Asymptotic stability of the Cauchy and Jensen functional equations

The aim of this note is to investigate the asymptotic stability behaviour of the Cauchy and Jensen functional equations. Our main results show that if these equations hold for large arguments with small error, then they are also valid everywhere with a new error term which is a constant multiple of the original error term. As consequences, we also obtain results of hyperstability character for these two functional equations

Some functional equations related to the characterizations of information measures and their stability

The main purpose of this paper is to investigate the stability problem of some functional equations that appear in the characterization problem of information measures.Comment: 36 pages. arXiv admin note: text overlap with arXiv:1307.0657, arXiv:1307.0631, arXiv:1307.0664, arXiv:1307.065

Set-valued orthogonal additivity

We study the set-valued Cauchy equation postulated for orthogonal vectors. We give its general solution as well as we look for selections of functions satisfying the equation

Remarks on the Cauchy functional equation and variations of it

This paper examines various aspects related to the Cauchy functional equation $f(x+y)=f(x)+f(y)$, a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as a one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation.Comment: To appear in Aequationes Mathematicae (important remark: the acknowledgments section in the official paper exists, but it appears before the appendix and not before the references as in the arXiv version); correction of a minor inaccuracy in Lemma 3.2 and the initial value proof of Theorem 2.1; a few small improvements in various sections; added thank

Interactive and Single Effects of Ectomycorrhiza Formation and Bacillus cereus on Metallothionein MT1 Expression and Phytoextraction of Cd and Zn by Willows

Single and joint ectomycorrhizal (+ Hebeloma mesophaeum) and bacterial (+ Bacillus cereus) inoculations of willows (Salix viminalis) were investigated for their potential and mode of action in the promotion of cadmium (Cd) and zinc (Zn) phytoextraction. Dual fungal and bacterial inoculations promoted the biomass production of willows in contaminated soil. Single inoculations either had no effect on the plant growth or inhibited it. All inoculated willows showed increased concentrations of nutritional elements (N, P, K and Zn) and decreased concentrations of Cd in the shoots. The lowest biomass production and concentration of Cd in the willows (+ B. cereus) were combined with the strongest expression of metallothioneins. It seems that biotic stress from bacterial invasion increased the synthesis of these stress proteins, which responded in decreased Cd concentrations. Contents of Cd and Zn in the stems of willows were combination-specific, but were always increased in dual inoculated plants. In conclusion, single inoculations with former mycorrhiza-associated B. cereus strains decreased the phytoextraction efficiency of willows by causing biotic stress. However, their joint inoculation with an ectomycorrhizal fungus is a very promising method for promoting the phytoextraction of Cd and Zn through combined physiological effects on the plant

Orthogonalities and functional equations

In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations

A review on the eco-epidemiology and clinical management of human granulocytic anaplasmosis and its agent in Europe

Anaplasma phagocytophilum is the agent of tick-borne fever, equine, canine and human granulocytic anaplasmosis. The common route of A. phagocytophilum transmission is through a tick bite, the main vector in Europe being Ixodes ricinus. Despite the apparently ubiquitous presence of the pathogen A. phagocytophilum in ticks and various wild and domestic animals from Europe, up to date published clinical cases of human granulocytic anaplasmosis (HGA) remain rare compared to the worldwide status. It is unclear if this reflects the epidemiological dynamics of the human infection in Europe or if the disease is underdiagnosed or underreported. Epidemiologic studies in Europe have suggested an increased occupational risk of infection for forestry workers, hunters, veterinarians, and farmers with a tick-bite history and living in endemic areas. Although the overall genetic diversity of A. phagocytophilum in Europe is higher than in the USA, the strains responsible for the human infections are related on both continents. However, the study of the genetic variability and assessment of the difference of pathogenicity and infectivity between strains to various hosts has been insufficiently explored to date. Most of the European HGA cases presented as a mild infection, common clinical signs being pyrexia, headache, myalgia and arthralgia. The diagnosis of HGA in the USA was recommended to be based on clinical signs and the patient’s history and later confirmed using specialized laboratory tests. However, in Europe since the majority of cases are presenting as mild infection, laboratory tests may be performed before the treatment in order to avoid antibiotic overuse. The drug of choice for HGA is doxycycline and because of potential for serious complication the treatment should be instituted on clinical suspicion alone