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Amberlyst 15 resin supported the tandem reaction of cinnamic acids with phenols under solvent-free reaction condition has been introduced to afford 4-aryl-3,4-dihydrocoumarin (neoflavanone) derivat...

Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role for studying rigidity properties of mechanism in material designs. In this paper, we design an algorithm which solves this problem. It is based on the computations of critical points as well as roadmaps for answering connectivity queries in real algebraic sets. This leads to a probabilistic algorithm of complexity $(nd)^{O(n\log(n))}$ for computing the real isolated points of real algebraic hypersurfaces of degree $d$. It allows us to solve in practice instances which are out of reach of the state-of-the-art., Conference paper ISSAC 2020

For genetic algorithms using a bit-string representation of length~$n$, the general recommendation is to take $1/n$ as mutation rate. In this work, we discuss whether this is really justified for multimodal functions. Taking jump functions and the $(1+1)$ evolutionary algorithm as the simplest example, we observe that larger mutation rates give significantly better runtimes. For the $\jump_{m,n}$ function, any mutation rate between $2/n$ and $m/n$ leads to a speed-up at least exponential in $m$ compared to the standard choice. The asymptotically best runtime, obtained from using the mutation rate $m/n$ and leading to a speed-up super-exponential in $m$, is very sensitive to small changes of the mutation rate. Any deviation by a small $(1 \pm \eps)$ factor leads to a slow-down exponential in $m$. Consequently, any fixed mutation rate gives strongly sub-optimal results for most jump functions. Building on this observation, we propose to use a random mutation rate $\alpha/n$, where $\alpha$ is chosen from a power-law distribution. We prove that the $(1+1)$ EA with this heavy-tailed mutation rate optimizes any $\jump_{m,n}$ function in a time that is only a small polynomial (in~$m$) factor above the one stemming from the optimal rate for this $m$. Our heavy-tailed mutation operator yields similar speed-ups (over the best known performance guarantees) for the vertex cover problem in bipartite graphs and the matching problem in general graphs. Following the example of fast simulated annealing, fast evolution strategies, and fast evolutionary programming, we propose to call genetic algorithms using a heavy-tailed mutation operator \emph{fast genetic algorithms}.

The introduction of N and S-nucleophiles into the 2H-4-azabenzimidazole system using the 1H-benzotriazole ring as a synthetic auxiliary, and also the syntheses of novel pentacyclic ring systems by application of the Graebe-Ullmann method, are described.

Nucleophilic introduction of the 1H-benzotriazole ring into spiro[2H-benzimidazole-2,1′-cyclohexane] and spiro[2H-imidazo[4,5-b]pyridine-2,1′-cyclohexane], followed by Graebe-Ullmann thermolysis, and also the reduction and formic acid cyclization of five spiro[cyclohexane-1,2′-2′H-imidazo[4,5-b]pyridine] derivatives to give 5-substituted 1H-imidazo[4′,5′:5,6]pyrido[1,2-a]benzimidazoles, are described.

The introduction of N and S-nucleophiles into the 2H-4-azabenzimidazole system using the 1H-benzotriazole ring as a synthetic auxiliary, and also the syntheses of novel pentacyclic ring systems by application of the Graebe-Ullmann method, are described.

International audience; We design a new algorithm for solving parametric systems of equations having finitely many complex solutions for generic values of theparameters. More precisely, let $\mathbf{f} = (f_1, \ldots, f_m)\subset \mathbb{Q}[\mathbf{y}][\mathbf{x}]$ with $\mathbf{y} = (y_1, \ldots, y_{t})$ and $\mathbf{x} = (x_1, \ldots, x_{n})$, $\mathcal{V}\subset \mathbb{C}^t \times \mathbb{C}^n$ be the algebraic set defined by the simultaneous vanishing of the $f_i$'s and $\pi$ be the projection $(\mathbf{y}, \mathbf{x}) \to \mathbf{y}$. Under the assumptions that $\mathbf{f}$ admits finitely many complex solutions when specializing $\mathbf{y}$ to generic values and that the ideal generated by $\mathbf{f}$ is radical, we solve the following algorithmic problem. On input $\mathbf{f}$, we compute {\em semi-algebraic formulas} defining open semi-algebraic sets$\mathcal{S}_1, \ldots, \mathcal{S}_{\ell}$ in the parameters' space $\mathbb{R}^t$ such that $\cup_{i=1}^{\ell} \mathcal{S}_i$ is dense in$\mathbb{R}^t$ and, for $1\leq i \leq \ell$, the number of real points in $\mathcal{V}\cap \pi^{-1}(\eta)$ is invariant when $\eta$ rangesover $\mathcal{S}_i$.This algorithm exploits special properties of some well chosen monomial bases in the quotient algebra $\mathbb{Q}(\mathbf{y})[\mathbf{x}] / I$ where $I\subset \mathbb{Q}(\mathbf{y})[\mathbf{x}]$ is the ideal generated by $\mathbf{f}$ in $\mathbb{Q}(\mathbf{y})[\mathbf{x}]$ as well as thespecialization property of the so-called Hermite matrices which represent Hermite's quadratic forms. This allows us to obtain``compact'' representations of the semi-algebraic sets $\mathcal{S}_i$ by means of semi-algebraic formulas encoding the signature of a givensymmetric matrix. When $\mathbf{f}$ satisfies extra genericity assumptions (such as regularity), we use the theory of Gr\"obner bases to derive complexitybounds both on the number of arithmetic operations in $\mathbb{Q}$ and the degree of the output polynomials. More precisely, letting $d$ bethe maximal degrees of the $f_i$'s and $\mathfrak{D} = n(d-1)d^n$, we prove that, on a generic input $\mathbf{f}=(f_1,\ldots,f_n)$, one cancompute those semi-algebraic formulas with $O\ {\widetilde{~}}\left (\binom{t+\mathfrak{D}}{t}\ 2^{3t}\ n^{2t+1} d^{3nt+2(n+t)+1} \right )$arithmetic operations in $\mathbb{Q}$ and that the polynomials involved in these formulas have degree bounded by $\mathfrak{D}$.We report on practical experiments which illustrate the efficiency of this algorithm, both on generic parametric systems and parametricsystems coming from applications since it allows us to solve systems which are out of reach on the current state-of-the-art.