In this paper, a class of semi-linear elliptic equations involving Hardy-Sobolev critical exponents has been investigated. This problem comes from the consideration of standing waves in the anisotropic Schr\"{o}dinger equation, also it is very important in the field of hydrodynamics, glaciology, quantum field theory and statistical mechanics. By a detailed estimation for the extremum function and using Mountain Pass Lemma with $\left( PS\right) _{c}$ conditions, the existence of positive solutions has been obtained. On the other hand, by establishing Pohozaev-type identity and using the properties of Bessel function, the nonexistence of positive solution also has been obtained. These results are extensions of E. Jannelli's research (\cite[Theorem 1.A-1.C]{EJ}).

γ-Aminobutyric acid (GABA) plays an important role in plant development and postharvest properties of fruits. However, studies on regulation of kiwifruit ripening by GABA are scarce. Here, we evaluted physicochemical characters and the expression profile of genes related to ripening in hardy kiwifruit treated with exogenous GABA compared to control during postharvest storage period. The results showed that (1) exogenous GABA treatment not only improved the fruit quality but also inhibited respiration rate and ethylene production, as well as reduced the enzyme activities of ACC oxidase (ACO) and ACC synthase (ACS), via down-regulating the expression of AaACO1 and AaACO3, AaACS1 and AaACS2 during kiwifruit storage period; (2) on the one hand, overexpression of AaGAD1 or AaGAD4 gene which is involved in GABA biosynthesis inhibited ethylene production by reducing the enzyme activities of ACO and ACS, whereas silencing of AaGAD1 or AaGAD4 gene led to the reverse effect. On the other hand, overexpression of AaGAD1 or AaGAD4 decreased the expression levels of AaACO1 and AaACO3, AaACS1 and AaACS2, whereas silencing of AaGAD1 or AaGAD4 gene increased the expression levels of these four genes. Above all, our findings demonstrate that exogenous GABA treatment could improve the storage quality and extend the shelf life of kiwifruit, Additionally, the content of GABA was participated in the regulation of ethylene biosynthesis at the molecular level.

A fundamental tool in mathematical physics is the logarithmic Sobolev inequality. A quantitative version proven by Carlen with a remainder involving the Fourier-Wiener transform is equivalent to an entropic uncertainty principle more general than the Heisenberg uncertainty principle. In the stability, the remainder is in terms of an entropy, not a metric. Recently, a stability result for H¹ was obtained by Dolbeault, Esteban, Figalli, Frank, and Loss in terms of an L^p norm. Afterwards, Brigati, Dolbeault, and Simonov discussed the stability problem involving a stronger norm. A full characterization with a necessary and sufficient condition to have H¹ convergence is identified in this paper. Moreover, an explicit H¹ bound via a moment assumption is shown. Also, the L^p stability of Dolbeault, Esteban, Figalli, Frank, and Loss is proven to be sharp.

In this paper, impulsive control on a single-species Markovian jumping ecosystem leads to a stability criterion, and the newly-obtained theorems improve the related existing results. Numerical examples illuminate the effectiveness.

Hardy Cross originally proposed a method for analysis of flow in networks of conduits or conductors in 1936. His method was the first really useful engineering method in the field of pipe network calculation. Only electrical analogs of hydraulic networks were used before the Hardy Cross method. A problem with the flow resistance versus the electrical resistance makes these electrical analog methods obsolete. The method by Hardy Cross is taught extensively at faculties and it still remains an important tool for analysis of looped pipe systems. Engineers today mostly use a modified Hardy Cross method which threats the whole looped network of pipes simultaneously (use of these methods without computers is practically impossible). A method from the Russian practice published during 1930s, which is similar to the Hardy Cross method, is described, too. Some notes from the life of Hardy Cross are also shown. Finally, an improved version of the Hardy Cross method, which significantly reduces number of iterations, is presented and discussed. Also we tested multi-point iterative methods which can be used as substitution for the Newton-Raphson approach used by Hardy Cross, but this approach didn’t reduce number of required iterations to reach the final balanced solution. Although, many new models have been developed since the time of Hardy Cross, main purpose of this paper is to illustrate the very beginning of modeling of gas and water pipe networks or ventilation systems.

The present paper presents findings from the follow-up of 185 of these nurses who had been involved in the front-line care of COVID-19 patients in urban centres in Northern England, during and after the “second wave” of COVID patients in November, 2020. Measures of adjustment, stress (including PTSD), and Maslowian self-actualization were added to earlier measures of personality, adjustment, work-life stress, and career intention. Cluster analysis identified three types in the 2020-2023 cohort: I ‘Actualizing Professionals’; (N=62); II ‘Strong Professionals’ (N=56); III ‘Highly Stressed Nurses’ (N=45). These latter very stressed nurses included all nurses (N=16) identified as having PTSD symptoms.

