THE COMPARISON OF GREEN FUNCTION FOR QUASI-LINEAR ELLIPTIC EQUATION

A generalization of the usual Green function to a kind of nonlinear elliptic equation of divergence form is discussed. The regularity and comparison principle of Green function in the sense of distribution are shown.

Existence Theorem for a Nonlinear Functional Integral Equation and an Initial Value Problem of Fractional Order in L₁(R₊)

The aim of this paper is to study the existence of integrable solutions of a nonlinear functional integral equation in the space of Lebesgue integrable functions on unbounded interval, L1(R+). As an application we deduce the existence of solution of an initial value problem of fractional order that be studied only on a bounded interval. The main tools used are Schauder fixed point theorem, measure of weak noncompactness, superposition operator and fractional calculus.

Analysis of a kind of Duffing oscillator system used to detect weak signals

The stability of the periodic solution of the Duffing oscillator system in the periodic phase state is proved by using the Yoshizaw theorem, which establishes a theoretical basis for using this kind of chaotic oscillator system to detect weak signals. The restoring force term of the system affects the weak-signal detection ability of the system directly, the quantitative relationship between the coefficients of the linear and nonlinear items of the restoring force of the Duffing oscillator system and the SNR in the detection of weak signals is obtained through a large number of simulation experiments, then a new restoring force function with better detection results is established.

TAUBERIAN THEOREMS FOR WEAK ALMOST CONVERGENT FUNCTIONS

The almost convergent function which was introduced by Raimi [6] and discussed by Ho [4], Das and Nanda [2, 3], is the continuous analogue of almost convergent sequences （see [5]）. In this paper, we establish the Tauberian conditions and the Cauchy criteria for weak almost convergent functions on R2＋ .

MULTIPLICATION OF WEAK FUNCTIONS

This paper is a contionuation of [1] and is concerned with multiplication of weak functions. Here the weak functions are treated as generalized expansions in Hermite functions in [2].

Evaluating neuromuscular electrical stimulation for preventing and managing intensive care unit-acquired weakness:Current evidence and future directions

Intensive care unit-acquired weakness(ICU-AW)is a prevalent issue in critical care,leading to significant muscle atrophy and functional impairment.Aiming to address this,Neuromuscular Electrical Stimulation(NMES)has been explored as a therapy.This systematic review assesses NMES's safety and effectiveness in enhancing functional capacity and mobility in pre-and post-cardiac surgery patients.NMES was generally safe and feasible,with intervention sessions varying in frequency and duration.Improvements in muscle strength and 6-minute walking test distances were observed,particularly in preoperative settings,but postoperative benefits were inconsistent.NMES showed promise in preventing muscle loss and improving strength,although its impact on overall functional capacity remained uncertain.Challenges such as short ICU stays and body composition affecting NMES efficacy were noted.NMES also holds potential for other conditions like cerebral palsy and stroke.Further research is needed to optimize NMES protocols and better understand its full benefits in preventing ICU-AW and improving patient outcomes.

Global existence of weak solutions to a prey-predator model with strong cross-diffusion

Using finite differences and entropy inequalities, the global existence of weak solutions to a multidimensional parabolic strongly coupled prey-predator model is obtained. The nonnegativity of the solutions is also shown.

Convergence Phenomenon with Fourier Series of tg(x2)and Alike

The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .

On the Functional Empirical Process and Its Application to the Mutual Influence of the Theil-Like Inequality Measure and the Growth

We set in this paper a coherent theory based on functional empirical processes that allows to consider both the poverty and the inequality indices in one Gaussian field in which the study of the influence of the one over the other is done. We use the General Poverty Index (GPI), that is a class of poverty indices gathering the most common ones and a functional class of inequality measures including the Entropy Measure, the Mean Logarithmic Deviation, the different inequality measures of Atkinson, Champernowne, Kolm and Theil called Theil-Like Inequality Measures (TLIM). Our results are given in a unified approach with respect to the two classes instead of their particular elements. We provide the asymptotic laws of the variations of each class over two given periods and the ratio of the variation and derive confidence intervals for them. Although the variances may seem somehow complicated, we provide R codes for their computations and apply the results for the pseudo-panel data for Senegalwith a simple analysis.

Functional Weak Laws for the Weighted Mean Losses or Gains and Applications

In this paper, we show that many risk measures arising in Actuarial Sciences, Finance, Medicine, Welfare analysis, etc. are gathered in classes of Weighted Mean Loss or Gain (WMLG) statistics. Some of them are Upper Threshold Based (UTH) or Lower Threshold Based (LTH). These statistics may be time-dependent when the scene is monitored in the time and depend on specific functions w and d. This paper provides time-dependent and uniformly functional weak asymptotic laws that allow temporal and spatial studies of the risk as well as comparison among statistics in terms of dependence and mutual influence. The results are particularized for usual statistics like the Kakwani and Shorrocks ones that are mainly used in welfare analysis. Data-driven applications based on pseudo-panel data are provided.

Increasing role of post-intensive care syndrome in quality of life of intensive care unit survivors

In this editorial we comment on the detrimental consequences that post-intensive care syndrome(PICS)has in the quality of life of intensive care unit(ICU)survivors,highlighting the importance of early onset of multidisciplinary rehabilitation from within the ICU.Although,the syndrome was identified and well described early in 2012,more awareness has been raised on the long-term PICS related health problems by the increased number of coronavirus disease 2019 ICU survivors.It is well outlined that the syndrome affects both the patient and the family and is described as the appearance or worsening of impairment in physical,cognitive,or mental health as consequence of critical illness.PICS was described in order:(1)To raise awareness among clinicians,researchers,even the society;(2)to highlight the need for a multilevel screening of these patients that starts from within the ICU and continues after discharge;(3)to present preventive strategies;and(4)to offer guidelines in terms of rehabilitation.An early multidisci-plinary approach is the key element form minimizing the incidence of PICS and its consequences in health related quality of life of both survivors and their families.

