A survey of the representations of rational ruled surfaces

The rational ruled surface is a typical modeling surface in computer aided geometric design. A rational ruled surface may have different representations with respective advantages and disadvantages. In this paper, the authors revisit the representations of ruled surfaces including the parametric form, algebraic form, homogenous form and Pl¨ucker form. Moreover, the transformations between these representations are proposed such as parametrization for an algebraic form, implicitization for a parametric form, proper reparametrization of an improper one and standardized reparametrization for a general parametrization. Based on these transformation algorithms, one can give a complete interchange graph for the different representations of a rational ruled surface. For rational surfaces given in algebraic form or parametric form not in the standard form of ruled surfaces, the characterization methods are recalled to identify the ruled surfaces from them.Agencia Estatal de Investigació

Smooth parametric surfaces and n-sided patches

The theory of 'geometric continuity' within the subject of CAGD is reviewed. In particular, we are concerned with how parametric surface patches for CAGD can be pieced together to form a smooth Ck surface. The theory is applied to the problem of filling an n-sided hole occurring within a smooth rectangular patch complex. A number of solutions to this problem are surveyed

In-situ resonant band engineering of solution-processed semiconductors generates high performance n-type thermoelectric nano-inks.

Thermoelectric devices possess enormous potential to reshape the global energy landscape by converting waste heat into electricity, yet their commercial implementation has been limited by their high cost to output power ratio. No single "champion" thermoelectric material exists due to a broad range of material-dependent thermal and electrical property optimization challenges. While the advent of nanostructuring provided a general design paradigm for reducing material thermal conductivities, there exists no analogous strategy for homogeneous, precise doping of materials. Here, we demonstrate a nanoscale interface-engineering approach that harnesses the large chemically accessible surface areas of nanomaterials to yield massive, finely-controlled, and stable changes in the Seebeck coefficient, switching a poor nonconventional p-type thermoelectric material, tellurium, into a robust n-type material exhibiting stable properties over months of testing. These remodeled, n-type nanowires display extremely high power factors (~500 µW m-1K-2) that are orders of magnitude higher than their bulk p-type counterparts

On the base point locus of surface parametrizations: formulas and consequences

This paper shows that the multiplicity of the base point locus of a projective rational surface parametrization can be expressed as the degree of the content of a univariate resultant. As a consequence, we get a new proof of the degree formula relating the degree of the surface, the degree of the parametrization, the base point multiplicity and the degree of the rational map induced by the parametrization. In addition, we extend both formulas to the case of dominant rational maps of the projective plane and describe how the base point loci of a parametrization and its reparametrizations are related. As an application of these results, we explore how the degree of a surface reparametrization is affected by the presence of base points.Agencia Estatal de Investigació

