Strings of congruent primes in short intervals

Soit $p_1 = 2, p_2 = 3, p_3 = 5,\ldots$ la suite des nombres premiers, et soient $q \ge 3$ et $a$ des entiers premiers entre eux. R\'ecemment, Daniel Shiu a d\'emontr\'e une ancienne conjecture de Sarvadaman Chowla. Ce dernier a conjectur\'e qu'il existe une infinit\'e de couples $p_n,p_{n+1}$ de premiers cons\'ecutifs tels que $p_n \equiv p_{n+1} \equiv a \bmod q$. Fixons $\epsilon > 0$. Une r\'ecente perc\'ee majeure, de Daniel Goldston, J\`anos Pintz et Cem Y{\i}ld{\i}r{\i}m, a \'et\'e de d\'emontrer qu'il existe une suite de nombres r\'eels $x$ tendant vers l'infini, tels que l'intervalle $(x,x+\epsilon\log x]$ contienne au moins deux nombres premiers $\equiv a \bmod q$. \'Etant donn\'e un couple de nombres premiers $\equiv a \bmod q$ dans un tel intervalle, il pourrait exister un nombre premier compris entre les deux qui n'est pas $\equiv a \bmod q$. On peut d\'eduire que soit il existe une suite de r\'eels $x$ tendant vers l'infini, telle que $(x,x+\epsilon\log x]$ contienne un triplet $p_n,p_{n+1},p_{n+2}$ de nombres premiers cons\'ecutifs, soit il existe une suite de r\'eels $x$, tendant vers l'infini telle que l'intervalle $(x,x+\epsilon\log x]$ contienne un couple $p_n,p_{n+1}$ de nombres premiers tel que $p_n \equiv p_{n+1} \equiv a \bmod q$. On pense que les deux \'enonc\'es sont vrais, toutefois on peut seulement d\'eduire que l'un d'entre eux est vrai, sans savoir lequel. Dans la premi\`ere partie de cette th\`ese, nous d\'emontrons que le deuxi\`eme \'enonc\'e est vrai, ce qui fournit une nouvelle d\'emonstration de la conjecture de Chowla. La preuve combine des id\'ees de Shiu et de Goldston-Pintz-Y{\i}ld{\i}r{\i}m, donc on peut consid\'erer que ce r\'esultat est une application de leurs m\'thodes. Ensuite, nous fournirons des bornes inf\'erieures pour le nombre de couples $p_n,p_{n+1}$ tels que $p_n \equiv p_{n+1} \equiv a \bmod q$, $p_{n+1} - p_n < \epsilon\log p_n$, avec $p_{n+1} \le Y$. Sous l'hypoth\`ese que $\theta$, le \og niveau de distribution \fg{} des nombres premiers, est plus grand que $1/2$, Goldston-Pintz-Y{\i}ld{\i}r{\i}m ont r\'eussi \`a d\'emontrer que $p_{n+1} - p_n \ll_{\theta} 1$ pour une infinit\'e de couples $p_n,p_{n+1}$. Sous la meme hypoth\`ese, nous d\'emontrerons que $p_{n+1} - p_n \ll_{q,\theta} 1$ et $p_n \equiv p_{n+1} \equiv a \bmod q$ pour une infinit\'e de couples $p_n,p_{n+1}$, et nous prouverons \'egalement un r\'esultat quantitatif. Dans la deuxi\`eme partie, nous allons utiliser les techniques de Goldston-Pintz-Y{\i}ld{\i}r{\i}m pour d\'emontrer qu'il existe une infinit\'e de couples de nombres premiers $p,p'$ tels que $(p-1)(p'-1)$ est une carr\'e parfait. Ce resultat est une version approximative d'une ancienne conjecture qui stipule qu'il existe une infinit\'e de nombres premiers $p$ tels que $p-1$ est une carr\'e parfait. En effet, nous d\'emontrerons une borne inf\'erieure sur le nombre d'entiers naturels $n \le Y$ tels que $n = \ell_1\cdots \ell_r$, avec $\ell_1,\ldots,\ell_r$ des premiers distincts, et tels que $(\ell_1-1)\cdots (\ell_r-1)$ est une puissance $r$-i\`eme, avec $r \ge 2$ quelconque. \'Egalement, nous d\'emontrerons une borne inf\'erieure sur le nombre d'entiers naturels $n = \ell_1\cdots \ell_r \le Y$ tels que $(\ell_1+1)\cdots (\ell_r+1)$ est une puissance $r$-i\`eme. Finalement, \'etant donn\'e $A$ un ensemble fini d'entiers non-nuls, nous d\'emontrerons une borne inf\'erieure sur le nombre d'entiers naturels $n \le Y$ tels que $\prod_{p \mid n} (p+a)$ est une puissance $r$-i\`eme, simultan\'ement pour chaque $a \in A$.

