Truth Table Invariant Cylindrical Algebraic Decomposition by Regular Chains

A new algorithm to compute cylindrical algebraic decompositions (CADs) is presented, building on two recent advances. Firstly, the output is truth table invariant (a TTICAD) meaning given formulae have constant truth value on each cell of the decomposition. Secondly, the computation uses regular chains theory to first build a cylindrical decomposition of complex space (CCD) incrementally by polynomial. Significant modification of the regular chains technology was used to achieve the more sophisticated invariance criteria. Experimental results on an implementation in the RegularChains Library for Maple verify that combining these advances gives an algorithm superior to its individual components and competitive with the state of the art

A Poly-algorithmic Approach to Quantifier Elimination

Cylindrical Algebraic Decomposition (CAD) was the first practical means for doing real quantifier elimination (QE), and is still a major method, with many improvements since Collins' original method. Nevertheless, its complexity is inherently doubly exponential in the number of variables. Where applicable, virtual term substitution (VTS) is more effective, turning a QE problem in $n$ variables to one in $n-1$ variables in one application, and so on. Hence there is scope for hybrid methods: doing VTS where possible then using CAD. This paper describes such a poly-algorithmic implementation, based on the second author's Ph.D. thesis. The version of CAD used is based on a new implementation of Lazard's recently-justified method, with some improvements to handle equational constraints

Metal-Organic Frameworks in Germany: from Synthesis to Function

Metal-organic frameworks (MOFs) are constructed from a combination of inorganic and organic units to produce materials which display high porosity, among other unique and exciting properties. MOFs have shown promise in many wide-ranging applications, such as catalysis and gas separations. In this review, we highlight MOF research conducted by Germany-based research groups. Specifically, we feature approaches for the synthesis of new MOFs, high-throughput MOF production, advanced characterization methods and examples of advanced functions and properties

In Situ X-ray Diffraction Investigation of the Crystallisation of Perfluorinated CeIV-Based Metal–Organic Frameworks with UiO-66 and MIL-140 Architectures

We report on the results of an in situ synchrotron powder X-ray diffraction study of the crystallisation in aqueous medium of two recently discovered perfluorinated CeIV-based metal–organic frameworks (MOFs), analogues of the already well investigated ZrIV-based UiO-66 and MIL-140A, namely, F4_UiO-66(Ce) and F4_MIL-140A(Ce). The two MOFs were originally obtained in pure form in similar conditions, using ammonium cerium nitrate and tetrafluoroterephthalic acid as reagents, and small variations of the reaction parameters were found to yield mixed phases. Here, we investigate the crystallisation of these compounds, varying parameters such as temperature, amount of the protonation modulator nitric acid and amount of the coordination modulator acetic acid. When only HNO3 is present in the reaction environment, only F4_MIL-140A(Ce) is obtained. Heating preferentially accelerates nucleation, which becomes rate determining below 57 °C. Upon addition of AcOH to the system, alongside HNO3, mixed-phased products are obtained. F4_UiO-66(Ce) is always formed faster, and no interconversion between the two phases occurs. In the case of F4_UiO-66(Ce), crystal growth is always the rate-determining step. A higher amount of HNO3 favours the formation of F4_MIL-140A(Ce), whereas increasing the amount of AcOH favours the formation of F4_UiO-66(Ce). Based on the in situ results, a new optimised route to achieving a pure, high-quality F4_MIL-140A(Ce) phase in mild conditions (60 °C, 1 h) is also identified

Applying machine learning to the problem of choosing a heuristic to select the variable ordering for cylindrical algebraic decomposition

Cylindrical algebraic decomposition(CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. When using CAD, there is often a choice for the ordering placed on the variables. This can be important, with some problems infeasible with one variable ordering but easy with another. Machine learning is the process of fitting a computer model to a complex function based on properties learned from measured data. In this paper we use machine learning (specifically a support vector machine) to select between heuristics for choosing a variable ordering, outperforming each of the separate heuristics.Comment: 16 page