Optimised design of systems and processes using algebraic inequalities

A method for optimising the design of systems and processes has been introduced that consists of interpreting the left- and the right-hand side of a correct algebraic inequality as the outputs of two alternative design configurations delivering the same required function. In this way, on the basis of an algebraic inequality, the superiority of one of the configurations is established. The proposed method opens wide opportunities for enhancing the performance of systems and processes and is very useful for design in general. The method has been demonstrated on systems and processes from diverse application domains. The meaningful interpretation of an algebraic inequality based on a single-variable sub-additive function led to developing a light-weight design for a supporting structure based on cantilever beams. The interpretation of a new algebraic inequality based on a multivariable sub-additive function led to a method for increasing the kinetic energy absorbing capacity during inelastic impact. The interpretation of a new inequality has been used for maximising the mass of deposited substance during electrolysis.

An Application of Quantum Logic to Experimental Behavioral Science

In 1933, Kolmogorov synthesized the basic concepts of probability that were in general use at the time into concepts and deductions from a simple set of axioms that said probability was a σ-additive function from a boolean algebra of events into [0, 1]. In 1932, von Neumann realized that the use of probability in quantum mechanics required a different concept that he formulated as a σ-additive function from the closed subspaces of a Hilbert space onto [0,1]. In 1935, Birkhoff & von Neumann replaced Hilbert space with an algebraic generalization. Today, a slight modification of the Birkhoff-von Neumann generalization is called “quantum logic”. A central problem in the philosophy of probability is the justification of the definition of probability used in a given application. This is usually done by arguing for the rationality of that approach to the situation under consideration. A version of the Dutch book argument given by de Finetti in 1972 is often used to justify the Kolmogorov theory, especially in scientific applications. As von Neumann in 1955 noted, and his criticisms still hold, there is no acceptable foundation for quantum logic. While it is not argued here that a rational approach has been carried out for quantum physics, it is argued that (1) for many important situations found in behavioral science that quantum probability theory is a reasonable choice, and (2) that it has an arguably rational foundation to certain areas of behavioral science, for example, the behavioral paradigm of Between Subjects experiments.

Meaningful interpretation of algebraic inequalities to achieve uncertainty and risk reduction

The paper develops an important method related to using algebraic inequalities for uncertainty and risk reduction and enhancing systems performance. The method consists of creating relevant meaning for the variables and different parts of the inequalities and linking them with real physical systems or processes. The paper shows that inequalities based on multivariable sub-additive functions can be interpreted meaningfully and the generated new knowledge used for optimising systems and processes in diverse areas of science and technology. In this respect, an interpretation of the Bergström inequality, which is based on a sub-additive function, has been used to increase the accumulated strain energy in components loaded in tension and bending. The paper also presents an interpretation of the Chebyshev’s sum inequality that can be used to avoid the risk of overestimation of returns from investments and an interpretation of a new algebraic inequality that can be used to construct the most reliable series-parallel system. The meaningful interpretation of other algebraic inequalities yielded a highly counter-intuitive result related to assigning devices of different types to missions composed of identical tasks. In the case where the probabilities of a successful accomplishment of a task, characterising the devices, are unknown, the best strategy for a successful accomplishment of the mission consists of selecting randomly an arrangement including devices of the same type. This strategy is always correct, irrespective of existing uknown interdependencies among the probabilities of successful accomplishment of the tasks characterising the devices.

Some remarks on the stability of the Cauchy equation and completeness

It was proved in Forti and Schwaiger (C R Math Acad Sci Soc R Can 11(6):215–220, 1989), Schwaiger (Aequ Math 35:120–121, 1988) and with different methods in Schwaiger (Developments in functional equations and related topics. Selected papers based on the presentations at the 16th international conference on functional equations and inequalities, ICFEI, Bȩdlewo, Poland, May 17–23, 2015, Springer, Cham, pp 275–295, 2017) that under the assumption that every function defined on suitable abelian semigroups with values in a normed space such that the norm of its Cauchy difference is bounded by a constant (function) is close to some additive function, i.e., the norm of the difference between the given function and that additive function is also bounded by a constant, the normed space must necessarily be complete. By Schwaiger (Ann Math Sil 34:151–163, 2020) this is also true in the non-archimedean case. Here we discuss the situation when the bound is a suitable non-constant function.

A study on some generalized multiplicative and generalized additive arithmetic functions

In this paper by an arithmetic function we shall mean a real-valued function on the set of positive integers. We recall the definitions of some common arithmetic convolutions. We also recall the definitions of a multiplicative function, a generalized multiplicative function (or briefly a GM-function), an additive function and a generalized additive function (or briefly a GA-function). We shall study in details some properties of GM-functions as well as GA-functions using some particular arithmetic convolutions namely the Narkiewicz’s A-product and the author’s B-product. We conclude our discussion with some examples.

The Functional Equation max{χ(xy),χ(xy-1)}=χ(x)χ(y) on Groups and Related Results

This research paper focuses on the investigation of the solutions χ:G→R of the maximum functional equation max{χ(xy),χ(xy−1)}=χ(x)χ(y), for every x,y∈G, where G is any group. We determine that if a group G is divisible by two and three, then every non-zero solution is necessarily strictly positive; by the work of Toborg, we can then conclude that the solutions are exactly the e|α| for an additive function α:G→R. Moreover, our investigation yields reliable solutions to a functional equation on any group G, instead of being divisible by two and three. We also prove the existence of normal subgroups Zχ and Nχ of any group G that satisfy some properties, and any solution can be interpreted as a function on the abelian factor group G/Nχ.

Functional Features of Footnotes in Literary Translation

The article provides a functional analysis of footnotes in literary translation. A literary text, besides the textual, comprises metatextual, intertextual and, possibly, paratextual elements. The paratextual elements can be also introduced while rendering a literary text in another language, in this case they can be added by translator or editor. Paratext comprises a number of components, footnotes among them, namely, the documentary ones which are subjected to analysis in this article. Concentrating upon the functional features of footnotes, we argue that they are rooted in their formal and semantic-cognitive characteristics. The formal – graphic – features of the footnotes under study are as follows: they are always placed at the foot of a page below a line, marked mostly by a figure (in some cases by an asterisk), printed in small font (sometimes in italic font). All these perform the attractive function, drawing the reader's attention to the footnote. The semantic-cognitive features of footnotes come in two basic varieties and one minor one. The footnotes of the two basic groups provide information about a textual element, that of universal or culturally or lingvoculturally specific character thus performing the explanatory function. The footnotes of the third - minor - group provide information about an intertextual element, in particular, about the authorship of the translation of this element. Thus they perform the additive function.

Method of Selecting a Radar Sample Preferably

A new method of selecting a radar sample preferably has been developed. The methods is based on the representation in the form of an additive function of the estimated functional of each normalized indicator characterizing the radar. For each variant of the compared samples, which makes it possible to evaluate the effectiveness of the radar with the help of a single integral indicator, and also to eliminate the uncertainty in the choice of the radar by taking into account additional indicators and to take into account the cost radar to the maximum of the preference function

Stability of Functional Equations and Properties of Groups

Investigating Hyers–Ulam stability of the additive Cauchy equation with domain in a group G, in order to obtain an additive function approximating the given almost additive one we need some properties of G, starting from commutativity to others more sophisticated. The aim of this survey is to present these properties and compare, as far as possible, the classes of groups involved.