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• Polytechnic University of Catalonia

At present, Convolutional Neural Networks (ConvNets) achieve remarkable performance in image classification tasks. However, current ConvNets cannot guarantee the capabilities of mammalian visual systems such as invariance to contrast and illumination changes. Some ideas for overcoming the illumination and contrast variations must usually be tuned m...

Convolutional Neural Networks (ConvNets) at present achieve remarkable performance in image classification tasks. However, current ConvNets cannot guarantee the capabilities of the mammalian visual systems such as invariance to contrast and illumination changes. Some ideas to overcome the illumination and contrast variations usually have to be tune...

Deep learning (DL) is attracting considerable interest as it currently achieves remarkable performance in many branches of science and technology. However, current DL cannot guarantee capabilities of the mammalian visual systems such as lighting changes. This paper proposes a deterministic entry layer capable of classifying images even with low-con...

Signal representation and processing are the backbone of mathematically-aided engineering. Among the myriad of ideas and results in that realm, many sorts of algorithms and techniques capable of learning from experience have taken the stage in the last decades, with a crescendo of great successes in a variety of fronts in recent years. This paper p...

The intention of this collection agrees with the purposes of the homonymous mini-symposium (MS) at ICIAM-2019, which were to overview the essentials of geometric calculus (GC) formalism, to report on state-of-the-art applications showcasing its advantages and to explore the bearing of GC in novel approaches to deep learning. The first three contrib...

The 40-year old McEliece public-key crypto-system is revisited with the help of recently developed resources: an improved Peterson–Gorenstein–Zierler decoder for alternant error-correcting codes; PyECC, a purely Python CAS; a package of PyECC functional utilities for the computations involved in defining, coding and decoding error-correcting codes;...

Deep learning models have been particularly successful with image recognition using Convolutional Neural Networks (CNN). However, the learning of a contrast invariance and rotation equivariance response may fail even with very deep CNNs or by large data augmentations in training. We were inspired by the V1 visual features of the mammalian visual sy...

The main purpose is to present a biographical portrayal of Antonio Campillo López. In addition to the most relevant aspects of his life and academic career, we offer a panoramic view of his scientific work in fields such as singularity theory, commutative algebra, algebraic geometry, or coding theory. We also stress that his endeavors have been clo...

In this paper, we first review the classical Petterson–Gorenstein–Zierler decoding algorithm for the class of alternant codes, which includes Reed–Solomon, Bose–Chaudhuri–Hocquenghem and classical Goppa codes. Afterwards, we present an improvement of the method to find the number of errors and the error-locator polynomial. Finally, we illustrate th...

In this note we present a fast algorithm that finds for any $r$ the number $N_r$ of $\mathbb{F}_{q^r}$ rational points on a smooth absolutely irreducible curve $C$ defined over $\mathbb{F}_{q}$ assuming that we know $N_1,\cdots,N_g$, where $g$ is the genus of $C$. The proof of its validity is given in detail and its working are illustrated with sev...

This paper is devoted to present the Mathematics Grammar Library, a system for multilingual mathematical text processing. We explain the context in which it originated, its current design and functionality and the current development goals. We also present two prototype services and comment on possible future applications in the area of artificial...

The aim of this paper is to review the Mathematics Grammar Library (MGL) approach to multilingual mathematics and to present gfsage, a prototype of a multilingual interface to Sage.

The goal of this chapter is the study of some basic concepts and constructions pertaining to block error-correcting codes, and of their more salient properties, with a strong emphasis on all the relevant algorithmic and computational aspects.

The goal of this paper is to present a broad picture of Emili Elizalde’s unfolding as a person and as a researcher in physics and in mathematics. In addition to biographical information, we include his answers to a number of questions on his experience as a researcher and his role as a leading figure in his fields of expertise.

In this note we obtain, phrased in present day geometric and computational frameworks, the characteristic numbers of the family Unod of non–degenerate nodal plane cubics in P3, first obtained by Schubert in his Kalk¨ul der abz¨ahlenden Geometrie. The main geometric contribution is a detailed study of a variety Xnod, which is a compactification of t...

El projecte consta de dues parts. La primera EVAM (Eina Virtual per a l’Autoaprenentatge de les Matemàtiques) és un conjunt de material interactiu accesible via Web per tal de consolidar i ampliar els coneixements de matemàtiques de l’estudiant. El projecte fa referència a temes de matemàtiques que majoritàriament, l’estudiant ha vist abans d’entra...

In addition to the current state of knowledge about the learning of mathematics and its aims in today�s society, the main purpose of this paper is discussing ways of improving the process of learning, and especially, in that regard, the role of e-learning technologies. We chart the situation of e-learning mathematics as of December, 2005, including...

In this paper we obtain some fundamental numbers of the family Unod of non-degenerate nodal cubics in P3 involving, in addition to the characteristic conditions, other fundamental conditions, as for example that the node lies on a plane. Some of these numbers were first obtained by Schubert in his Kalkül der abzählenden Geometrie. In our approach w...

The formula χ=3μ−3b+5νχ=3μ−3b+5ν in Corollary 2.2 in [Hernández, X., Miret, J., Xambó, S., 2007. Computing the characteristic numbers of the variety of nodal plane cubics in P3P3. Journal of Symbolic Computation 42 (1–2), 192–202] should be χ=−9μ−3b+5νχ=−9μ−3b+5ν. This mistake does not affect the rest of the paper. We also take the opportunity to s...

