arXiv subjects
Publications and source records attributed to Franck Ramaharo.
Given a 4-tangle shadow, we concatenate it with itself n times and form a knot by applying a closure operation that connects each top endpoint to the corresponding bottom endpoint on the same side without introducing any crossings. We then compute the Kauffman bracket polynomial for the resulting knot using a states matrix defined with respect to the basis of the Kauffman 4-strand diagram monoid.
We derive the Kauffman bracket polynomial for the shadow of the Celtic link $CK_4^{2n}$ using two complementary approaches. The first approach uses a recursive relation within the Celtic framework of Gross and Tucker, based on diagrammatic identities. The second approach makes use of a 4-tangle algebraic framework: a fundamental tangle is concatenated with itself n times to form an iterated composite tangle, and the Kauffman bracket polynomial is computed by decomposing the state space with respect to the basis elements of the 4-strand diagram monoid.
We investigate the predictive power of different machine learning algorithms to nowcast Madagascar's gross domestic product (GDP). We trained popular regression models, including linear regularized regression (Ridge, Lasso, Elastic-net), dimensionality reduction model (principal component regression), k-nearest neighbors algorithm (k-NN regression), support vector regression (linear SVR), and tree-based ensemble models (Random forest and XGBoost regressions), on 10 Malagasy quarterly macroeconomic leading indicators over the period 2007Q1--2022Q4, and we used simple econometric models as a benchmark. We measured the nowcast accuracy of each model by calculating the root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE). Our findings reveal that the Ensemble Model, formed by aggregating individual predictions, consistently outperforms traditional econometric models. We conclude that machine learning models can deliver more accurate and timely nowcasts of Malagasy economic performance and provide policymakers with additional guidance for data-driven decision making.
The aim of this note is to identify the factors influencing renewable energy consumption in Madagascar. We tested 12 features covering macroeconomic, financial, social, and environmental aspects, including economic growth, domestic investment, foreign direct investment, financial development, industrial development, inflation, income distribution, trade openness, exchange rate, tourism development, environmental quality, and urbanization. To assess their significance, we assumed a linear relationship between renewable energy consumption and these features over the 1990-2021 period. Next, we applied different machine learning feature selection algorithms classified as filter-based (relative importance for linear regression, correlation method), embedded (LASSO), and wrapper-based (best subset regression, stepwise regression, recursive feature elimination, iterative predictor weighting partial least squares, Boruta, simulated annealing, and genetic algorithms) methods. Our analysis revealed that the five most influential drivers stem from macroeconomic aspects. We found that domestic investment, foreign direct investment, and inflation positively contribute to the adoption of renewable energy sources. On the other hand, industrial development and trade openness negatively affect renewable energy consumption in Madagascar.
We compute the Kauffman bracket polynomial of the numerator and denominator closures of A + A + ... + A ( A is repeated n times), where A is a 2-tangle shadow that has at most 4 crossings.
We give the connection between three polynomials that generate triangles in The On-Line Encyclopedia of Integer Sequences (A123192, A137396 and A300453). We show that they are related with the bracket polynomial for the (2,n)-torus knot
We construct an integer polynomial whose coefficients enumerate the Kauffman states of the two-bridge knot with Conway's notation C(n,r).
We compute the Kauffman bracket polynomial of the three-lead Turk's head, the chain sinnet and the figure-eight chain shadow diagrams. Each of these knots can in fact be constructed by repeatedly concatenating the same 3-tangle, respectively, then taking the closure. The bracket is then evaluated by expressing the state diagrams of the concerned 3-tangle by means of the Kauffman monoid diagram's elements.
We collect statistics which consist of the coefficients in the expansion of the generating polynomials that count the Kauffman states associated with certain classes of pretzel knots having n tangles, of r half-twists respectively.
The present paper is concerned with the enumeration of the state diagrams for some classes of knot shadows endowed with the usual connected sum operation. We focus on shadows that are recursively generated by knot shadows with up to 3 crossings, and for which the enumeration problem is solved with the help of generating polynomials.
We take advantage of the properties of the Pell numbers to construct an integer version of the Jerusalem square fractal.
We enumerate the state diagrams of the twist knot shadow which consist of the disjoint union of two trivial knots. The result coincides with the maximal number of regions into which the plane is divided by a given number of circles. We then establish a bijection between the state enumeration and this particular partition of the plane by means of binary words.
We split the crossings of the foil knot and enumerate the resulting states with a generating polynomial. Unexpectedly, the number of such states which consist of two components are given by the lazy caterer's sequence. This sequence describes the maximum number of planar regions that is obtained with a given number of straight lines. We then establish a bijection between this partition of the plane and the concerned foil splits sequence.