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The goal of this paper is to present a critique on the incorrect use of the Lipschitz definition concerning the first variable and/or the second variable in the literature on the system of variational inequalities by many authors. The possible impact of this paper is rather important, it questions the results of different authors, particularly when taking into account that some of these papers are published in quite good mathematical journals. As a result, not only that the proofs are wrong, but also the credibility of the theorems themselves is compromised. In addition, this paper illustrate, using a counterexample, that there is an error in setting up first variable definition and the results obtained in listed references do not hold up in H × H.

The goal of this work is demonstrating, through the gradient observation of a of type linear ( -systems), the possibility for reducing the effect of any disturbances (pollution, radiation, infection, etc.) asymptotically, by a suitable choice of related actuators of these systems. Thus, a class of ( -system) was developed based on finite time ( -system). Furthermore, definitions and some properties of this concept -system and asymptotically gradient controllable system ( -controllable) were stated and studied. More precisely, asymptotically gradient efficient actuators ensuring the weak asymptotically gradient compensation system ( -system) of known or unknown disturbances are examined. Consequently, under convenient hypothesis, the existence and the uniqueness of the control of type optimal, guaranteeing the asymptotically gradient compensation system ( -system), are shown and proven. Finally, an approach that leads to a Mathematical approximation algorithm is explored.

The concept of convexity and its various generalizations is important for quantitative and qualitative studies in operations research or applied mathematics. Recently, E-convex sets and functions were introduced with important implications across numerous branches of mathematics. By relaxing the definition of convex sets and functions, a new concept of semi-EE-convex functions was introduced, and its properties are discussed. It has been demonstrated that if a function f:M→Rf:M→R is semi-EE-convex on an EE-convex set M⊂RnM⊂Rn then, f(E(x))≤f(x)f(E(x))≤f(x) for each x∈Mx∈M. This article discusses the inverse of this proposition and presents some results for convex functions.

Althought Malassezia spp. have been involved in various pathologies, they are an integral part of the cutaneous, gut, oral, ears, nose and throat (ENT) mycobiota. Since Malassezia are difficult to grow in culture, unexhaustive molecular biology methods have been developed to detect them. The aim of the study was to evaluate an in-house pan- Malassezia quantitative polymerase chain reaction (panM-qPCR) on various clinical human samples and determine Malassezia burden in various human mycobiota. The panM-qPCR was designed to target the repeated 28S rDNA gene from all Malassezia species. We used the assay to quantify the Malassezia burden on 361 samples from 161 subjects (80 skin swabs from 10 healthy volunteers (HV), 13 samples from 2 seborrheic dermatitis patients (SD), 90 skin samples from 19 burned patients, 119 stool samples from 89 immunocompromised patients, 59 ENT samples from 41 patients). For HV, the amount of Malassezia was different according to the swabbed areas. Quantification cycle (Cq) in SD is lower than in HV. In burned patients, Cq was significantly lower compared to HV. In stool samples, 6.7% were positive for Malassezia spp. with a high Cq. For the ENT area, a higher proportion of positive specimens were detected in ear samples than in nose samples. Our findings emphasized the importance of qPCR, confirming elevated Malassezia spp. levels on individuals' faces and scalps, increased burden in SD patients and in severely burnt patients than in HV. The pan-MqPCR appears to be a promising tool for studying Malassezia in various human mycobiota. [ABSTRACT FROM AUTHOR]

It is well known that convexity theory and its generalization play an important role and have many applications in mathematical programming. In the previous article [2], the authors studied E-convex and semi-E-convex functions. In the present article, we deal with a class of E-quasiconvex functions and give a weak condition for a lower semi-continuous function on ℝn to be an E-quasiconvex function. [ABSTRACT FROM AUTHOR]

In this paper, we show that the results obtained in [7]: On a system of general mixed variational inequalities are incorrect. Also we suggest and analyze new approximation schemes 4.1 for solving the (SGMVID) which were introduced by M.A. Noor in [7]. [ABSTRACT FROM AUTHOR]

We propose a nonsmooth dynamic system integrating production and inventory where the items may deteriorate and the demand is stock-dependent. We aim to derive the optimal production rate. In our first model, backorders are not allowed, while in the second model they are. Using optimal control, necessary optimality conditions are obtained for general forms of the cost, demand, and deterioration rates and closed form solutions are derived for specific forms of these rates. Numerical simulations are presented and sensitivity of the solutions are examined.

Kernel functions play an important role in the complexity analysis of the interior point methods (IPMs) for linear optimization (LO). In this paper, an interior-point algorithm for LO based on a new parametric kernel function is proposed. By means of some simple analysis tools, we prove that the primal-dual interior-point algorithm for solving LO problems meets ..., iteration complexity bound for large-update methods with the special choice of its parameters. [ABSTRACT FROM AUTHOR]

In this paper, we propose a primal–dual interior point method for Linear Complementarity Problem (LCP) based on a new parameterized kernel function. The investigation according to it yields the best-known iteration bound O (n log ⁡ (n) log ⁡ (n ϵ)) for large-update algorithm and thus improves the iteration bound obtained in Bai et al. (SIAM J Optim. 2004;15:101–128) for large-update algorithm. Finally, we present few numerical results to demonstrate the efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]