In this paper we obtain some fundamental numbers of the family View the U_nod of non-degenerate nodal cubics in P^3 involving, in addition to the characteristic conditions, other fundamental conditions, as for example that the node lies on a plane. Some of these numbers were first obtained by Schubert in his Kalkül der abzählenden Geometrie. In our...

L�objecte d�aquest article és donar una perspectiva general de l�impacte científic i humà de Sir Michael Atiyah. La presentació de les principals idees i resultats amb què ha contribuït al coneixement matemàtic es complementa amb informacions sobre les persones que, per una raó o una altra, han tingut o tenen un paper rellevant en la seva trajectòr...

In this note we obtain, phrased in present day geometric and computational frameworks, the characteristic numbers of the family Unod of non-degenerate nodal plane cubics in P3, first obtained by Schubert in his Kalkül der abzählenden Geometrie. The main geometric contribution is a detailed study of a variety Xnod, which is a compactification of the...

In this note we obtain, phrased in present day geometric and computational frameworks, the characteristic numbers of the family U_nod of non-degenerate nodal plane cubics in P^3, first obtained by Schubert in his Kalkül der abzählenden Geometrie. The main geometric contribution is a detailed study of a variety X_nod, which is a compactification of...

This sketch is fairly complete. The verification is completely reduced in Sects. 2–3 to proving 4 lemmas. Their proofs are sketched in Sects. 4–7, and the new ideas are emphasized. Also, the enumerative significance of the number is fully treated.

We show how to compute all fundamental numbers for plane cuspidal cubics. This updates and extends the work of Schubert on this subject. In our approach we need a far more precise description of the first order degenerations (13 in all) than that given by Schubert and this is obtained by proving a number of key geometric relations that are satisfie...

Without Abstract

The technical barriers hindering the development of on-line learning in the sciences have disappeared in 2005. It is now possible and practical to teach on-line via videoconferencing supported by systems like Festoon. Students are ready to embrace synchronous and asynchronous on-line learning at the university level. Instructors are still largely e...

The real challenge of e-Learning is to produce content that brings a general improvement in the way students learn and teachers teach. For that, “intelligent” interactivity is the single most important feature that such content should have. But the design and production of content with this kind of interactivity has turned out to be harder than exp...

We outline the results of our revisiting Hermann Schubert's work on the enumerative geometry of cuspidal cubics in ℙ^3 (Sec. 23 of his Kalkül der abzählenden Geometrie, Teubner, [1879]. Rep. in 1979 by Springer-Verlag). There are three main aspects that we would like to point to. First, we describe the spaces parameterizing cuspidal cubics in ℙ, as...

In this chapter we will introduce the class of alternant codes over F q by means of a (generalized) control matrix. This matrix looks like the control matrix of a general Reed-Solomon code (Example 1.26), but it is defined over an extension F q m of F q and the h i need not be related to the αi. Lower bounds for the minimum distance and the dimensi...

This chapter is devoted to the presentation of some of the basic ideas and results of the theory of finite fields that are used in the theory of error-correcting codes. It is a self-contained exposition, up to a few elementary ideas on rings and polynomials (for the convenience of the reader, the latter are summarized below). On the computational s...

Cyclic codes are linear codes that are invariant under cyclic permutations of the components of its vectors. These codes have a nice algebraic structure (after reinterpreting vectors as univariate polynomials) which favors its study and use in a particularly effective way.

L'objecte d'aquest treball és explicar com es poden implementar d'una manera efectiva, mitjançant el programa de manipulació simbòlica OMEGA, algunes de les construccions i operacions més importants de la teoria de codis correctors algebraics. Per als codis alternants, la classe més important que considerem, i que inclou els codis BCH, RS i de Gopp...

This is the second volume of the proceedings of the third European Congress of Mathematics. Volume I presents the speeches delivered at the Congress, the list of lectures, and short summaries of the achievements of the prize winners as well as papers by plenary and parallel speakers. The second volume collects articles by prize winners and speakers...

An abstract is not available.

If V_{d,δ} denotes the variety of irreducible plane curves of degree d with exactly δ nodes as singularities, Diaz and Harris (1986) have conjectured that Pic(V_{d,δ}) is a torsion group. In this note we study rational equivalence on some families of singular plane curves and we prove, in particular, that Pic(V_{d,1}) is a finite group, so that the...

In this paper we study the Chow groups of schemes for which the class map to Borel-Moore homology is an isomorphism. Then we determine the Chow groups of the scheme CopkP n parametrizing finite coplanary subschemes of lenght k ofP n and of the variety of «complete S-tuples» of Le Barz.

In this paper we prove that the variety of complete cuspidal cubics is smooth in codimension one and that there are no first order degenerations other than Schubert's 13. We also establish a number of properties of cuspidal cubics that give a geometric understanding of the “Stammzahlen” tables of Schubert.

Starting with an introduction to the computational problems posed by algorithms in the domain of error-correcting codes, and the description of some aspects of the experience extracted from a first try (in previous papers) toward giving a general solution of such problems (language Ω 0 and its interpreter ω 0 ), the main goal of this paper is to in...

Contiene: Geometría clásica; Geometría afín; Geometría métrica; Geometría proyectiva; Cónicas y cuádricas.

Aquest projecte dissenya un curs de matemàtiques usant els diferents recursos de Moodle. Els temes que es tracten en el curs corresponen bàsicament a les assignatures de Càlcul i Matemàtiques de diferents titulacions a diversos centres de la UPC, concretament l’E.T.S. d’Enginyers de Camins, Canals i Ports, la Facultat de Matemàtiques i Estadística...

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