COMMON FIXED POINT OF GENERALIZED WEAKLY CONTRACTIVE MAPS IN PARTIAL METRIC SPACES

In this paper, using the context of complete partial metric spaces, some common fixed point results of maps that satisfy the generalized （ψ, Ф）-weak contractive conditions are obtained. Our results generalize, extend, unify, enrich and complement many existing results in the literature. Example are given showing the validaty of our results.

Beta Distribution of Surface Elevation of Random Waves

A probability density function (PDF) is derived of beta distribution with both lambda (3) skewness) and lambda (4) (kurtosis) as parameters for weakly nonlinear wave surface elevation by use of a method recently proposed by Srokosz. This PDF not only has a simpler form than the well-known Gram-Charlier Series PDF derived by Longuet-Higgins, but also overcomes an obvious shortcoming of the latter that when the series is unsuitably truncated, the resulting PDF is locally negative. To test the derived beta PDF, laboratorial experiments of wind waves are conducted. The experimental data indi cate that the theoretical requirements of the parameters in the beta PDF are fulfilled. The experimental results show that the present PDF is in better agreement with the measured data than the beta PDF only including parameter lambda (3), and also than the Gram-Charlier Series PDF truncated up to the term of H-6.

GENERALIZED WEAK FUNCTIONS

In this paper the authors introduce some new ideas on generalized numbers and generalized weak functions. They prove that the product of any two weak functions is a generalized weak function. So in particular they solve the problem of the multiplication of two generalized functions.

TLR signaling that induces weak inflammatory response and SHIP1 enhances osteogenic functions

Toll-like receptor （TLR）-mediated inflammatory response could negatively affect bone metabolism. In this study, we determined how osteogenesis is regulated during inflammatory responses that are downstream of TLR signaling. Human primary osteoblasts were cultured in collagen gels. Pam3CSK4 （P3C） and Escherichia coli lipopolysaccharide （EcLPS） were used as TLR2 and TLR4 ligand respectively. Porphyromonas gingivalis LPS having TLR2 activity with either TLR4 agonism （Pg1690） or TLR4 antagonism （Pg1449） and mutant E. coli LPS （LPxE/LPxF/WSK） were used. IL-lp, SH2-containing inositol phosphatase-1 （SHIP1） that has regulatory roles in osteogenesis, alkaline phosphatase and mineralization were analyzed. 3α-Aminocholestane （3AC） was used to inhibit SHIP1. Our results suggest that osteoblasts stimulated by P3C, poorly induced IL-1β but strongly upregulated SHIP1 and enhanced osteogenic mediators. On the contrary, EcLPS significantly induced IL-1β and osteogenic mediators were not induced. While Pg1690 downmodulated osteogenic mediators, Pg1449 enhanced osteogenic responses, suggesting that TLR4 signaling annuls osteogenesis even with TLR2 activity. Interestingly, mutant E. coli LPS that induces weak inflammation upregulated osteogenesis, but SHIP1 was not induced. Moreover, inhibiting SHIP1 significantly upregulated TLR2-mediated inflammatory response and downmodulated osteogenesis. In conclusion, these results suggest that induction of weak inflammatory response through TLR2 （with SHIP1 activity） and mutant TLR4 ligands could enhance osteogenesis.

OPTIMALITY CONDITIONS AND DUALITY RESULTS FOR NONSMOOTH VECTOR OPTIMIZATION PROBLEMS WITH THE MULTIPLE INTERVAL-VALUED OBJECTIVE FUNCTION

In this paper, both Fritz John and Karush-Kuhn-Tucker necessary optimality conditions are established for a （weakly） LU-efficient solution in the considered nonsmooth multiobjective programming problem with the multiple interval-objective function. Further, the sufficient optimality conditions for a （weakly） LU-efficient solution and several duality results in Mond-Weir sense are proved under assumptions that the functions constituting the considered nondifferentiable multiobjective programming problem with the multiple interval- objective function are convex.

Two New Characterizations of H^p(D) Functions

In this paper,two new characterizations of H^p(D) functions are given.

MORE ABOUT MELLIN TRANSFORM IN WEAK FUNCTIONS AND MUNTS FORMULA

This paper is a further continuation of the paper [1]. In the present paper Mellin transform and Miints formula of weak functions in complex domain will be treated.

GLOBAL WEAK SHARP MINIMA AND COMPLETENESS OF METRIC SPACE

A sufficient condition on the existence of a global weak sharp minima for general function in metric space is established. A characterization for convex function to have global weak sharp minima is also presented, which generalized Burke and Ferris' result to infinite dimensional space. A characterization of the completeness of a metric space is given by the existence of global weak sharp minima.

LIMITING WEAK-TYPE BEHAVIORS FOR CERTAIN LITTLEWOOD-PALEY FUNCTIONS

In this paper, we establish the following limiting weak-type behaviors of Littlewood-Paley g-function g_φ: for nonnegative function f∈ L^1(R^n),■and ■where f_t(x) =t^(-n)f(t^(-1) x) for t > 0. Meanwhile, the corresponding results for Marcinkiewicz integral and its fractional version with kernels satisfying L_α~q-Dini condition are also given.