Avian sex chromosomes evolved after the divergence birds and crocodilians from their common ancestor, so they are much younger than the better-studied chromosomes of mammals. It has long been recognized that there may have been several stages to the evolution of avian sex chromosomes. For example, the CHD1 undergoes recombination in paleognaths but not neognaths. Genome assemblies have suggested there may be variation in the timing of barriers to recombination among Neognathae, but there remains little understanding of the extent of this variability. Here, we look at partial sequences of ATP5F1A, which is on the avian Z and W chromosomes. It is known that recombination of this gene has independently ceased in Galliformes, Anseriformes, and at least five neoavian orders, but whether there are other independent cessations of recombination among Neoaves is not understood. We used a combination of data extracted from published chromosomal-level genomes with data collected using PCR and cloning to identify Z and W copies in 22 orders. Our results suggest there may be at least 19 independent cessations of recombination within Neognathae, and 3 clades that may still be undergoing recombination (or have only recently ceased recombination). Analyses of ATP5F1A protein sequences revealed an increased amino acid substitution rate for W chromosome gametologs, suggesting relaxed purifying selection on the W chromosome. Supporting this hypothesis, we found that the increased substitution rate was particularly pronounced for buried residues, which are expected to be more strongly constrained by purifying selection. This highlights the dynamic nature of avian sex chromosomes, and that this level of variation among clades means they should be a good system to understand sex chromosome evolution.

We study C*-algebraic versions of following conjectures/theorems: (1) Bieberbach conjecture (de Branges theorem) (2) Robertson conjecture (3) Lebedev-Milin conjecture (4) Zalcman conjecture (5) Krzyz conjecture (6) Corona conjecture (Carleson theorem). We prove that the C*-algebraic Bieberbach Conjecture for invertible coefficients is true for second degree C*-algebraic polynomials.

It has been known that dimensional constants such as ℏ, c, G, e, and k are merely human constructs whose values and units vary depending on the chosen system of measurement. Therefore, the time variation of dimensional constants lacks operational significance due to their dependence on them. It is well-structured and represents a valid discussion. However, this fact only becomes a meaningful debate within the context of a static or present universe. As well-established theoretically and observationally, the current universe is undergoing accelerated expansion, wherein dimensional quantities, like the wavelength of light, also experience redshift phenomena elongating over cosmic time. In other words, in an expanding universe, dimensional quantities of physical parameters vary with cosmic time. From this perspective, there exists the possibility that dimensional constants, such as the speed of light, could vary with the expansion of the universe. In this review paper, we contemplate under what circumstances the speed of light may change or remain constant over cosmic time, and discuss the potential for distinguishing these cases observationally.