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This work presents a perception system applied to robotic manipulation, that is able to assist in navigation, household waste classification and collection in outdoor environments. This system is made up of optical tactile sensors, RGBD cameras and a LiDAR. These sensors are integrated on a mobile platform with a robot manipulator and a robotic gripper. Our system is divided in three software modules, two of them are vision-based and the last one is tactile-based. The vision-based modules use CNNs to localize and recognize solid household waste, together with the grasping points estimation. The tactile-based module, which also uses CNNs and image processing, adjusts the gripper opening to control the grasping from touch data. Our proposal achieves localization errors around 6 %, a recognition accuracy of 98% and ensures the grasping stability the 91% of the attempts. The sum of runtimes of the three modules is less than 750 ms., Comment: in Spanish language

The accumulation of litter is increasing in many places and is consequently becoming a problem that must be dealt with. In this paper, we present a manipulator robotic system to collect litter in outdoor environments. This system has three functionalities. Firstly, it uses colour images to detect and recognise litter comprising different materials. Secondly, depth data are combined with pixels of waste objects to compute a 3D location and segment three-dimensional point clouds of the litter items in the scene. The grasp in 3 Degrees of Freedom (DoFs) is then estimated for a robot arm with a gripper for the segmented cloud of each instance of waste. Finally, two tactile-based algorithms are implemented and then employed in order to provide the gripper with a sense of touch. This work uses two low-cost visual-based tactile sensors at the fingertips. One of them addresses the detection of contact (which is obtained from tactile images) between the gripper and solid waste, while another has been designed to detect slippage in order to prevent the objects grasped from falling. Our proposal was successfully tested by carrying out extensive experimentation with different objects varying in size, texture, geometry and materials in different outdoor environments (a tiled pavement, a surface of stone/soil, and grass). Our system achieved an average score of 94% for the detection and Collection Success Rate (CSR) as regards its overall performance, and of 80% for the collection of items of litter at the first attempt., Comment: Article sent for final publication

Noble gases are accreted to the giant planets as part of the gas component of the planet-forming disk. While heavier noble gases can separate from the evolution of the hydrogen-rich gas, helium is thought to remain at the protosolar H/He ratio Yproto~0.27-0.28. However, spacecraft observations revealed a depletion in helium in the atmospheres of Jupiter, Saturn, and Uranus. For the gas giants, this is commonly seen as indication of H/He phase separation at greater depths. Here, we apply predictions of the H/He phase diagram and three H/He-EOS to compute the atmospheric helium mass abundance Yatm as a result of H/He phase separation. We obtain a strong depletion Yatm<0.1 for the ice giants if they are adiabatic. Introducing a thermal boundary layer at the Z-poor/Z-rich compositional transition with a temperature increase of up to a few 1000 K, we obtain a weak depletion in Uranus as observed. Our results suggest dissimilar internal structures between Uranus and Neptune. An accurate in-situ determination of their atmospheric He/H ratio would help to constrain their internal structures. This is even more true for Saturn, where we find that any considered H/He phase diagram and H/He-EOS would be consistent with any observed value. However, some H/He-EOS and phase diagram combinations applied to both Jupiter and Saturn require an outer stably-stratified layer at least in one of them., Comment: accepted to Space Science Reviews

A flag $C_0 \subsetneq C_1 \cdots \subsetneq C_s \subsetneq {\mathbb F}_q^n $ of linear codes is said to be self-orthogonal if the duals of the codes in the flag satisfy $C_{i}^\perp=C_{s-i}$, and it is said to satisfy the isometry-dual property with respect to an isometry vector ${\bf x}$ if $C_i^\perp={\bf x} C_{s-i}$ for $i=1, \dots, s$. We characterize complete (i.e. $s=n$) flags with the isometry-dual property by means of the existence of a word with non-zero coordinates in a certain linear subspace of ${\mathbb F}_q^n$. For flags of algebraic geometry (AG) codes we prove a so-called translation property of isometry-dual flags and give a construction of complete self-orthogonal flags, providing examples of self-orthogonal flags over some maximal function fields. At the end we characterize the divisors giving the isometry-dual property and the related isometry vectors showing that for each function field there is only a finite number of isometry vectors and that they are related by cyclic repetitions.

