directly assess the impact of pollution on the skin and to test the efficacy of anti-pollution products.To address this problem, two different chambers were developed for the reproducible exposure to realistic air pollutant concentrations. One chamber for the exclusive use of excised skin and hair samples, the second chamber for ex vivo and in vivo measurements. Measurements of nicotine next to the investigated skin area allow conclusions to be drawn on the particle concentration to which the skin is exposed. Electron paramagnetic resonance spectroscopy, which enables the detection of free radicals in different systems, was applied to assess the hazard potential of pollution in the skin. A direct proof of the formation of free radicals in the skin by the model pollutant cigarette smoke could be demonstrated. An additional application of UV irradiation even increased the formation of free radicals in the skin seven-fold (sum parameter). Depending on the question of interest, the use of different spin probes allows various assessments of the radical formation in skin: the amount of radicals but also the antioxidant status of the microenvironment can be estimated.Using two exposure chambers, the direct formation of oxidative stress by cigarette smoke on ex vivo skin, with and without additional UV exposure, could be reproducibly examined. This measurement method is promising for the assessment of anti-pollution products and could allow a direct causal connection between pollutant, effect on the skin and the protective function of skin care products.

Hereditarily Structurally Complete Intermediate Logics: Citkin’s Theorem Via Duality

A deductive system is said to be structurally complete if its admissible rules are derivable. In addition, it is called hereditarily structurally complete if all its extensions are structurally complete. Citkin (1978) proved that an intermediate logic is hereditarily structurally complete if and only if the variety of Heyting algebras associated with it omits five finite algebras. Despite its importance in the theory of admissible rules, a direct proof of Citkin’s theorem is not widely accessible. In this paper we offer a self-contained proof of Citkin’s theorem, based on Esakia duality and the method of subframe formulas. As a corollary, we obtain a short proof of Citkin’s 2019 characterization of hereditarily structurally complete positive logics.

An Angular Method to Calculate the Electric Field of a Long Cylindrical Shell

The electric field inside a uniformly charged long cylindrical shell is a standard example discussed in introductory university physics. The field is zero inside the cylinder; outside of the cylinder, the field behaves like a long wire at the center carrying the same amount of charge per unit length, analogous to that of a uniformly charged spherical shell, typically referred to as Newton’s shell theorem. It is usually proven with the use of Gauss’s law and cylindrical symmetry of the electric field. A more direct proof by integration is lacking at this level. In fact, students often have difficulties visualizing intuitively why the field at off-center points is cancelled out completely and behaves like a line charge at the center on the outside.

Eternal universe in dynamic equilibrium

Big Bang cosmology is problematic because of the hypothetical beginning that is not in accord with the conservation of energy. Furthermore, it is based on the interpretation of astronomical data that is questionable. Cosmic microwave background radiation (CMB) is not direct proof of the existence of the recombination period in some remote physical past. Cosmological redshift can be seen as the “tired light effect” proposed by Zwicky [Proc. Natl. Acad. Sci. U. S. A. 15, 773 (1929)]. Based on direct reading of astronomical data, here we introduce a model of the universe, which predicts that in Active Galactic Nuclei (AGN) in the centers of galaxies, matter is transforming back into the elementary particles in the form of huge jets that are throwing elementary particles into the intergalactic space and so creating “fresh material” for new stars formation. This process occurring in AGNs, invoked by our model, has no beginning, and it is in permanent dynamic equilibrium, non-created, which means the universe is eternal. The entropy of the universe as a whole is constant.

Ridge Polariton Laser: Different from a Semiconductor Edge-Emitting Laser

We experimentally demonstrate the difference between a ridge polariton laser, and a conventional edge-emitting ridge laser operating under electron-hole population inversion. The horizontal laser cavities are 20 -- 60 $\mu$m long GaN etched ridge structures with vertical Bragg reflectors. We investigate the laser threshold under optical pumping and assess quantitatively the effect of a varying optically-pumped length. The laser effect is achieved for an exciton reservoir length of just 15% of the cavity length, which would not be possible in a conventional ridge laser, with an inversion-less polaritonic gain about 10 times larger than in equivalent GaN lasers. The modelling of the cavity free spectral range demonstrates the polaritonic nature of the modes.

A direct proof of Müller’s result on automorphisms of finite p-groups

A direct proof of Müller’s result on automorphisms of finite p-groups

Directional extremal statistics for Ginibre eigenvalues

We consider the eigenvalues of a large dimensional real or complex Ginibre matrix in the region of the complex plane where their real parts reach their maximum value. This maximum follows the Gumbel distribution and that these extreme eigenvalues form a Poisson point process as the dimension asymptotically tends to infinity. In the complex case, these facts have already been established by Bender [Probab. Theory Relat. Fields 147, 241 (2010)] and in the real case by Akemann and Phillips [J. Stat. Phys. 155, 421 (2014)] even for the more general elliptic ensemble with a sophisticated saddle point analysis. The purpose of this article is to give a very short direct proof in the Ginibre case with an effective error term. Moreover, our estimates on the correlation kernel in this regime serve as a key input for accurately locating maxRSpec(X) for any large matrix X with i.i.d. entries in the companion paper [G. Cipolloni et al., arXiv:2206.04448 (2022)].

