The distribution of k-tuples of reduced residues

En 1940, Paul Erdős énonça une conjecture sur la distribution des classes inversibles modulo un entier. La présente thèse étudie la distribution des k-uplets de classes inversibles propose une preuve de la conjecture d'Erdős étendue au cas des k-uplets.

Irrégularités dans la distribution des nombres premiers et des suites plus générales dans les progressions arithmétiques

Le sujet principal de cette thèse est la distribution des nombres premiers dans les progressions arithmétiques, c'est-à-dire des nombres premiers de la forme $qn+a$, avec $a$ et $q$ des entiers fixés et $n=1,2,3,\dots$ La thèse porte aussi sur la comparaison de différentes suites arithmétiques par rapport à leur comportement dans les progressions arithmétiques. Elle est divisée en quatre chapitres et contient trois articles. Le premier chapitre est une invitation à la théorie analytique des nombres, suivie d'une revue des outils qui seront utilisés plus tard. Cette introduction comporte aussi certains résultats de recherche, que nous avons cru bon d'inclure au fil du texte. Le deuxième chapitre contient l'article \emph{Inequities in the Shanks-Rényi prime number race: an asymptotic formula for the densities}, qui est le fruit de recherche conjointe avec le professeur Greg Martin. Le but de cet article est d'étudier un phénomène appelé le <>, qui s'observe dans les <>. Chebyshev a observé qu'il semble y avoir plus de premiers de la forme $4n+3$ que de la forme $4n+1$. De manière plus générale, Rubinstein et Sarnak ont montré l'existence d'une quantité $\delta(q;a,b)$, qui désigne la probabilité d'avoir plus de premiers de la forme $qn+a$ que de la forme $qn+b$. Dans cet article nous prouvons une formule asymptotique pour $\delta(q;a,b)$ qui peut être d'un ordre de précision arbitraire (en terme de puissance négative de $q$). Nous présentons aussi des résultats numériques qui supportent nos formules. Le troisième chapitre contient l'article \emph{Residue classes containing an unexpected number of primes}. Le but est de fixer un entier $a\neq 0$ et ensuite d'étudier la répartition des premiers de la forme $qn+a$, en moyenne sur $q$. Nous montrons que l'entier $a$ fixé au départ a une grande influence sur cette répartition, et qu'il existe en fait certaines progressions arithmétiques contenant moins de premiers que d'autres. Ce phénomène est plutôt surprenant, compte tenu du théorème des premiers dans les progressions arithmétiques qui stipule que les premiers sont équidistribués dans les classes d'équivalence $\bmod q$. Le quatrième chapitre contient l'article \emph{The influence of the first term of an arithmetic progression}. Dans cet article on s'intéresse à des irrégularités similaires à celles observées au troisième chapitre, mais pour des suites arithmétiques plus générales. En effet, nous étudions des suites telles que les entiers s'exprimant comme la somme de deux carrés, les valeurs d'une forme quadratique binaire, les $k$-tuplets de premiers et les entiers sans petit facteur premier. Nous démontrons que dans chacun de ces exemples, ainsi que dans une grande classe de suites arithmétiques, il existe des irrégularités dans les progressions arithmétiques $a\bmod q$, avec $a$ fixé et en moyenne sur $q$.

Remarks on the classification of a pair of commuting semilinear operators

Gelfand and Ponomarev [I.M. Gelfand, V.A. Ponomarev, Remarks on the classification of a pair of commuting linear transformations in a finite dimensional vector space, Funct. Anal. Appl. 3 (1969) 325-326] proved that the problem of classifying pairs of commuting linear operators contains the problem of classifying k-tuples of linear operators for any k. We prove an analogous statement for semilinear operators. (C) 2011 Elsevier Inc. All rights reserved.

