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The aim of this study was to present a simple method for finding the asymptotic relations for products of elements of some positive real sequences. The main reason to carry out this study was the result obtained by Alzer and Sandor concerning an estimation of a sequence of the product of the first k primes.

The presented paper is strictly connected, among others, with the paper On the sum of some alternating series, Comp. Math. Appl. (2011), written by Wituła and Słota. A problem concerning the form of determinants formulated in the cited paper is solved here. Next, the obtained result is adapted to solve some system of linear equations and the description of the sum of alternating series.

The paper presents original methods of calculating integrals of selected trigonometric rational functions. [ABSTRACT FROM AUTHOR]

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Undoubtedly, one of the most powerful applications that allow symbolic computations is Wolfram Mathematica. However, it turns out that sometimes Mathematica does not give the desired result despite its continuous improvement. Moreover, these gaps are not filled by many authors of books and tutorials. For example, our attempts to obtain a compact symbolic description of the roots of polynomials or coefficients of a polynomial with known roots using Mathematica have often failed and they still fail. Years of our work with theory, computations, and different kinds of applications in the area of polynomials indicate that an application 'offering' the user alternative methods of solving a given problem would be extremely useful. Such an application would be valuable not only for people who look for solutions to very specific problems but also for people who need different descriptions of solutions to known problems than those given by classical methods. Therefore, we propose the development of an application that would be not only a program doing calculations but also containing an interactive database about polynomials. In this paper, we present examples of methods and information which could be included in the described project. [ABSTRACT FROM AUTHOR]

We consider threshold group testing – a generalization of group testing , which asks to identify a set of positive individuals in a population, by performing tests on pools of elements. Each test is represented by a subset Q of individuals and its output is yes if Q contains at least one positive element and no otherwise. Threshold group testing is the natural generalization, introduced by P. Damaschke in 2005, arising when we are given a threshold t > 0 and the answer to a test Q is yes if Q contains at least t positives and no otherwise. We give upper and lower bounds for this general problem, showing a complexity separation with the classical group testing. Next, we introduce a further generalization in which the goal is minimizing not only the number of tests, but also the number of thresholds which is related to the accuracy of the tests. [ABSTRACT FROM AUTHOR]

Herbal medicines with antipyretic action have a long tradition of use in Polish phytotherapy. Over the years, they have proven both their efficacy and safety in a feverishness, which very often have accompanied the upper respiratory tract infections in children. Leading active ingredient of these products is the willow bark (Cortex salicis), which has been the active ingredient in both, the herbal mixture Pyrosan placed on the market in the 50's of last century and Pyrosal syrup marketed in the year 1989. Other herbal substances used in such preparations are elder flower, lime flower and meadowsweet flower. The usage of a pharmaceutical form of syrup is prerequisite when the main group of the patients are children. Such products are continuously in use for nearly 30 years in Poland.