In this work, a new rational approximation scheme, based on the recently developed Transformational High Dimensional Model Representation (THDMR) approximation method is developed. As an initial step to the construction of a rational approximation for multivariate functions via THDMR, this paper focuses on the general theoretical background of the method and gives explicit formulae for the computation of such approximants. The performance of the technique is shown by several examples both in univariate and bivariate cases.
Borate is an essential material to numerous industries and even to individual countries’ economies, defense, and politics. Almost all industries need borates for production, and almost everybody needs their products. Borate is a compound that contains or supplies boric oxide (B2O3). Among the minerals that contain boric oxide, there are only four minerals significant from an economic standpoint, namely borax (tincal), colemanite, ulexite, and kernite. Turkey has almost 70% of all known reserves in the world. Therefore, borates and their products could be one of the main topics for sustainable development in the whole world. The recent development and pursuit of new boron-consuming technologies and alternative products to existing borate-consuming products introduce additional uncertainty to the sustainability of boron minerals. Therefore, the European Union (EU) Commission also declared borate one of the 30 critical raw materials. Turkey is a prosperous country in terms of boron reserves, and it exports almost 96% of borates’ production. In order to better understand the relation between borate minerals and borate products, a material flow analysis (MFA) study has been carried out within the content of this work in order to update the data about the current status of boron. For this purpose, a system has been established that shows the flow of boron material. The extraction, enrichment, and refining processes of boron products are drawn. The results indicate that about 41% of extracted colemanite ore is converted into refined borate, about 31% of tincal ore is converted to refined borate, and 4% of tincal ore is converted to end-usage products, such as detergent. The correctness of the data and the sensitivity of the processes are all estimated values. The results can help in the development of boron sustainability and boron production strategies. The MFA study on tincal and colemanite ore may be an example of boron studies in different countries.
The construction of (near-)minimal cubature formulae on the disk is still a complicated subject on which many results have been published. We restrict ourselves to the case of radial weight functions and make use of a recent connection between cubature and the concept of multivariate spherical orthogonal polynomials to derive a new system of equations defining the nodes and weights of (near-)minimal rules for general degree m = 2n − 1, n ≥ 2. The approach encompasses all previous derivations. The new system is small and may consist of only (n + 1) 2 /4 equations when n is odd and n(n + 2)/4 equations when n is even. It is valid for general n and has a Prony-like structure. It may admit a unique solution (such as for n = 3) or an infinity of solutions (such as for n = 7). In Section 2, the new approach is described, whereas the new system is derived in Sections 3 and 4. All well-known (near-)minimal cubature rules can be reobtained. Some typical illustrations of how this works are given in Section 5. We expect that this unifying theory will shed new light on the topic of cubature, in particular with respect to the discovery of new bounds on the number of nodes and their connection with the zeros of multivariate orthogonal polynomials.
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