"What" and "Where" in the intraparietal sulcus : an fMRI study of object identity and location in visual short-term memory

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

Insights into the function of short interspersed degenerated retroposons in the protozoan parasite Leishmania

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

La réductin des dictionnaires bilingues (français-anglais) au XVIIe siècle : l'exemple du Short Dictionary (1684) de Guy Miège

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Mechanical Properties of Short-Isora-Fiber-Reinforced Natural Rubber Composites: Effects of Fiber Length, Orientation, and Loading; Alkali Treatment;and Bonding Agent

A series of short-isora-fiber-reinforced natural rubber composites were prepared by the incorporation of fibers of different lengths (6, 10, and 14 mm) at 15 phr loading and at different concentrations (10, 20, 30, and 40 phr) with a 10 mm fiber length. Mixes were also prepared with 10 mm long fibers treated with a 5% NaOH solution. The vulcanization parameters, processability, and stress-strain properties of these composites were analyzed. Properties such as tensile strength, tear strength, and tensile modulus were found to be at maximum for composites containing longitudinally oriented fibers 10 mm in length. Mixes containing fiber loadings of 30 phr with bonding agent (resorcinol-formaldehyde [RF] resin) showed mechanical properties superior to all other composites. Scanning electron microscopy (SEM) studies were carried out to investigate the fiber surface morphology, fiber pullout, and fiber-rubber interface. SEM studies showed that the bonding between the fiber and rubber was improved with treated fibers and with the use of bonding agent.

Effect of processing parameters on the mechanical properties of short Kevlar aramid fibre-thermoplastic polyurethane composite

The effect of various processing parameters, such as nip gap, friction ratio and roll temperature, on the tensile properties of short Kevlar aramid fibre-thermoplastic polyurethane composite has been investigated and the tensile and tear fracture surfaces have been characterised using a scanning electron microscope. A nip gap of 0.45 mm, a friction ratio of 1.15 and a roll temperature of 62°C was found to give optimum mechanical properties. Scanning electron microscopy study revealed a higher extent of fibre orientation in the milling direction in the above condition.

Scanning Electron Microscopy Studies on Tear and Wear Failure of Short Kevlar Fiber-Thermoplastic- Polyurethane Composite

Tear and wear properties of short kevlar fiber, thermoplastic polcurethane (TPU) composite with respect to fiber loading-and fiber onentation has been studied and the fracture surfaces were examined under scanning electron microscope (SEM). Tear strength first decreased up to 20 phr fiber loading and then gradually increased with increasing fiber loading. Anisotropy in tear strength was evident beyond a fiber loading of 20 phr. Tear fracture surface of unfilled TPU showed sinusoidal folding characteristics of high strength matrix. At low fiber loading the tear failure was mainly due to fibermatrix failure whereas at higher fiber loading the failure occurred by fiber breakage. Abrasion loss shows a continuous rise with increasing fiber loading, the loss in the transverse orientation of fibers being higher than that in the longitudinal orientation. The abraded surface showed lone cracks and ridges parallel to the direction of abrasion indicating an abrasive wear mechanism. In the presence of fber the abrasion loss was mainly due to fiber low.

Self adhesion of short Kevlar fibre -thermoplastic polyurethane composite

The self adhesion behaviour of thermoplastic polyurethane (TPU) in itself and its composite with short Kevlar fibre with respect to contact time, temperature, pressure, and fibre loading has been studied. The adhesion strength showed two linear increments of different slopes with respect to the square root of time: with temperature and pressure of contact, the adhesion strength was improved. The maximum strength was obtained with 20 phr of short fibre in only one of the mating substrates in the peel test sample. The duration for wetting and diffusion was shifted to longer time intervals with fibres loaded in both the substrates.

Studies on the Effect of an External Flaw on the Tensile Strength of Short Kevlar Fiber-Thermoplastic Polyurethane Composites

The effect of an external flaw on the tensile strength of short kevlar fiber-thermoplastic composites has been studied with respect to fiber content, fiber orientation, location of the external flaw, and the temperature of test. The composites showed a three-step reduction in tensile strength with increasing flaw size. The critical flaw-length region was shifted to higher flaw-size levels with increasing fiber content. With increasing temperature, the critical flaw length was increased in the case of unfilled TPU, whereas it remained more or less constant in the case of short kevlar fiber-filled-TPU composite.