In this work, we study the perception problem for garments using tools from computational topology: the identification of their geometry and position in space from point-cloud samples, as obtained e.g. with 3D scanners. We present a reconstruction algorithm based on a direct topological study of the sampled textile surface that allows us to obtain a cellular decomposition of it via a Morse function. No intermediate triangulation or local implicit equations are used, avoiding reconstruction-induced artifices. No a priori knowledge of the surface topology, density or regularity of the point-sample is required to run the algorithm. The results are a piecewise decomposition of the surface as a union of Morse cells (i.e. topological disks), suitable for tasks such as noise-filtering or mesh-independent reparametrization, and a cell complex of small rank determining the surface topology. This algorithm can be applied to smooth surfaces with or without boundary, embedded in an ambient space of any dimension., Comment: 40 pages, 17 figures, 1 table, 1 algorithm, 1 appendix

Tracking a magnetic field in real-time with an atomic magnetometer presents significant challenges, primarily due to sensor non-linearity, the presence of noise, and the need for one-shot estimation. To address these challenges, we propose a comprehensive approach that integrates measurement, estimation and control strategies. Specifically, this involves implementing a quantum non-demolition measurement based on continuous light-probing of the atomic ensemble. The resulting photocurrent is then directed into an Extended Kalman Filter to produce instantaneous estimates of the system's dynamical parameters. These estimates, in turn, are utilised by a Linear Quadratic Regulator, whose output is applied back to the system through a feedback loop. This procedure automatically steers the atomic ensemble into a spin-squeezed state, yielding a quantum enhancement in precision. Furthermore, thanks to the feedback proposed, the atoms exhibit entanglement even when the measurement data is discarded. To prove that our approach constitutes the optimal strategy in realistic scenarios, we derive ultimate bounds on the estimation error applicable in the presence of both local and collective decoherence, and show that these are indeed attained. Additionally, we demonstrate for large ensembles that the EKF not only reliably predicts its own estimation error in real time, but also accurately estimates spin-squeezing at short timescales., Comment: 26 pages (incl. 13 pages of appendices), 10 figures. Comments are more than welcome!

We prove that, for each fixed genus, the portion of semigroups of that genus belonging to infinite chains in the semigroup tree approaches 0 as the genus grows to infinite. This means that most numerical semigroups have a finite number of descendants in the semigroup tree. This problem has been open since 2009.

One major problem in the study of numerical semigroups is determining the growth of the semigroup tree. In the present work, infinite chains of numerical semigroups in the semigroup tree, firstly introduced in Bras-Amor\'os and Nulygin (2009), are studied. Computational results show that these chains are rare, but without them the tree would not be infinite. It is proved that for each genus $g\geq 5$ there are more semigroups of that genus not belonging to infinite chains than semigroups belonging. The reference Bras-Amor\'os and Bulygin (2009) presented a characterization of the semigroups that belong to infinite chains in terms of the coprimality of the left elements of the semigroup as well as a result on the cardinality of the set of infinite chains to which a numerical semigroup belongs in terms of the primality of the greatest common divisor of these left elements. We revisit these results and fix an imprecision on the cardinality of the set of infinite chains to which a semigroup belongs in the case when the greatest common divisor of the left elements is a prime number. We then look at infinite chains in subtrees with fixed multiplicity. When the multiplicity is a prime number there is only one infinite chain in the tree of semigroups with such multiplicity. When the multpliplicity is $4$ or $6$ we prove a self-replication behavior in the subtree and prove a formula for the number of semigroups in infinite chains of a given genus and multiplicity $4$ and $6$, respectively.

We introduce a new class of numerical semigroups, which we call the class of {\it acute} semigroups and we prove that they generalize symmetric and pseudo-symmetric numerical semigroups, Arf numerical semigroups and the semigroups generated by an interval. For a numerical semigroup $\Lambda=\{\lambda_0<\lambda_1<\dots\}$ denote $\nu_i=\#\{j\mid\lambda_i-\lambda_j\in\Lambda\}$. Given an acute numerical semigroup $\Lambda$ we find the smallest non-negative integer $m$ for which the order bound on the minimum distance of one-point Goppa codes with associated semigroup $\Lambda$ satisfies $d_{ORD}(C_i)(:=\min\{\nu_j\mid j>i\})=\nu_{i+1}$ for all $i\geq m$. We prove that the only numerical semigroups for which the sequence $(\nu_i)$ is always non-decreasing are ordinary numerical semigroups. Furthermore we show that a semigroup can be uniquely determined by its sequence $(\nu_i)$., Comment: Published in IEEE Trans. Inf. Theory, 2004. MR2094884. arXiv admin note: substantial text overlap with arXiv:1706.09765

We give a graphical reinterpretation of the seeds algorithm to explore the tree of numerical semigroups. We then exploit the seeds algorithm to find all the Eliahou semigroups of genus up to 65. Since all these semigroups satisfy the Wilf conjecture, this shows that the Wilf conjecture holds up to genus 65.