Changing representation of curves and surfaces: exact and approximate methods

Το κύριο αντικείμενο μελέτης στην παρούσα διατριβή είναι η αλλαγή αναπαράστασης γεωμετρικών αντικειμένων από παραμετρική σε αλγεβρική (ή πεπλεγμένη) μορφή. Υπολογίζουμε την αλγεβρική εξίσωση παρεμβάλλοντας τους άγνωστους συντελεστές του πολυωνύμου δεδομένου ενός υπερσυνόλου των μονωνύμων του. Το τελευταίο υπολογίζεται απο το Newton πολύτοπο της αλγεβρικής εξίσωσης που υπολογίζεται από μια πρόσφατη μέθοδο πρόβλεψης του συνόλου στήριξης της εξίσωσης. H μέθοδος πρόβλεψης του συνόλου στήριξης βασίζεται στην αραιή (ή τορική) απαλοιφή: το πολύτοπο υπολογίζεται από το Newton πολύτοπο της αραιής απαλοίφουσας αν θεωρίσουμε την παραμετροποίηση ως πολυωνυμικό σύστημα. Στα μονώνυμα που αντιστοιχούν στα ακέραια σημεία του Newton πολυτόπου δίνονται τιμές ώστε να σχηματίσουν έναν αριθμητικό πίνακα. Ο πυρήνα του πίνακα αυτού, διάστασης 1 σε ιδανική περίπτωση, περιέχει τους συντελεστές των μονωνύμων στην αλγεβρική εξίσωση. Υπολογίζουμε τον πυρήνα του πίνακα είτε συμβολικά είτε αριθμητικά εφαρμόζοντας την μέθοδο του singular value decomposition (SVD). Προτείνουμε τεχνικές για να διαχειριστούμε την περίπτωση ενός πολυδιάστατου πυρήνα το οποίο εμφανίζεται όταν το προβλεπόμενο σύνολο στήριξης είναι ένα υπερσύνολο του πραγματικού. Αυτό δίνει έναν αποτελεσματικό ευαίσθητο-εξόδου αλγόριθμο υπολογισμού της αλγεβρικής εξίσωσης. Συγκρίνουμε διαφορετικές προσεγγίσεις κατασκευής του πίνακα μέσω των λογισμικών Maple και SAGE. Στα πειράματά μας χρησιμοποιήθηκαν ρητές καμπύλες και επιφάνειες καθώς και NURBS. Η μέθοδός μας μπορεί να εφαρμοστεί σε πολυώνυμα ή ρητές παραμετροποιήσεις επίπεδων καμπυλών ή (υπερ)επιφανειών οποιασδήποτε διάστασης συμπεριλαμβανομένων και των περιπτώσεων με παραμετροποίηση σεσημεία βάσης που εγείρουν σημαντικά ζητήματα για άλλες μεθόδους αλγεβρικοποίησης. Η μέθοδος έχει τον εξής περιορισμό: τα γεωμετρικά αντικείμενα πρέπει να αναπαριστώνται από βάσεις μονωνύμων που στην περίπτωση τριγωνομετρικών παραμετροποιήσεων θα πρέπει να μπορούν να μετασχηματιστούν σε ρητές συναρτήσεις. Επιπλέον η τεχνική που προτείνουμε μπορεί να εφαρμοστεί σε μη γεωμετρικά προβλήματα όπως ο υπολογισμόςτης διακρίνουσας ενός πολυωνύμου με πολλές μεταβλητές ή της απαλοίφουσας ενός συστήματος πολυωνύμων με πολλές μεταβλητές.The main object of study in our dissertation is the representation change of the geometric objects from the parametric form to implicit. We compute the implicit equation interpolating the unknown coefficients of the implicit polynomial given a superset of its monomials. The latter is derived from the Newton polytope of the implicit equation obtained by the recently developed method for support prediction. The support prediction method we use relies on sparse (or toric) elimination: the implicit polytope is obtained from the Newton polytope of the sparse resultant of the system in parametrization, represented as polynomials. The monomials that correspond to the lattice points of the Newton polytope are suitably evaluated to build a numeric matrix, ideally of corank 1. Its kernel contains their coefficients in the implicit equation. We compute kernel of the matrix either symbolically, or numerically, applying singular value decomposition (SVD). We propose techniques for handling the case of the multidimensional kernel space, caused by the predicted support being a superset of the actual. This yields an efficient, output-sensitive algorithm for computing the implicit equation. We compare different approaches for constructing the matrix in Maple and SAGE software. In our experiments we have used classical algebraic curves and surfaces as well as NURBS. Our method can be applied to polynomial or rational parametrizations of planar curves or (hyper)surfaces of any dimension including cases of parameterizations with base points which raise important issues for other implicitization methods. The method has its limits: geometric objects have to be presented using monomial basis; in the case of trigonometric parametrizations they have to be convertible to rational functions. Moreover, the proposed technique can be applied for nongeometric problems such as the computation of the discriminant of a multivariate polynomial or the resultant of a system of multivariate polynomials

Computing the μ-bases of algebraic monoid curves and surfaces

The μ-basis is a developing algebraic tool to study the expressions of rational curves and surfaces. It can play a bridge role between the parametric forms and implicit forms and show some advantages in implicitization, inversion formulas and singularity computation. However, it is difficult and there are few works to compute the μ-basis from an implicit form. In this paper, we derive the explicit forms of μ-basis for implicit monoid curves and surfaces, including the conics and quadrics which are particular cases of these entities. Additionally, we also provide the explicit form of μ-basis for monoid curves and surfaces defined by any rational parametrization (not necessarily in standard proper form). Our technique is simply based on the linear coordinate transformation and standard forms of these curves and surfaces. As a practical application in numerical situation, if an exact multiple point can not be computed, we can consider the problem of computing “approximate μ-basis” as well as the error estimation.Agencia Estatal de Investigació

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4

System- and Data-Driven Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This first volume focuses on real-time control theory, data assimilation, real-time visualization, high-dimensional state spaces and interaction of different reduction techniques