Abstract:
The goal of this paper is to study the boundedness, the persistence, and the asymptotic behavior of the positive solutions of the system of two difference equations of exponential form: where , and are positive constants and the initial values are positive real values. Also, we determine the rate of convergence of a solution that converges to the equilibrium of this system. 1. Introduction In [1], the authors studied the boundedness, the asymptotic behavior, the periodicity, and the stability of the positive solutions of the difference equation: where are positive constants and the initial values are positive numbers. Motivated by the above paper we will extend the above difference equation to a system of difference equations; our goal will be to investigate the boundedness, the persistence, and the asymptotic behavior of the positive solutions of the following system of exponential form: where are positive constants and the initial values are positive real values. Difference equations and systems of difference equations of exponential form can be found in [2–6]. Moreover, as difference equations have many applications in applied sciences, there are many papers and books that can be found concerning the theory and applications of difference equations; see [7–9] and the references cited therein. 2. Global Behavior of Solutions of System (2) In the first lemma we study the boundedness and persistence of the positive solutions of (2). Lemma 1. Every positive solution of (2) is bounded and persists. Proof. Let be an arbitrary solution of (2). From (2) we can see that In addition, from (2) and (3) we get Therefore, from (3) and (4) the proof of the lemma is complete. In order to prove the main result of this section, we recall the next theorem without its proof. See [10, 11]. Theorem 2. Let and be a continuous functions such that the following hold:(a) is decreasing in both variables and is decreasing in both variables for each ;(b)if is a solution of then and . Then the following system of difference equations, has a unique equilibrium and every solution of the system (7) with converges to the unique equilibrium . In addition, the equilibrium is globally asymptotically stable. Now we state the main theorem of this section. Theorem 3. Consider system (2). Suppose that the following relation holds true: Then system (2) has a unique positive equilibrium and every positive solution of (2) tends to the unique positive equilibrium as . In addition, the equilibrium is globally asymptotically stable. Proof. We consider the functions where It is easy to see that ,

Abstract:
Antimicrobial consumption is one of the major contributing factors facilitating the development and maintenance of bacteria exhibiting antimicrobial resistance. Plasmid-mediated quinolone resistance (PMQR) genes, such as the qnr family, can be horizontally transferred and contribute to reduced susceptibility to fluoroquinolones. We performed an observational study, investigating the copy number of PMQR after antimicrobial therapy. We enrolled 300 children resident in Ho Chi Minh City receiving antimicrobial therapy for acute respiratory tract infections (ARIs). Rectal swabs were taken on enrollment and seven days subsequently, counts for Enterobacteriaceae were performed and qnrA, qnrB and qnrS were quantified by using real-time PCR on metagenomic stool DNA. On enrollment, we found no association between age, gender or location of the participants and the prevalence of qnrA, qnrB or qnrS. Yet, all three loci demonstrated a proportional increase in the number of samples testing positive between day 0 and day 7. Furthermore, qnrB demonstrated a significant increase in copy number between paired samples (p<0.001; Wilcoxon rank-sum), associated with non-fluoroquinolone combination antimicrobial therapy. To our knowledge, this is the first study describing an association between the use of non-fluoroquinolone antimicrobials and the increasing relative prevalence and quantity of qnr genes. Our work outlines a potential mechanism for the selection and maintenance of PMQR genes and predicts a strong effect of co-selection of these resistance determinants through the use of unrelated and potentially unnecessary antimicrobial regimes.

Abstract:
Let $\F_q$ be a finite field of order $q$ and $P$ be a polynomial in $\F_q[x_1, x_2]$. For a set $A \subset \F_q$, define $P(A):=\{P(x_1, x_2) | x_i \in A \}$. Using certain constructions of expanders, we characterize all polynomials $P$ for which the following holds \vskip2mm \centerline{\it If $|A+A|$ is small, then $|P(A)|$ is large.} \vskip2mm The case $P=x_1x_2$ corresponds to the well-known sum-product problem.

Abstract:
We discuss a structural approach to subset-sum problems in additive combinatorics. The core of this approach are Freiman-type structural theorems, many of which will be presented through the paper. These results have applications in various areas, such as number theory, combinatorics and mathematical physics.

Abstract:
Computing the first few singular vectors of a large matrix is a problem that frequently comes up in statistics and numerical analysis. Given the presence of noise, exact calculation is hard to achieve, and the following problem is of importance: \vskip2mm \centerline {\it How much a small perturbation to the matrix changes the singular vectors ?} \vskip2mm Answering this question, classical theorems, such as those of Davis-Kahan and Wedin, give tight estimates for the worst-case scenario. In this paper, we show that if the perturbation (noise) is random and our matrix has low rank, then better estimates can be obtained. Our method relies on high dimensional geometry and is different from those used an earlier papers.

Abstract:
Let $K$ be a smooth convex set with volume one in $\BBR^d$. Choose $n$ random points in $K$ independently according to the uniform distribution. The convex hull of these points, denoted by $K_n$, is called a {\it random polytope}. We prove that several key functionals of $K_n$ satisfy the central limit theorem as $n$ tends to infinity.

Abstract:
Finding a hidden partition in a random environment is a general and important problem, which contains as subproblems many famous questions, such as finding a hidden clique, finding a hidden coloring, finding a hidden bipartition etc. In this paper, we provide a simple SVD algorithm for this purpose, answering a question of McSherry. This algorithm is very easy to implement and works for sparse graphs with optimal density.

Abstract:
Background The relationships between the infecting dengue serotype, primary and secondary infection, viremia and dengue severity remain unclear. This cross-sectional study examined these interactions in adult patients hospitalized with dengue in Ha Noi. Methods and Findings 158 patients were enrolled between September 16 and November 11, 2008. Quantitative RT-PCR, serology and NS1 detection were used to confirm dengue infection, determine the serotype and plasma viral RNA concentration, and categorize infections as primary or secondary. 130 (82%) were laboratory confirmed. Serology was consistent with primary and secondary infection in 34% and 61%, respectively. The infecting serotype was DENV-1 in 42 (32%), DENV-2 in 39 (30%) and unknown in 49 (38%). Secondary infection was more common in DENV-2 infections (79%) compared to DENV-1 (36%, p<0.001). The proportion that developed dengue haemorrhagic fever (DHF) was 32% for secondary infection compared to 18% for primary infection (p = 0.14), and 26% for DENV-1 compared to 28% for DENV-2. The time until NS1 and plasma viral RNA were undetectable was shorter for DENV-2 compared to DENV-1 (p≤0.001) and plasma viral RNA concentration on day 5 was higher for DENV-1 (p = 0.03). Plasma viral RNA concentration was higher in secondary infection on day 5 of illness (p = 0.046). We didn't find an association between plasma viral RNA concentration and clinical severity. Conclusion Dengue is emerging as a major public health problem in Ha Noi. DENV-1 and DENV-2 were the prevalent serotypes with similar numbers and clinical presentation. Secondary infection may be more common amongst DENV-2 than DENV-1 infections because DENV-2 infections resulted in lower plasma viral RNA concentrations and viral RNA concentrations were higher in secondary infection. The drivers of dengue emergence in northern Viet Nam need to be elucidated and public health measures instituted.