Regulation of pyridoxal $5\prime$-phosphate biosynthesis in {\it Escherichia coli\/} K-12

Vitamin B$\sb6$ (or pyridoxal 5$\sp\prime$-phosphate, PLP) is an essential, ubiquitous coenzyme that affects many aspects of amino acid and cellular metabolism in all organisms. The goal of this thesis is to examine the regulation of PLP biosynthesis in Escherichia coli K-12. First, PdxH oxidase is a PLP biosynthetic enzyme, which uses molecular oxygen as an electron acceptor under aerobic assay conditions. To test if facultative anaerobic E. coli uses another enzyme to replace the function of PdxH oxidase anaerobically, suppressors of a pdxH null mutant were isolated anaerobically after 2-aminopurine or spontaneous mutagenesis. Only one specific bypass mutation in another PLP biosynthetic gene pdxJ was found, suggesting that PdxH oxidase is able to function anaerobically and PdxT utilizes D-1-deoxyxyulose as a substrate. Second, regulation of the serC (pdxF)-aroA operon, which is involved the biosynthesis of L-serine, PLP and aromatic compounds was examined. A serC (pdxF) single gene transcript and a serC (pdXf)-aroA cotranscript initiated at P$\sb{serC\ (pdxF)}$ upstream of serC (pdxF) were detected. The expression of the operon is activated by leucine responsive regulatory protein (LRP) and repressed by cAMP receptor protein-cAMP complex (CRP$\cdot$cAMP) at the transcriptional level. LRP activates the operon by directly binding to the upstream consensus box. Binding of CRP$\cdot$cAMP to the upstream CRP box diminishes the activation effect of LRP. However, deletion of the CRP box did not affect the repression of CRP$\cdot$cAMP, suggesting that CRP$\cdot$cAMP may repress the operon indirectly by stimulating the activity or level of an unidentified repressor. The overall effect of this regulation is to maximize the expression of the operon when the cells are growing in minimal-glucose medium. In addition, the binding and the transcription of P$\sb{serC\ (pdxF)}$ by RNA polymerase require a supercoiled circular DNA, indicating that DNA supercoiling affects the transcription of the operon. Third, regulation of another PLP biosynthetic gene gapB was also examined. gapB is activated by CRP$\cdot$cAMP and repressed by catabolic repressor activator protein (CRA). However, the activation of CRP$\cdot$cAMP is epistatic to the repression of CRA. Due to the CRA repression, gapB was expressed at a low level in all the media tested, suggesting that it may be the rate-limiting step of PLP biosynthesis. In summary, unlike genes in many biosynthetic pathways, PLP biosynthetic genes are regulated by global regulators that are important for carbon and amino acid metabolism, instead of the end product(s) of the pathway. ^

A prime-boost immunisation regimen using recombinant BCG and Pr55 gag virus-like particle vaccines based on HIV type 1 subtype C successfully elicits Gag-specific responses in baboons

Mycobacterium bovis BCG is considered an attractive live bacterial vaccine vector. In this study, we investigated the immune response of baboons to a primary vaccination with recombinant BCG (rBCG) constructs expressing the gag gene from a South African HIV-1 subtype C isolate, and a boost with HIV-1 subtype C Pr55 gag virus-like particles (Gag VLPs). Using an interferon enzyme-linked immunospot assay, we show that although these rBCG induced only a weak or an undetectable HIV-1 Gag-specific response on their own, they efficiently primed for a Gag VLP boost, which strengthened and broadened the immune responses. These responses were predominantly CD8+ T cell-mediated and recognised similar epitopes as those targeted by humans with early HIV-1 subtype C infection. In addition, a Gag-specific humoral response was elicited. These data support the development of HIV-1 vaccines based on rBCG and Pr55 gag VLPs. © 2009 Elsevier Ltd. All rights reserved.

