The unsatisfiability threshold revisited

The problem of determining the unsatisfiability threshold for random 3-SAT formulas consists in determining the clause to variable ratio that marks the experimentally observed abrupt change from almost surely satisfiable formulas to almost surely unsatisfiable. Up to now, there have been rigorously established increasingly better lower and upper bounds to the actual threshold value. In this paper, we consider the problem of bounding the threshold value from above using methods that, we believe, are of interest on their own right. More specifically, we show how the method of local maximum satisfying truth assignments can be combined with results for the occupancy problem in schemes of random allocation of balls into bins in order to achieve an upper bound for the unsatisfiability threshold less than 4.571. In order to obtain this value, we establish a bound on the q-binomial coefficients (a generalization of the binomial coefficients). No such bound was previously known, despite the extensive literature on q-binomial coefficients. Finally, to prove our result we had to establish certain relations among the conditional probabilities of an event in various probabilistic models for random formulas. It turned out that these relations were considerably harder to prove than the corresponding ones for unconditional probabilities, which were previously known.     

Variability of dissolution rates at constant undersaturation

Dissolution processes play an important role in marble weathering. Investigation of the dissolution kinetics of powdered Pentelic marble, a calcitic natural stone, showed that the rates measured at constant undersaturation decreased with time. It was found that the rate of dissolution decreased sharply to about 50% of the initial value before 5% of the initially suspended powder was dissolved. Past a dissolution extent corresponding to 25% with respect to the suspended solid, the rates of dissolution at the same constant undersaturation reached a stable value corresponding to 20-25% of the initial rate measured. The reduction in the rates of dissolution up to 25% of mass loss was in agreement with the reduction in the total surface area exposed to the solution, suggesting a dependence of the rate on the number of active sites on the surface of the dissolving material. The decrease in the dissolution rates did not depend on the undersaturation of the solution or on the physical characteristics of the particles such as specific surface area, porosity, and particle size distribution. Equal mass losses, with respect to the initially suspended solid, of different samples exhibited the same relative decrease in rate regardless of the conditions of the aqueous solutions.     

Theoretical studies of carbocations in ion pairs. 8. Search for anchimeric assistance in the ionization of 2-butyl cation precursors

The structures of carbocations formed in the ionization of 2-butyl precursors were investigated by high level ab initio MO calculations on the reaction of 2-butyl fluoride (1) with borane, which gives a C4H9+ cation paired with trihydrofluoroborate (FBH3-, A). Two conformations of the "open," secondary cation (2) in the ion pair resulted from two conformations of 1, with F gauche and trans to C4 (2-g and 2-t, respectively). No anchimeric assistance by hydrogen (in 1-g) or methyl (in 1-t) was evidenced. In fact, attempts at optimizing the geometry of the H-bridged (3) and methyl-bridged (6) cations at short interionic distances (d) led to the corresponding conformations of 2. Upon ion separation, proton transfer from 2 to the anion occurred at intermediate interionic distances, consonant with experimental observations in trifluoroacetic acid. Elimination was prevented by addition of a lithium cation to the ion pair, i.e., running computations on triple ions (2.A.Li+). Cation 6 became an energy minimum beyond d = 2.5 A and 3 beyond 2.8 A. Cation 2-g was still the most stable isomer at d = 3.2 A, which was greater than the interionic distance in the crystals of the isomeric tert-butyl cation salts (3-3.1 A). Thus, spectral determinations of 2-butyl cations in the solid state should be interpreted with 2-g as the main component of the ion mixture. When the ions became separated (d >/= 4 A), only the bridged ions were energy minima. In this process, bridging did not occur opposite to the leaving group to assist the ionization, but on the same side with it, being controlled by the electrostatic interaction with the anion, as it departed from the vicinity of the cation. Such behavior was also noted in the ionization of the 3-methyl-2-butyl homolog.     

Warfarin Sodium Stability in Oral Formulations

Warfarin sodium is a low-dose pharmaceutical blood thinner that exists in two forms: the clathrate form and the amorphous form. In commercially available warfarin sodium oral suspension, the active pharmaceutical ingredient (API) is added in the amorphous state. This study investigates the apparent instability of the commercially available warfarin liquid oral formulation using Raman and IR spectroscopy, X-ray diffraction, differential scanning calorimetry, UV spectroscopy, and optical microscopy. Warfarin, not its sodium salt, was identified as the undissolved solid existing in the suspension. This was found to be due to the dissociation of sodium salt and the protonation of the warfarin ion in the liquid phase, which triggered the crystallization of the sparingly soluble unsalted form. The coexistence of protonated and unprotonated warfarin ions in the supernatant, as detected by Raman and UV spectroscopy, confirmed this assumption. Study of the dissolution of warfarin sodium amorphous salt and crystalline sodium clathrate in the placebo and pure water verified the results. The effect of pH and temperature on warfarin precipitation was also explored.     

