How large are the level sets of the Takagi function?

Let T be Takagi's continuous but nowhere-differentiable function. This paper considers the size of the level sets of T both from a probabilistic point of view and from the perspective of Baire category. We first give more elementary proofs of three recently published results. The first, due to Z. Buczolich, states that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. The second, due to J. Lagarias and Z. Maddock, states that the average number of points in a level set is infinite. The third result, also due to Lagarias and Maddock, states that the average number of local level sets contained in a level set is 3/2. In the second part of the paper it is shown that, in contrast to the above results, the set of ordinates y with uncountably infinite level sets is residual, and a fairly explicit description of this set is given. The paper also gives a negative answer to a question of Lagarias and Maddock by showing that most level sets (in the sense of Baire category) contain infinitely many local level sets, and that a continuum of level sets even contain uncountably many local level sets. Finally, several of the main results are extended to a version of T with arbitrary signs in the summands.Comment: Added a new Section 5 with generalization of the main results; some new and corrected proofs of the old material; 29 pages, 3 figure

Bunyavirus requirement for endosomal K+ reveals new roles of cellular ion channels during infection

In order to multiply and cause disease a virus must transport its genome from outside the cell into the cytosol, most commonly achieved through the endocytic network. Endosomes transport virus particles to specific cellular destinations and viruses exploit the changing environment of maturing endocytic vesicles as triggers to mediate genome release. Previously we demonstrated that several bunyaviruses, which comprise the largest family of negative sense RNA viruses, require the activity of cellular potassium (K+) channels to cause productive infection. Specifically, we demonstrated a surprising role for K+ channels during virus endosomal trafficking. In this study, we have used the prototype bunyavirus, Bunyamwera virus (BUNV), as a tool to understand why K+ channels are required for progression of these viruses through the endocytic network. We report three major findings: First, the production of a dual fluorescently labelled bunyavirus to visualize virus trafficking in live cells. Second, we show that BUNV traffics through endosomes containing high [K+] and that these K+ ions influence the infectivity of virions. Third, we show that K+ channel inhibition can alter the distribution of K+ across the endosomal system and arrest virus trafficking in endosomes. These data suggest high endosomal [K+] is a critical cue that is required for virus infection, and is controlled by cellular K+ channels resident within the endosome network. This highlights cellular K+ channels as druggable targets to impede virus entry, infection and disease

Mutational Analyses of the Influenza A Virus Polymerase Subunit PA Reveal Distinct Functions Related and Unrelated to RNA Polymerase Activity

Influenza A viral polymerase is a heterotrimeric complex that consists of PA, PB1, and PB2 subunits. We previously reported that a di-codon substitution mutation (G507A-R508A), denoted J10, in the C-terminal half of PA had no apparent effect on viral RNA synthesis but prevented infectious virus production, indicating that PA may have a novel role independent of its polymerase activity. To further examine the roles of PA in the viral life cycle, we have now generated and characterized additional mutations in regions flanking the J10 site from residues 497 to 518. All tested di-codon mutations completely abolished or significantly reduced viral infectivity, but they did so through disparate mechanisms. Several showed effects resembling those of J10, in that the mutant polymerase supported normal levels of viral RNA synthesis but nonetheless failed to generate infectious viral particles. Others eliminated polymerase activity, in most cases by perturbing the normal nuclear localization of PA protein in cells. We also engineered single-codon mutations that were predicted to pack near the J10 site in the crystal structure of PA, and found that altering residues K378 or D478 each produced a J10-like phenotype. In further studies of J10 itself, we found that this mutation does not affect the formation and release of virion-like particles per se, but instead impairs the ability of those particles to incorporate each of the eight essential RNA segments (vRNAs) that make up the viral genome. Taken together, our analysis identifies mutations in the C-terminal region of PA that differentially affect at least three distinct activities: protein nuclear localization, viral RNA synthesis, and a trans-acting function that is required for efficient packaging of all eight vRNAs