Synthesis of biodegradable polymer-mesoporous silica composite microspheres for DNA prime-protein boost vaccination

DNA vaccines or proteins are capable of inducing specific immunity; however, the translation to the clinic has generally been problematic, primarily due to the reduced magnitude of immune response and poor pharmacokinetics. Herein we demonstrate a composite microsphere formulation, composed of mesoporous silica spheres (MPS) and poly(d,l-lactide-co-glycolide) (PLGA), enables the controlled delivery of a prime-boost vaccine via the encapsulation of plasmid DNA (pDNA) and protein in different compartments. Method with modified dual-concentric-feeding needles attached to a 40 kHz ultrasonic atomizer was studied. These needles focus the flow of two different solutions, which passed through the ultrasonic atomizer. The process synthesis parameters, which are important to the scale-up of composite microspheres, were also studied. These parameters include polymer concentration, feed flowrate, and volumetric ratio of polymer and pDNA-PEI/MPS-BSA. This fabrication technique produced composite microspheres with mean D[4,3] ranging from 6 to 34 μm, depending upon the microsphere preparation. The resultant physical morphology of composite microspheres was largely influenced by the volumetric ratio of pDNA-PEI/MPS-BSA to polymer, and this was due to the precipitation of MPS at the surface of the microspheres. The encapsulation efficiencies were predominantly in the range of 93-98% for pDNA and 46-68% for MPS. In the in vitro studies, the pDNA and protein showed different release kinetics in a 40 day time frame. The dual-concentric-feeding in ultrasonic atomization was shown to have excellent reproducibility. It was concluded that this fabrication technique is an effective method to prepare formulations containing a heterologous prime-boost vaccine in a single delivery system.

Improving the selectivity of engineered protease inhibitors: Optimizing the P2 prime residue using a versatile cyclic peptide library

Standard mechanism inhibitors are attractive design templates for engineering reversible serine protease inhibitors. When optimizing interactions between the inhibitor and target protease, many studies focus on the nonprimed segment of the inhibitor's binding loop (encompassing the contact β-strand). However, there are currently few methods for screening residues on the primed segment. Here, we designed a synthetic inhibitor library (based on sunflower trypsin inhibitor-1) for characterizing the P2′ specificity of various serine proteases. Screening the library against 13 different proteases revealed unique P2′ preferences for trypsin, chymotrypsin, matriptase, plasmin, thrombin, four kallikrein-related peptidases, and several clotting factors. Using this information to modify existing engineered inhibitors yielded new variants that showed considerably improved selectivity, reaching up to 7000-fold selectivity over certain off-target proteases. Our study demonstrates the importance of the P2′ residue in standard mechanism inhibition and unveils a new approach for screening P2′ substitutions that will benefit future inhibitor engineering studies.

Japanese encephalitis virus infection of mouse cell lines: ability to prime mice for generation of virus specific cytotoxic T lymphocytes and differences in CTL recognisable viral determinants

Ten different mouse cell lines were examined for Japanese encephalitis virus (JEV) infection in vitro and then tested for their ability to generate virus specific cytotoxic T lymphocytes (CTL). Among all cell lines examined, Neuro La (a neuroblastoma) was readily infected with JEV as examined by immunofluorescence and viral replication. Among other cells, P388D1, RAW 264.7 (Macrophage origin), Sp2/0 (B-cell Hybridoma), YAC-1 (T-cell lymphoma), and L929 (Fibroblast) were semipermissive to JEV infection. The cytopathic effects caused by progressive JEV infection varied from cell line to cell line. In the case of YAC-1 cells long-term viral antigen expression was observed without significant alterations in cell viability. Intermediate degrees of cytopathicity are seen in RAW 264.7 and L929 cells while infection of PS, Neuro 2a, P388D1 and Sp2/0 caused major viability losses. All infected cell lines were able to prime adult BALB/c (H-2(d)) mice for the generation of secondary JEV specific CTL. In contrast to YAC-1, the permissive neuroblastoma cell line Neuro 2a (H-2K(k)D(d)) was found to be least efficient in its ability to stimulate anti-viral CTL generation. Cold target competition studies demonstrated that both Neuro 2a and YAC-1 (H-2K(k)D(d)) cells expressed similar viral determinants that are recognised by CTL, suggesting that the reason for the lower ability of Neuro 2a to stimulate anti-viral CTL was not due to lack of viral CTL determinants. These findings demonstrate that a variety of mouse cell lines can be infected with Japanese encephalitis virus, and that these infected cells could be utilised to generate virus specific CTL in BALB/c mice.

