Loop invariants are properties of a program loop that hold before and after each iteration of the loop. They are often employed to verify programs and ensure that algorithms consistently produce correct results during execution. Consequently, the generation of invariants becomes a crucial task for loops. We specifically focus on polynomial loops, where both the loop conditions and assignments within the loop are expressed as polynomials. Although computing polynomial invariants for general loops is undecidable, efficient algorithms have been developed for certain classes of loops. For instance, when all assignments within a while loop involve linear polynomials, the loop becomes solvable. In this work, we study the more general case where the polynomials exhibit arbitrary degrees.

We consider the problem of answering connectivity queries on a real algebraic curve. The curve is given as the real trace of an algebraic curve, assumed to be in generic position, and being defined by some rational parametrizations. The query points are given by a zero-dimensional parametrization.

Cuspidal robots are robots with at least two inverse kinematic solutions that can be connected by a singularity-free path. Deciding the cuspidality of generic 3R robots has been studied in the past, but extending the study to six-degree-of-freedom robots can be a challenging problem. Many robots can be modeled as a polynomial map together with a real algebraic set so that the notion of cuspidality can be extended to these data.

In this paper, a data-adaptive method, namely, principal modes (PM) method, based on the spatially averaged temporal covariance of a time series of InSAR displacement measurement obtained from consecutive SAR acquisitions is proposed to retrieve the displacement signal for decorrelating targets. On wrapped interferogram time series, the PM method can highlight and restore coherent fringe patterns where they are more or less significantly hindered by decorrelation noise, whereas on unwrapped interferogram time series, the PM method provides a satisfactory separation of the displacement signal from the spatially correlated perturbations. A two-stage application of the PM method to both wrapped and unwrapped interferogram time series can significantly improve the retrieval of the displacement signal. Synthetic simulations are first performed to investigate the impact of the choice of the appropriate number of modes to retain in the empirical orthogonal function decomposition and of the time series size on the performance of the PM method, as well as to highlight the efficiency of the PM method. Then, the PM method is applied to time series of wrapped and unwrapped Sentinel 1 A/B interferograms over the Gorner glacier between October 2016 and April 2017. The main characteristics of the PM method, such as realistic assumptions, ease of implementation, and high efficiency, are highlighted.

In this paper, a data-adaptive method, namely Principal Modes (PM) method, based on the spatially averaged temporal covariance of a time series is proposed to extract the displacement signal from a time series of Sentinel 1 A/B inter-ferograms over the Gorner glacier during the period between October 2016 and April 2017. On unwrapped interferogram time series, the PM method provides a satisfactory separation of the displacement signal from the spatially correlated perturbations, while on wrapped interferogram time series, the PM method can highlight fringe patterns where they are more or less significantly hindered by the decorrelation noise.