Mechanically assisted electrochemical degradation of alumina-TiC composites

Alumina-TiC composite material is a tough ceramic composite with excellent hardness, wear resistance and oxidation resistance in dry and high-temperature conditions. In aqueous conditions, however, it is likely to be electrochemically active facilitating charge transfer processes due to the conductive nature of TiC. For application as an orthopedic biomaterial, it is crucial to assess the electrochemical behavior of this composite, especially under a combined mechanical and electrochemical environment. In this study, we examined the mechanically assisted electrochemical performance of alumina-TiC composite in an aqueous environment. The spontaneous electrochemical response to brushing abrasion was measured. Changes in the magnitude of electrochemical current with abrasion test conditions and possible causal relationship to the alteration in surface morphology were examined. Results showed that the alumina matrix underwent abrasive wear with evidence of microploughing and grain boundary damage. Chemical analysis revealed TiO2 formation in the abraded region, indicating oxidation of the conductive TiC domain. Furthermore, wear debris from alumina abrasion appeared to affect reaction kinetics at the composite-electrolyte interface. From this work, we established that the composite undergoes abrasion assisted electrochemical degradation even in gentle abrasive conditions and the severity of degradation is related to temperature and conditions of test environment

Phase stability of colloidal mixtures of spheres and rods

We determined the phase boundaries of aqueous mixtures containing colloidal rod-like fd-viruses and polystyrene spheres using diffusing-wave spectroscopy and compared the results with free volume theory predictions. Excluded volume interactions in mixtures of colloidal rods and spheres lead to mediated depletion interactions. The strength and range of this attractive interaction depend on the concentrations of the particles, the length L and diameter D of the rods, and the radius R of the spheres. At strong enough attraction, this depletion interaction leads to phase separation. We experimentally determined the rod and sphere concentrations where these phase transitions occur by systematically varying the size ratios L/R and D/R and the aspect ratio L/D. This was done by using spheres with different radii and modifying the effective diameter of the rods through either the ionic strength of the buffer or anchoring a polymeric brush to the surface of the rods. The observed phase transitions were from a binary fluid to a colloidal gas/liquid phase coexistence that occurred already at very low concentrations due to the depletion efficiency of highly anisotropic rods. The experimentally measured phase transitions were compared to phase boundaries obtained using free volume theory (FVT), a well established theory for calculating the phase behavior of colloidal particles mixed with depletants. We find good correspondence between the experimental phase transitions and the theoretical FVT model where the excluded volume of the rod-like depletants was explicitly accounted for in both the reservoir and the system

Erratum: Phase stability of colloidal mixtures of spheres and rods (Journal of Chemical Physics (2021) 154 (204906) DOI: 10.1063/5.0048809)

In this erratum, we show the correct version of Eq. (21) as follows: (Equation Presented) Equations (21) and (22) shown in Ref. 1 contained a typographical error, which results in a minor quantitative deviation of the obtained results when used. It is noted that all results shown and plotted in Ref. 1 were computed using the correct equations shown here

Phase stability of colloidal mixtures of spheres and rods

We determined the phase boundaries of aqueous mixtures containing colloidal rod-like \textit{fd}-viruses and polystyrene spheres using diffusing-wave spectroscopy and compared the results with free volume theory predictions. Excluded volume interactions in mixtures of colloidal rods and spheres lead to mediated depletion interactions. The strength and range of this attractive interaction depend on the concentrations of the particles, the length $L$ and diameter $D$ of the rods and the radius $R$ of the spheres. At strong enough attraction, this depletion interaction leads to phase separation. We experimentally determined the rod and sphere concentrations where these phase transitions occur by systematically varying the size ratios $L/R$ and $D/R$ and the aspect ratio $L/D$. This was done by using spheres with different radii and modifying the effective diameter of the rods through either the ionic strength of the buffer or anchoring a polymeric brush to the surface of the rods. The observed phase transitions were from a binary fluid to a colloidal gas/liquid phase coexistence which occurred already at very low concentrations due to the depletion efficiency of highly anisotropic rods. The experimentally measured phase transitions were compared to phase boundaries obtained using free volume theory (FVT), a well established theory for calculating the phase behaviour of colloidal particles mixed with depletants. We find good correspondence between the experimental phase transitions and the theoretical FVT model where the excluded volume of the rod-like depletants was explicitly accounted for in both the reservoir and the system