We present a revisit of the seeds algorithm to explore the semigroup tree. First, an equivalent definition of seed is presented, which seems easier to manage. Second, we determine the seeds of semigroups with at most three left elements. And third, we find the great-grandchildren of any numerical semigroup in terms of its seeds. The RGD algorithm is the fastest known algorithm at the moment. But if one compares the originary seeds algorithm with the RGD algorithm, one observes that the seeds algorithm uses more elaborated mathematical tools while the RGD algorithm uses data structures that are better adapted to the final C implementations. For genera up to around one half of the maximum size of native integers, the newly defined seeds algorithm performs significantly better than the RGD algorithm. For future compilators allowing larger native sized integers this may constitute a powerful tool to explore the semigroup tree up to genera never explored before. The new seeds algorithm uses bitwise integer operations, the knowledge of the seeds of semigroups with at most three left elements and of the great-grandchildren of any numerical semigroup, apart from techniques such as parallelization and depth first search as wisely introduced in this context by Fromentin and Hivert. The algorithm has been used to prove that there are no Eliahou semigroups of genus $66$, hence proving the Wilf conjecture for genus up to $66$. We also found three Eliahou semigroups of genus $67$. One of these semigroups is neither of Eliahou-Fromentin type, nor of Delgado's type. However, it is a member of a new family suggested by Shalom Eliahou., Comment: Accepted for publication at Mathematics of Computation

Adding tactile sensors to a robotic system is becoming a common practice to achieve more complex manipulation skills than those robotics systems that only use external cameras to manipulate objects. The key of tactile sensors is that they provide extra information about the physical properties of the grasping. In this paper, we implemented a system to predict and quantify the rotational slippage of objects in hand using the vision-based tactile sensor known as Digit. Our system comprises a neural network that obtains the segmented contact region (object-sensor), to later calculate the slippage rotation angle from this region using a thinning algorithm. Besides, we created our own tactile segmentation dataset, which is the first one in the literature as far as we are concerned, to train and evaluate our neural network, obtaining results of 95% and 91% in Dice and IoU metrics. In real-scenario experiments, our system is able to predict rotational slippage with a maximum mean rotational error of 3 degrees with previously unseen objects. Thus, our system can be used to prevent an object from falling due to its slippage., Comment: 3 pages, 4 figures, accepted at ICRA 2023 ViTac Workshop: Blending Virtual and Real Visuo-Tactile Perception

In this work, we introduce a collision model specifically tailored for the simulation of inextensible textiles. The model considers friction, contacts, and inextensibility constraints all at the same time without any decoupling. Self-collisions are modeled in a natural way that allows considering the thickness of cloth without introducing unwanted oscillations. The discretization of the equations of motion leads naturally to a sequence of quadratic problems with inequality and equality constraints. In order to solve these problems efficiently, we develop a novel active-set algorithm that takes into account past active constraints to accelerate the resolution of unresolved contacts. We put to a test the developed collision procedure with diverse scenarios involving static and dynamic friction, sharp objects, and complex-topology folding sequences. Finally, we perform an experimental validation of the collision model by comparing simulations with recordings of real textiles as given by a $\textit{Motion Capture System}$. The results are very accurate, having errors around 1 cm for DIN A2 textiles (42 x 59.4 cm) even in difficult scenarios involving fast and strong hits with a rigid object., Comment: 30 pages, 12 figures, 3 tables, 1 algorithm

Robotic manipulation of cloth is a highly complex task because of its infinite-dimensional shape-state space that makes cloth state estimation very difficult. In this paper we introduce the dGLI Cloth Coordinates, a low-dimensional representation of the state of a rectangular piece of cloth that allows to efficiently distinguish key topological changes in a folding sequence, opening the door to efficient learning methods for cloth manipulation planning and control. Our representation is based on a directional derivative of the Gauss Linking Integral and allows us to represent both planar and spatial configurations in a consistent unified way. The proposed dGLI Cloth Coordinates are shown to be more accurate in the classification of cloth states and significantly more sensitive to changes in grasping affordances than other classic shape distance methods. Finally, we apply our representation to real images of a cloth, showing we can identify the different states using a simple distance-based classifier., Comment: 24 pages, 34 references, 6 figures, 1 table