Studies on the Flammability of Short Kevlar Fiber-Thermoplastic Polyurethane Composite

The flammability of short Kevlar aramide fiber-thermoplastic polyurethane (TPU) has been investigated with respect to fiber loading and various flame retardant additives such as halogen containing polymers, antimony oxide/chlorine donor combination, zinc borate, and aluminum hydroxide. Smoke generation was reduced drastically, while the oxygen index was reduced marginally in the presence of short fibers. The best improvement in the oxygen index was obtained with antimony oxide/chlorinated paraffin wax combination, in the weight ratio 1:6. A 70 phr loading of aluminum hydroxide improved LOI and reduced smoke generation.

Characterization of Short Nylon-6 Fiber/Acrylonitrile Butadiene Rubber Composite by Thermogravimetry

The thermal degradation of short nylon-6 fiber reinforced acrylonitrile butadiene rubber (NBR) composites with and without epoxy-based bonding agent has been studied by thermogravimetric analysis (TGA). It was found that the onset of degradation shifted from 330.5 to 336.1°C in the presence of short nylon fiber, the optimum fiber loading being 20 phr. The maximum rate of degradation of the composites was lower than that of the unfilled rubber compound, and it decreased with increase in fiber concentration. The presence of epoxy resin-based bonding agent in the virgin elastomer and the composites improved the thermal stability. Results of kinetic studies showed that the degradation of NBR and the short nylon fiber reinforced composites followed first-order kinetics.

Rheological Characteristics of Short Nylon -6 Fiber Reinforced Styrene Butadiene Rubber Containing Epoxy Resin as Bonding Agent

The rheological characteristics of short Nylon-6 fiber-reinforced Styrene Butadiene rubber (SBR) in the presence of epoxy resin-based bonding agent were studied with respect to the effect of shear rate, fiber concentration , and temperature on shear viscosity and die swell using a capillary rheonzeter. All the composites containing bonding agent showed a pseudoplastic nature, which decreased with increasing temperature. Shear viscosity was increased in the presence of fibers. The temperature sensitivity of the SBR matrices was reduced on introduction of fibers. The temperature sensitivity of the melts was found to be lower at higher shear rates. Die swell was reduced in the presence of fibers. Relative viscosity of the composites increased with shear rate. In the presence of epoxy resin bonding agent the temperature sensitivity of the mixes increased. Die swell was larger in the presence of bonding agent.

Rheology of Short Nylon -6 Fiber Reinforced Styrene-Butadiene Rubber

The rheological characteristics of short Nylon-6 fiber reinforced styrene butadiene rubber (SBR) were studied using a capillary rheometer. The study was done with respect to the effect of shear rate, fiber concentration, and temperature on shear viscosity and die swell. All the melts showed pseudoplastic nature, which decreased with increasing temperature. Shear viscosity increased in the presence of fibers. Introduction of fiber reduces the temperature sensitivity of the rubber matrix. A reduction in die swell was found in presence of fibers.

Thermal Degradation of Short Nylon -6 Fiber-Reinforced Styrene Butadiene Rubber Composite

The thermal properties of short Nylon-6 fiber-reinforced Styrene butadiene rubber (SBR) composites were studied by Thermogravimetric Analysis (TGA). The effect of epoxy-based bonding agent on thermal degradation of the gum and the composites was also studied. The thermal stability of the SBR was enhanced in the presence of Nylon-6 fibers and the stability of the composites increased in the presence of bonding agent. The epoxy resin did not significantly change the thermal stability of SBR gum vulcanizate. Results of kinetic studies showed that the degradation of SBR and the short nylon fiber-reinforced composites with and without bonding agents followed first-order kinetics.

Cure Characteristics and Mechanical Properties of Short Nylon Fiber Reinforced Acrylonitrile Butadiene Rubber/Reclaimed Rubber Blends

Cure characteristics and mechanical properties of short nylon fiber reinforced acrylonitrile butadiene rubber-reclaimed rubber composites were studied. Minimum torque, (maximum-minimum) torque and cure rate increased with fiber concentration. Scorch time and cure time decreased by the addition of fibers. Properties like tensile strength, tear strength, elongation at break, abrasion loss and heat build up were studied in both orientations of fibers. Tensile and tear properties were enhanced by the addition of fibers and were higher in the longitudinal direction. Heat build up increased with fiber concentration and were higher in the longitudinal direction. Abrasion resistance was improved in presence of short fibers and was higher in the longitudinal direction. Resilience increased on the introduction of fibers. Compression set was higher for blends.