We say that a system {zmF(z)}m=0∞ is a Beurling system if F is an outer function. Beurling’s approximation theorem asserts that if F is an outer function from H2(D) then the system {zmF(z)}m=0∞ is complete in the space H2(D). We prove that a Beurling system with F∈Hp(D),1≤p<∞ is an M−bases in Hp(D) with an explicit dual system. Any function f∈Hp(D),1≤p<∞ can be expanded as a series by the system {zmF(z)}m=0∞. For different methods of summation we characterize outer functions F for which the expansion converges to f. Related results for weighted Hardy spaces in the unit disc are studied. Particularly we prove Rosenblum’s hypothesis.

We introduce a class of logarithmic wave equation. We study the global existence of week solution for this class of equation. We deal with the initial boundary value problem of this class. Using the Galerkin method and the Gross logarithmic Sobolev inequality we establish the main theorem of existence of week solution for this class of equation arising from Q-Ball Dynamic in particular.

Original and improved version of the Hardy Cross iterative method with related modifications are today widely used for calculation of fluid flow through conduits in loops-like distribution networks of pipes with known node fluid consumptions. Fluid in these networks is usually natural gas for distribution in the municipalities, water in waterworks or hot water in district heating system, air in the case of ventilation systems in buildings or mines, etc. Since, the resistances in these networks depend of flow, problem is not linear like in electrical circuits, and iterative procedure must be used. In both version of the Hardy Cross method, in original and in the improved one, initial results of calculation in iteration procedure is not flow, but rather the correction of flow. Unfortunately, these corrections should be added to or subtracted from a flow calculated in previous iteration according to complicate algebraic rules. After the here presented node-loop method, final results in each of the iterations is flow directly rather than flow correction. In that way complex algebraic scheme for sign of flow correction is avoided, while the final results still remain unchanged. Numbers of required iterations for the same results are comparable with the improved Hardy Cross method.

In this article we propose the integral, series and product representations for the Ramanujan zeta-function. We suggest a variant for the Conrey-Ghosh product for the entire Ramanujan zeta-function. We present some variants for the product for the Ramanujan $\Xi$-function. We prove that all of its zeros are real. Along the way we obtain the truth of the Riemann-Hardy conjecture.

This initial report of a longitudinal study of 192 English hospital nurses has measured Nursing Values (the 6Cs of nursing); Personality, Self-Esteem and Depression; Burnout Potential; Work-Life Balance Stress; ‘Hardy Personality’; and Intention to Leave Nursing. Correlational, component and cluster analysis identifies four groups: “The Soldiers” (N = 79) , with medium scores on most measures, who bravely ‘soldier on’ in their nursing roles, in the face of numerous financial cuts to the National Health Service, and worsening nurse-patient ratios; “Cheerful Professionals” (N = 54), coping successfully with nursing roles, and a variety of challenges, in upwardly mobile careers; “High Achievers” (N = 39), senior nurses with strong profiles of a ‘hardy personality’, and commitment to fundamental nursing values; “Highly Stressed, Potential Leavers” (N = 20), with indicators of significant psychological distress, and difficulty in coping with nursing role challenges. We propose a model of co-counselling and social support for this distressed group, by nurses who are coping more successfully with multiple challenges. We discuss the role of nurse educators in fostering nursing values, and developing and supporting ‘hardy personality’ and emotional resilience in recruits to nursing. This study is framed within the disciplinary approach of Critical Realism, which identifies the value basis for research and dialogue in developing strategies for social change.

In this paper, the author utilizes Saddle Theorem and variational methods to deduce existence of at least six stationary solutions for reaction-diffusion Gilpin-Ayala competition model (RDGACM). To obtain the global stabilization of the positive stationary solution of the RDGACM, the author designs a suitable impulsive event triggered mechanism (IETM) to derive the global exponential stability of the the positive stationary solution. It is worth mentioning that the new mechanism can exclude Zeno behavior and effectively reduce the cost of impulse control through event triggering mechanism. Besides, compared with existing literature, the restrictions on the parameters of the RDGACM are relaxed so that the methods used in existing literature can not be applied to the relaxed case of this paper, and so the author makes comprehensive use of Saddle Theorem, orthogonal decomposition of Sobolev space $H_0^1(\Omega)$ and variational methods to overcome the mathematical difficulty. Numerical examples show the effectiveness of the methods proposed in this paper.