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1. The size of the class of countable sequences of ordinals

   https://doi.org/10.1090/tran/8573  

   Chan, William ; Jackson, Stephen ; Trang, Nam ( January 2021 , Transactions of the American Mathematical Society)

   Assume . There is no injection of (the set of countable length sequences of countable ordinals) into (the class of length sequences of ordinals). There is no injection of (the powerset of ) into (the class of countable length sequences of ordinals). 

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2. Analysis of signal-to-noise ratio of angle of polarization and degree of polarization

   https://doi.org/10.1364/OSAC.422541  

   Chen, Yingkai ; Zhu, Zhongmin ; Liang, Zuodong ; Iannucci, Leanne_E ; Lake, Spencer_P ; Gruev, Viktor ( April 2021 , OSA Continuum)

   Recent advancements in nanofabrication technology has led to commercialization of single-chip polarization and color-polarization imaging sensors in the visible spectrum. Novel applications have arisen with the emergence of these sensors leading to questions about noise in the reconstructed polarization images. In this paper, we provide theoretical analysis for the input and output referred noise for the angle and degree of linear polarization information. We validated our theoretical model with experimental data collected from a division of focal plane polarization sensor. Our data indicates that the noise in the angle of polarization images depends on both incident light intensity and degree of linear polarization and is independent of the incident angle of polarization. However, noise in degree of linear polarization images depends on all three parameters: incident light intensity, angle and degree of linear polarization. This theoretical model can help guide the development of imaging setups to record optimal polarization information.

    

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3. Spectra of products of digraphs

   https://doi.org/10.13001/ela.2020.5243  

   Catral, Minerva ; Ciardo, Lorenzo ; Hogben, Leslie ; Reinhart, Carolyn ( January 2020 , The Electronic Journal of Linear Algebra)

   null (Ed.)

   A unified approach to the determination of eigenvalues and eigenvectors of specific matrices associated with directed graphs is presented. Matrices studied include the new distance matrix, with natural extensions to the distance Laplacian and distance signless Laplacian, in addition to the new adjacency matrix, with natural extensions to the Laplacian and signless Laplacian. Various sums of Kronecker products of nonnegative matrices are introduced to model the Cartesian and lexicographic products of digraphs. The Jordan canonical form is applied extensively to the analysis of spectra and eigenvectors. The analysis shows that Cartesian products provide a method for building infinite families of transmission regular digraphs with few distinct distance eigenvalues. 

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4. Torsion Invariants of complexes of groups

   Okun, Boris ; Schreve, K ( February 2024 , Duke)

   Hain, Richard (Ed.)

   Suppose a residually finite group G acts cocompactly on a contractible complex with strict fundamental domain Q, where the stabilizers are either trivial or have normal Z-subgroups. Let dQ be the subcomplex of Q with nontrivial stabilizers. Our main result is a computation of the homology torsion growth of a chain of finite index normal subgroups of G. We show that independent of the chain, the normalized torsion limits to the torsion of dQ shifted a degree. Under milder assumptions of acyclicity of nontrivial stabilizers, we show similar formulas for the mod p-homology growth. We also obtain formulas for the universal and the usual L^2-torsion of G in terms of the torsion of stabilizers and topology of dQ. In particular, we get complete answers for right-angled Artin groups, which shows that they satisfy a torsion analogue of Lück’s approximation theorem. 

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5. Hybridization kinetics of out-of-equilibrium mixtures of short RNA oligonucleotides

   https://doi.org/10.1093/nar/gkac784  

   Todisco, Marco ; Szostak, Jack W. ( September 2022 , Nucleic Acids Research)

   Hybridization and strand displacement kinetics determine the evolution of the base paired configurations of mixtures of oligonucleotides over time. Although much attention has been focused on the thermodynamics of DNA and RNA base pairing in the scientific literature, much less work has been done on the time dependence of interactions involving multiple strands, especially in RNA. Here we provide a study of oligoribonucleotide interaction kinetics and show that it is possible to calculate the association, dissociation and strand displacement rates displayed by short oligonucleotides (5nt–12nt) that exhibit no expected secondary structure as simple functions of oligonucleotide length, CG content, ΔG of hybridization and ΔG of toehold binding. We then show that the resultant calculated kinetic parameters are consistent with the experimentally observed time dependent changes in concentrations of the different species present in mixtures of multiple competing RNA strands. We show that by changing the mixture composition, it is possible to create and tune kinetic traps that extend by orders of magnitude the typical sub-second hybridization timescale of two complementary oligonucleotides. We suggest that the slow equilibration of complex oligonucleotide mixtures may have facilitated the nonenzymatic replication of RNA during the origin of life.

    

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