Band 3-mediated Plasmodium vivax invasion is associated with transcriptional variation in PvTRAg genes.

The Plasmodium vivax reticulocyte invasion process is still poorly understood, with only a few receptor-ligand interactions identified to date. Individuals with the Southeast Asian ovalocytosis (SAO) phenotype have a deletion in the band 3 protein on the surface of erythrocytes, and are reported to have a lower incidence of clinical P. vivax malaria. Based on this observation, band 3 has been put forward as a receptor for P. vivax invasion, although direct proof is still lacking. In this study, we combined functional ex vivo invasion assays and transcriptome sequencing to uncover a band 3–mediated invasion pathway in P. vivax and potential band 3 ligands. Invasion by P. vivax field isolates was 67%-71% lower in SAO reticulocytes compared with non-SAO reticulocytes. Reticulocyte invasion was decreased by 40% and 27%-31% when blocking with an anti-band 3 polyclonal antibody and a PvTRAg38 peptide, respectively. To identify new band 3 receptor candidates, we mRNA-sequenced schizont-stage isolates used in the invasion assays, and observed high transcriptional variability in multigene and invasion-related families. Transcriptomes of isolates with low or high dependency on band 3 for invasion were compared by differential expression analysis, which produced a list of band 3 ligand candidates with high representation of PvTRAg genes. Our ex vivo invasion assays have demonstrated that band 3 is a P. vivax invasion receptor and confirm previous in vitro studies showing binding between PvTRAg38 and band 3, although the lower and variable inhibition levels observed suggest the involvement of other ligands. By coupling transcriptomes and invasion phenotypes from the same isolates, we identified a list of band 3 ligand candidates, of which the overrepresented PvTRAg genes are the most promising for future research.

N-band photonic Hopf insulators based on 2D microring lattices.

Hopf insulators are topological insulators whose topological behavior arises from the nontrivial mapping from a 3D sphere to a 2D sphere, known as the Hopf map. The Hopf map, typically encountered in the study of spinor and Skyrmion systems, is classified topologically by an integer invariant called the Hopf index. Here we show that, owing to the periodic circulation of light inside each microring, a 2D lattice of microring resonators can emulate an N-band photonic Hopf insulator with nontrivial Hopf index. In particular, we show by numerical computation and direct analytical proof that the N-band Hopf index of the microring lattice is identical to its winding number. The result shows that the Hopf index is an alternative topological invariant for classifying 2D microring photonic lattices and establishes a correspondence between the Hopf insulator phase and the anomalous Floquet insulator phase of the lattice. More generally, our work shows that 2D microring lattices can provide a versatile nanophotonic platform for studying non-Abelian topological photonic systems.

Backward Itô–Ventzell and stochastic interpolation formulae

We present a novel backward Itô–Ventzell formula and an extension of the Alekseev–Gröbner interpolating formula to stochastic flows. We also present some natural spectral conditions that yield direct and simple proofs of time uniform estimates of the difference between the two stochastic flows when their drift and diffusion functions are not the same, yielding what seems to be the first results of this type for this class of anticipative models. We illustrate the impact of these results in the context of diffusion perturbation theory, comparisons for solutions of stochastic differential equations, interacting diffusions and discrete time approximations.

Automated real-time EEG sleep spindle detection for brain-state-dependent brain stimulation.

Sleep spindles are a hallmark electroencephalographic feature of non-rapid eye movement sleep, and are believed to be instrumental for sleep-dependent memory reactivation and consolidation. However, direct proof of their causal relevance is hard to obtain, and our understanding of their immediate neurophysiological consequences is limited. To investigate their causal role, spindles need to be targeted in real-time with sensory or non-invasive brain-stimulation techniques. While fully automated offline detection algorithms are well established, spindle detection in real-time is highly challenging due to their spontaneous and transient nature. Here, we present the real-time spindle detector, a robust multi-channel electroencephalographic signal-processing algorithm that enables the automated triggering of stimulation during sleep spindles in a phase-specific manner. We validated the real-time spindle detection method by streaming pre-recorded sleep electroencephalographic datasets to a real-time computer system running a Simulink® Real-Time™ implementation of the algorithm. Sleep spindles were detected with high levels of Sensitivity (~83%), Precision (~78%) and a convincing F1-Score (~81%) in reference to state-of-the-art offline algorithms (which reached similar or lower levels when compared with each other), for both naps and full nights, and largely independent of sleep scoring information. Detected spindles were comparable in frequency, duration, amplitude and symmetry, and showed the typical time-frequency characteristics as well as a centroparietal topography. Spindles were detected close to their centre and reliably at the predefined target phase. The real-time spindle detection algorithm therefore empowers researchers to target spindles during human sleep, and apply the stimulation method and experimental paradigm of their choice.