We propose a novel approach to distinguish table vs non-table ideals by using different machine learning algorithms. We introduce the reader to table ideals, assuming some knowledge on commutative algebra and describe their main properties. We create a data set containing table and non-table ideals, and we use a feedforward neural network model, a decision tree and a graph neural networks for the classification. Our results indicate that there exists an algorithm to distinguish table ideals from non-table ideals, and we prove it along some novel results on table ideals., Comment: 29 figures

We review a lattice construction arising from quaternion algebras over number fields and use it to obtain some known extremal and densest lattices in dimensions 8 and 16. The benefit of using quaternion algebras over number fields is that the dimensionality of the construction problem is reduced by 3/4. We explicitly construct the $E_8$ lattice (resp. $E_8^2$ and $\Lambda_{16}$) from infinitely many quaternion algebras over real quadratic (resp. quartic) number fields and we further present a density result on such number fields. By relaxing the extremality condition, we also provide a source for constructing even unimodular lattices in any dimension multiple of $8$.

For Reed-Solomon codes, the key equation relates the syndrome polynomial---computed from the parity check matrix and the received vector---to two unknown polynomials, the locator and the evaluator. The roots of the locator polynomial identify the error positions. The evaluator polynomial, along with the derivative of the locator polynomial, gives the error values via the Forney formula. The Berlekamp-Massey algorithm efficiently computes the two unknown polynomials. This chapter shows how the key equation, the Berlekamp-Massey algorithm, the Forney formula, and another formula for error evaluation due to Horiguchi all generalize in a natural way to one-point codes. The algorithm presented here is based on K\"otter's adaptation of Sakata's algorithm., Comment: M. E. O'Sullivan, M. Bras-Amor\'os, The Key Equation for One-Point Codes, Chapter 3 of Advances in Algebraic Geometry Codes, World Scientific, E. Mart\'inez-Moro, C. Munuera, D. Ruano (eds.), vol. 5, pp. 99-152, 2008. ISBN 978-981-279-400-0

We determine the Weierstrass semigroup $H(P_\infty,P_1,\ldots,P_m)$ at several rational points on the maximal curves which cannot be covered by the Hermitian curve introduced by Tafazolian, Teher\'an-Herrera, and Torres. Furthermore, we present some conditions to find pure gaps. We use this semigroup to obtain AG codes with better relative parameters than comparable one-point AG codes arising from these curves.

Let $F_q$ be a finite field. A flag of $F_q$-linear codes $C_0\subsetneq C_1\subsetneq\dots\subsetneq C_s$ is said to satisfy the isometry-dual property if there exists a vector $x\in(F_q^*)^n$ such that $C_i=x\cdot C_{s-i}^\perp$, where $C_i^\perp$ denotes the dual code of $C_i$. Consider $F/F_q$ a function field and let $P$ and $Q_1,\ldots,Q_t$ be rational places of $F$. Let the divisor $D$ be the sum of pairwise different places of $F$ such that $P, Q_1,\dots,Q_t$ are not in $supp(D)$. In a previous work we investigated the existence of flags of two-point codes $C(D,a_0P+bQ_1)\subsetneq C(D,a_1P+bQ_1))\subsetneq\dots\subsetneq C(D,a_sP+bQ_1)$ satisfying the isometry-dual property for a non-negative integer $b$ and an increasing sequence of positive integers $a_0,\dots,a_s$. While for one-point codes (i.e. for $b=0$) there is only need to analyze positive integers $a$, for the case of $(t+1)$-point codes, the integers $a$ may be negative. We extend our previous results in different directions. On one hand to the case of negative integers $a$ and $b$, and on the other hand we extend our results to flags of $(t+1)$-point codes $C(D,a_0P+\sum_{i=1}^t\beta_iQ_i)\subsetneq C(D, a_1P+\sum_{i=1}^t\beta_iQ_i))\subsetneq\dots\subsetneq C(D, a_sP+\sum_{i=1}^t\beta_iQ_i)$ for any tuple of (either positive or negative) integers $\beta_1,\dots,\beta_t$ and for an increasing sequence of (either positive or negative) integers $a_0,\dots,a_s$. We apply the obtained results to the broad class of Kummer extensions defined by affine equations of the form $y^m=f(x)$, for $f(x)$ a separable polynomial of degree $r$, where $gcd(r, m)=1$. In particular, depending on the place $P$ and for $D$ an $Aut(F_q(x, y)/F_q(x))$-invariant sum of rational places of $F$ such that $P,Q_i\notin supp(D)$, we obtain necessary and sufficient conditions on $m$ and $\beta_i$'s such that the flag has the isometry-dual property., Comment: Accepted at SIAGA, the SIAM Journal on Applied Algebra and Geometry