Weak $$A_\infty $$ Weights and Weak Reverse Hölder Property in a Space of Homogeneous Type

In the Euclidean setting, the Fujii–Wilson-type \(A_\infty \) weights satisfy a reverse Hölder inequality (RHI), but in spaces of homogeneous type the best-known result has been that \(A_\infty \) weights satisfy only a weak reverse Hölder inequality. In this paper, we complement the results of Hytönen, Pérez and Rela and show that there exist both \(A_\infty \) weights that do not satisfy an RHI and a genuinely weaker weight class that still satisfies a weak RHI. We also show that all the weights that satisfy a weak RHI have a self-improving property, but the self-improving property of the strong reverse Hölder weights fails in a general space of homogeneous type. We prove most of these purely non-dyadic results using convenient dyadic systems and techniques.     

Quantitative weighted estimates for rough homogeneous singular integrals

We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space L ²(w), we obtain a bound that is quadratic in A ₂ constant \({\left[ w \right]_{{A_2}}}\). We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on the Dini norm of the kernels is crucial to control the summability of the series expansion of the rough operator. We conclude with applications and conjectures related to weighted bounds for powers of the Beurling transform.     

Singular convolution integrals with operator-valued kernel

We study operators \(f\mapsto Kf\) of the form \((Kf)(t)=\int_{{\bf R}^{n}} k(t-s)f(s) {\rm d}s\), where f is a vector-valued function and k an operator-valued singular kernel. Sufficient conditions for boundedness on L ^p -spaces of UMD-valued functions are given in terms of a Hörmander-type condition involving R-boundedness. The results cover large classes of kernels and also provide new proofs of recent operator-valued Fourier multiplier theorems. Moreover, they give new information about families of singular integral operators.     

1,8-Dimercuri-6-Phenyl-1H-Carbazole as a Monofacial Dinuclear Organometallic Nucleobase

A C-nucleoside with 6-phenyl-1H-carbazole as the base moiety has been synthesized and incorporated in the middle of an oligonucleotide. Mercuration of this modified residue at positions 1 and 8 gave the first example of an oligonucleotide featuring a monofacial dinuclear organometallic nucleobase. The dimercurated oligonucleotide formed stable duplexes with unmodified oligonucleotides placing either cytosine, guanine, or thymine opposite to the organometallic nucleobase. A highly stabilizing (ΔT_m=7.3 °C) Hg^II-mediated base pair was formed with thymine. According to DFT calculations performed at the PBE0DH level of theory, this base pair is most likely dinuclear, with the two Hg^II ions coordinated to O2 and O4 of the thymine base.     

On the Carleson duality

As a tool for solving the Neumann problem for divergence-form equations, Kenig and Pipher introduced the space ${\mathcal{X}}$ of functions on the half-space, such that the non-tangential maximal function of their L ₂ Whitney averages belongs to L ₂ on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of ${\mathcal{X}}$ , and characterize the pointwise multipliers from ${\mathcal{X}}$ to L ₂ on the half-space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to L _p generalizations of the space ${\mathcal{X}}$ . Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.     

Non-homogeneous Tb Theorem and Random Dyadic Cubes on Metric Measure Spaces

We prove a Tb theorem on quasimetric spaces equipped with what we call an upper doubling measure. This is a property that encompasses both the doubling measures and those satisfying the upper power bound ??(B(x,r))??Cr ^d . Our spaces are only assumed to satisfy the geometric doubling property: every ball of radius r can be covered by at most N balls of radius r/2. A key ingredient is the construction of random systems of dyadic cubes in such spaces.     

5‐Mercuricytosine: An Organometallic Janus Nucleobase

The base‐pairing properties of 5‐mercuricytosine have been explored at the monomer level by NMR titrations and at the oligonucleotide level by melting temperature measurements. The NMR studies revealed a relatively high affinity for guanine, hypoxanthine, and uridine, that is, bases that are deprotonated upon coordination of Hg^II. Within an oligonucleotide duplex, 5‐mercuricytosine formed Hg^II‐mediated base pairs with thymine and guanine. In the former case, the duplex formed was as stable as the respective duplex comprising solely Watson–Crick base pairs. Based on detailed thermodynamic analysis of the melting curves, the stabilization by the Hg^II‐mediated base pairs may be attributed to a comparatively low entropic penalty of hybridization.     

Nucleated polymerization with secondary pathways. III. Equilibrium behavior and oligomer populations

We explore the long-time behavior and equilibrium properties of a system of linear filaments growing through nucleated polymerisation. We show that the length distribution for breakable filaments evolves through two well defined limiting cases: first, a steady state distribution determined by the balance of breakage and elongation is reached; upon monomer depletion at the end of the growth phase, an equilibrium length distribution biased towards smaller filament fragments emerges. We furthermore compute the time evolution of the concentration of small oligomeric filament fragments. For frangible filaments, oligomers are present both at early times and at equilibrium, whereas in the absence of fragmentation, oligomers are only present in significant quantities at the beginning of the polymerisation reaction. Finally, we discuss the significance of these results for the biological consequences of filamentous protein aggregation.     

Fountain-inspired erasure coding for real-time traffic

An erasure correction strategy based on fountain coding is proposed for traffic with real-time requirements. A sliding window marks the range of non-expired data. Each new block entering the window is once sent as such, followed by probabilistically sending a repair packet. The repair packets are formed as a random combination of the blocks in the current window using a degree distribution as in LT coding. The performance of the method with a given channel loss probability is analyzed using a Markov chain model. The state space, however, has to be truncated for computational tractability. The truncation error is verified to be small enough by simulations. By using the analytical model the optimal degree distribution is found to be of single-degree type. The performance of the proposed scheme is compared with deterministic settings, in which repair packets are sent after fixed number of systematic packets. Further comparison is made against Raptor coding, and we note that using the presented strategy can result in better performance in some situations.