The Importance of Nutraceuticals in COVID-19: What’s the Role of Resveratrol?

Since COVID-19 has affected global public health, there has been an urgency to find a solution to limit both the number of infections, and the aggressiveness of the disease once infected. The main characteristic of this infection is represented by a strong alteration of the immune system which, day by day, increases the risk of mortality, and can lead to a multiorgan dysfunction. Because nutritional profile can influence patient’s immunity, we focus our interest on resveratrol, a polyphenolic compound known for its immunomodulating and anti-inflammatory properties. We reviewed all the information concerning the different roles of resveratrol in COVID-19 pathophysiology using PubMed and Scopus as the main databases. Interestingly, we find out that resveratrol may exert its role through different mechanisms. In fact, it has antiviral activity inhibiting virus entrance in cells and viral replication. Resveratrol also improves autophagy and decreases pro-inflammatory agents expression acting as an anti-inflammatory agent. It regulates immune cell response and pro-inflammatory cytokines and prevents the onset of thrombotic events that usually occur in COVID-19 patients. Since resveratrol acts through different mechanisms, the effect could be enhanced, making a totally natural agent particularly effective as an adjuvant in anti COVID-19 therapy.     

An investigation of paint from a mural in the church of Sainte Madeleine, Manas, France

The pigment in brown paint samples taken from a church in the Drôme region of France has been shown to be almost pure 6‐bromoindigo and 6,6′‐dibromoindigo. The composition of the pigment was established by comparison with 6‐bromoindigo and 6,6′‐dibromoindigo standards using atmospheric pressure photoionization combined with liquid chromatography and tandem mass spectrometry. The brown paint samples were taken from a frieze of ca 20 brown images having symmetric tree‐like forms composed of five overlapping filled circles representing foliage. The tree‐like images, simple rather than artistic, had a metallic luster. The molar ratio of 6‐bromoindigo and 6,6′‐dibromoindigo in the brown paint pigment (98:2) is remarkably similar to that of shellfish purple from the Mediterranean Murex brandaris (96.5:3.5) thus, it is possible that the origin of the indigoid compounds in the brown paint pigment is the eastern coast of the Mediterranean Sea. As the production of Tyrian purple ended with the fall of Byzantium in 1453, purple pigment had to have been produced prior to this time. We have conjectured about the circumstances that may account for the use of ‘purple’ in this manner in the Drôme region and how the pigment was transported there from the eastern coast of the Mediterranean Sea during the time of the Crusades. Copyright © 2011 John Wiley & Sons, Ltd.     

Comparative Study of Sample Carriers for the Identification of Volatile Compounds in Biological Fluids Using Raman Spectroscopy

Vibrational spectroscopic techniques and especially Raman spectroscopy are gaining ground in substituting the officially established chromatographic methods in the identification of ethanol and other volatile substances in body fluids, such as blood, urine, saliva, semen, and vaginal fluids. Although a couple of different carriers and substrates have been employed for the biochemical analysis of these samples, most of them are suffering from important weaknesses as far as the analysis of volatile compounds is concerned. For this reason, in this study three carriers are proposed, and the respective sample preparation methods are described for the determination of ethanol in human urine samples. More specifically, a droplet of the sample on a highly reflective carrier of gold layer, a commercially available cuvette with a mirror to enhance backscattered radiation sealed with a lid, and a home designed microscope slide with a cavity coated with gold layer and covered with transparent cling film have been evaluated. Among the three proposed carriers, the last one achieved a quick, simple, and inexpensive identification of ethanol, which was used as a case study for the volatile compound, in the biological samples. The limit of detection (LoD) was found to be 1.00 μL/mL, while at the same time evaporation of ethanol was prevented.     

Upper Bounds and Asymptotics for the q-Binomial Coefficients

In this article, we a derive an upper bound and an asymptotic formula for the q -binomial, or Gaussian, coefficients. The q -binomial coefficients, that are defined by the expression are a generalization of the binomial coefficients, to which they reduce as q tends toward 1. In this article, we give an expression that captures the asymptotic behavior of these coefficients using the saddle point method and compare it with an upper bound for them that we derive using elementary means. We then consider as a case study the case q =1+ z / m , z <0, that was actually encountered by the authors before in an application stemming from probability and complexity theory. We show that, in this case, the asymptotic expression and the expression for the upper bound differ only in a polynomial factor; whereas, the exponential factors are the same for both expressions. In addition, we present some numerical calculations using MAPLE (a computer program for performing symbolic and numerical computations), that show that both expressions are close to the actual value of the coefficients, even for moderate values of m .