Galois Theory and Solvable Equations of Prime Degree

In this article we review classical and modern Galois theory with historical evolution and prove a criterion of Galois for solvability of an irreducible separable polynomial of prime degree over an arbitrary field k and give many illustrative examples.

On the distribution of the signs of the conjugates of the cyclotomic units in the maximal real subfield of the qth cyclotomic field, q A prime

Let F = Ǫ(ζ + ζ ^–1) be the maximal real subfield of the cyclotomic field Ǫ(ζ) where ζ is a primitive qth root of unity and q is an odd rational prime. The numbers u₁=-1, u_k=(ζ^k-ζ^-k)/(ζ-ζ^-1), k=2,…,p, p=(q-1)/2, are units in F and are called the cyclotomic units. In this thesis the sign distribution of the conjugates in F of the cyclotomic units is studied.

Let G(F/Ǫ) denote the Galoi's group of F over Ǫ, and let V denote the units in F. For each σϵ G(F/Ǫ) and μϵV define a mapping sgn_σ: V→GF(2) by sgn_σ(μ) = 1 iff σ(μ) ˂ 0 and sgn_σ(μ) = 0 iff σ(μ) ˃ 0. Let {σ₁, ... , σ_p} be a fixed ordering of G(F/Ǫ). The matrix M_q=(sgn_σj(v_i) ) , i, j = 1, ... , p is called the matrix of cyclotomic signatures. The rank of this matrix determines the sign distribution of the conjugates of the cyclotomic units. The matrix of cyclotomic signatures is associated with an ideal in the ring GF(2) [x] / (x^p+ 1) in such a way that the rank of the matrix equals the GF(2)-dimension of the ideal. It is shown that if p = (q-1)/ 2 is a prime and if 2 is a primitive root mod p, then M_q is non-singular. Also let p be arbitrary, let ℓ be a primitive root mod q and let L = {i | 0 ≤ i ≤ p-1, the least positive residue of defined by ℓⁱ mod q is greater than p}. Let H_q(x) ϵ GF(2)[x] be defined by H_q(x) = g. c. d. ((Σ xⁱ/I ϵ L) (x+1) + 1, x^p + 1). It is shown that the rank of M_q equals the difference p - degree H_q(x).

Further results are obtained by using the reciprocity theorem of class field theory. The reciprocity maps for a certain abelian extension of F and for the infinite primes in F are associated with the signs of conjugates. The product formula for the reciprocity maps is used to associate the signs of conjugates with the reciprocity maps at the primes which lie above (2). The case when (2) is a prime in F is studied in detail. Let T denote the group of totally positive units in F. Let U be the group generated by the cyclotomic units. Assume that (2) is a prime in F and that p is odd. Let F₍₂₎ denote the completion of F at (2) and let V₍₂₎ denote the units in F₍₂₎. The following statements are shown to be equivalent. 1) The matrix of cyclotomic signatures is non-singular. 2) U∩T = U². 3) U∩F²₍₂₎ = U². 4) V₍₂₎/ V₍₂₎² = ˂v₁ V₍₂₎²˃ ʘ…ʘ˂v_p V₍₂₎²˃ ʘ ˂3V₍₂₎²˃.

The rank of M_q was computed for 5≤q≤929 and the results appear in tables. On the basis of these results and additional calculations the following conjecture is made: If q and p = (q -1)/ 2 are both primes, then M_q is non-singular.

