Research patterns in algebraic, geometric and analytical fields - part I-II

Nowadays, mathematical models are becoming significant because they are increasingly addressed to other areas of sciences and technology. Our aim is to explain how commutative and computational algebra, linear and non-linear problems, variational methods, algebraic graph theory and combinatorics could help applied science to give solutions on real facts in different sectors. The proposal covers recent developments in these contexts and checks to translate or evaluate theoretical results into concrete examples. One topic within this minisymposium concerns the algebraic theory of the edge ideals, i.e. monomial ideals generated in degree two that descend from the lines of a graph. Connectivity in graphs is actively considered for shaping many types of relations and processes in networks such as transports, telecommunications and codes; moreover, they are fundamental in interdisciplinary contexts such as medical, chemical, physical and biological ones. In the same vein, generalized numerical semigroups are utilized. The generalization of some concepts, typical of numerical semigroups, to a higher dimension, for example, characterize generator systems for the subclass of symmetric semigroups or find relations among the main invariants, is useful in optimization, statistics, and in security, for instance, to encrypt messages or transmit confidential information. We also discuss about Groebner bases theory which represents a universal tool for those kinds of problems that can be modelled by equation systems as well as a powerful implementation method in combinatorial algebra, theoretical physics and engineering since many problems in these fields are dealt with polynomials. In addition, referring to Riemannian geometry, we apply to important theories of mathematics and physics the variational principle, which consists in selecting the best among a variety of objects in a space by taking an appropriate function on it having minima or maxima as the best objects. A typical model of calculus of variations in geometry is the geodesic, roughly speaking, the shortest curve between two points in the space of all curves, in physics is the search for the minima of some function such as the length. From this idea, establishing the connections among differential geometry and field theories supports for universal development of scientific research and technologies, and benefits people living standards. We are also interested in searching for parameters of practical interest in microwave engineering, namely cutoff wavenumbers of waveguides and resonant frequencies of cavities. They are often computed by evaluating the non linear eigenvalues for which the related nonlinear eigenproblem admits a non trivial solution. Since the procedure usually exploited to this purpose is very time consuming, it is enhanced the approach by means of Group theory, exploiting the symmetries of the domain on which the problem is defined. So, it is possible to obtain a block diagonal matrix representation for this eigenproblem, which allows to accomplish a remarkable reduction of the computational requirement involved in its resolution.