A novel dual-fluorescence strategy for functionally validating microRNA targets in 3-prime untranslated regions: regulation of the inward rectifier potassium channel Kir2.1 by miR-212.

Gene targeting by microRNAs is important in health and disease. We developed a functional assay for identifying microRNA targets and applied it to the K+ channel Kir2.1 (KCNJ2) which is dysregulated in cardiac and vascular disorders. The 3'UTR was inserted downstream of the mCherry red fluorescent protein coding sequence in a mammalian expression plasmid. MicroRNA sequences were inserted into the pSM30 expression vector which provides enhanced green fluorescent protein as an indicator of microRNA expression. HEK293 cells were co-transfected with the mCherry-3'UTR plasmid and a pSM30-based plasmid with a microRNA insert. The principle of the assay is that functional targeting of the 3'UTR by the microRNA results in a decrease in the red/green fluorescence intensity ratio as determined by automated image analysis. The method was validated with miR-1, a known downregulator of Kir2.1 expression, and was used to investigate targeting of the Kir2.1 3'UTR by miR-212. Red/green ratio was lower in miR-212-expressing cells compared to non-targeting controls, an effect that was attenuated by mutating the predicted target site. MiR-212 also reduced inward rectifier current and Kir2.1 protein in HeLa cells. This novel assay has several advantages over traditional luciferase-based assays including larger sample size, amenability to time course studies and adaptability to high-throughput screening.

K-Theory of Azumaya Algebras over Schemes

Let X be a connected, noetherian scheme and A{script} be a sheaf of Azumaya algebras on X, which is a locally free O{script}-module of rank a. We show that the kernel and cokernel of K(X) ? K(A{script}) are torsion groups with exponent a for some m and any i = 0, when X is regular or X is of dimension d with an ample sheaf (in this case m = d + 1). As a consequence, K(X, Z/m) ? K(A{script}, Z/m), for any m relatively prime to a. © 2013 Copyright Taylor and Francis Group, LLC.

Predictive Value of Depressive Symptoms for All-Cause Mortality: Findings From the PRIME Belfast Study Examining the Role of Inflammation and Cardiovascular Risk Markers

OBJECTIVES: To improve understanding about the potential underlying biological mechanisms in the link between depression and all-cause mortality and to investigate the role that inflammatory and other cardiovascular risk factors may play in the relationship between depressive symptoms and mortality.

METHODS: Depression and blood-based biological markers were assessed in the Belfast PRIME prospective cohort study (N = 2389 men, aged 50-59 years) in which participants were followed up for 18 years. Depression was measured using the 10-item Welsh Pure Depression Inventory. Inflammation markers (C-reactive protein [CRP], neopterin, interleukin [IL]-1 receptor antagonist [IL-1Ra], and IL-18) and cardiovascular-specific risk factors (N-terminal pro-b-type natriuretic peptide, midregion pro-atrial natriuretic peptide, midregion pro-adrenomedullin, C-terminal pro-endothelin-1 [CT-proET]) were obtained at baseline. We used Cox proportional hazards modeling to examine the association between depression and biological measures in relation to all-cause mortality and explore the mediating effects.

RESULTS: During follow-up, 418 participants died. Higher levels of depressive symptoms were associated with higher levels of CRP, IL-1Ra, and CT-proET. After adjustment for socioeconomic and life-style risk factors, depressive symptoms were significantly associated with all-cause mortality (hazard ratio = 1.10 per scale unit, 95% confidence interval = 1.04-1.16). This association was partly explained by CRP (7.3%) suggesting a minimal mediation effect. IL-1Ra, N-terminal pro-b-type natriuretic peptide, midregion pro-atrial natriuretic peptide, midregion pro-adrenomedullin, and CT-proET contributed marginally to the association between depression and subsequent mortality.

CONCLUSIONS: Inflammatory and cardiovascular risk markers are associated with depression and with increased mortality. However, depression and biological measures show additive effects rather than a pattern of meditation of biological factors in the association between depression and mortality.