scholarly journals Minimal codewords arising from the incidence of points and hyperplanes in projective spaces

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Daniele Bartoli ◽  
Lins Denaux
Keyword(s):  
Projective Space ◽  
Linear Combination ◽  
Projective Spaces ◽  
Sufficient Condition ◽  
Small Weight ◽  
The Past ◽  
Small Weight Codewords

<p style='text-indent:20px;'>Over the past few years, the codes <inline-formula><tex-math id="M1">\begin{document}$ {\mathcal{C}}_{n-1}(n,q) $\end{document}</tex-math></inline-formula> arising from the incidence of points and hyperplanes in the projective space <inline-formula><tex-math id="M2">\begin{document}$ {\rm{PG}}(n,q) $\end{document}</tex-math></inline-formula> attracted a lot of attention. In particular, small weight codewords of <inline-formula><tex-math id="M3">\begin{document}$ {\mathcal{C}}_{n-1}(n,q) $\end{document}</tex-math></inline-formula> are a topic of investigation. The main result of this work states that, if <inline-formula><tex-math id="M4">\begin{document}$ q $\end{document}</tex-math></inline-formula> is large enough and not prime, a codeword having weight smaller than roughly <inline-formula><tex-math id="M5">\begin{document}$ \frac{1}{2^{n-2}}q^{n-1}\sqrt{q} $\end{document}</tex-math></inline-formula> can be written as a linear combination of a few hyperplanes. Consequently, we use this result to provide a graph-theoretical sufficient condition for these codewords of small weight to be minimal.</p>

Branched Coverings and Minimal Free Resolution for Infinite-Dimensional Complex Spaces

2003 ◽  
Vol 10 (1) ◽  
pp. 37-43
Author(s):  
E. Ballico
Keyword(s):  
Projective Space ◽  
Projective Spaces ◽  
Free Resolution ◽  
Complete Intersections ◽  
Vanishing Theorem ◽  
Sheaf Cohomology ◽  
Minimal Free Resolution ◽  
Branched Coverings ◽  
Infinite Dimensional ◽  
Complex Spaces

Abstract We consider the vanishing problem for higher cohomology groups on certain infinite-dimensional complex spaces: good branched coverings of suitable projective spaces and subvarieties with a finite free resolution in a projective space P(V ) (e.g. complete intersections or cones over finitedimensional projective spaces). In the former case we obtain the vanishing result for H 1. In the latter case the corresponding results are only conditional for sheaf cohomology because we do not have the corresponding vanishing theorem for P(V ).

scholarly journals Projective Reconstruction in Algebraic Vision

2019 ◽  
Vol 63 (3) ◽  
pp. 592-609
Author(s):  
Atsushi Ito ◽  
Makoto Miura ◽  
Kazushi Ueda
Keyword(s):  
Projective Space ◽  
Arbitrary Dimension ◽  
Projective Spaces ◽  
Rational Maps ◽  
Projective Reconstruction ◽  
Reconstruction Theorem

AbstractWe discuss the geometry of rational maps from a projective space of an arbitrary dimension to the product of projective spaces of lower dimensions induced by linear projections. In particular, we give an algebro-geometric variant of the projective reconstruction theorem by Hartley and Schaffalitzky.

Blocking Sets and Skew Subspaces of Projective Space

1980 ◽  
Vol 32 (3) ◽  
pp. 628-630 ◽  
Author(s):  
Aiden A. Bruen
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Projective Spaces ◽  
Higher Dimensions ◽  
Blocking Set ◽  
Blocking Sets ◽  
Partial Spreads ◽  
Dimension One

In what follows, a theorem on blocking sets is generalized to higher dimensions. The result is then used to study maximal partial spreads of odd-dimensional projective spaces.Notation. The number of elements in a set X is denoted by |X|. Those elements in a set A which are not in the set Bare denoted by A — B. In a projective space Σ = PG(n, q) of dimension n over the field GF(q) of order q, ┌d(Ωd, Λd, etc.) will mean a subspace of dimension d. A hyperplane of Σ is a subspace of dimension n — 1, that is, of co-dimension one.A blocking set in a projective plane π is a subset S of the points of π such that each line of π contains at least one point in S and at least one point not in S. The following result is shown in [1], [2].

scholarly journals Combining different ECG derived respiration tracking methods to create an optimal reconstruction of the breathing pattern

2015 ◽  
Vol 1 (1) ◽  
pp. 54-57 ◽  
Author(s):  
Gustavo Lenis ◽  
Felix Conz ◽  
Olaf Dössel
Keyword(s):  
Linear Combination ◽  
Breathing Pattern ◽  
Accurate Estimation ◽  
The Past ◽  
Optimal Linear Combination ◽  
Fixed Set ◽  
Optimal Linear ◽  
Respiration Signal ◽  
Electrocardiogram Ecg ◽  
Ecg Derived Respiration

AbstractECG derived respiration (EDR) is a technique applied to estimate the respiration signal using only the electrocardiogram (ECG). Different approaches have been proposed in the past on how respiration could be gained from the ECG. However, in many applications only one of them is used while the others are not considered at all. In this paper, we propose a new algorithm for the optimal linear combination of different EDR methods in order to create a more accurate estimation. Using two well known databases, it was statistically shown that an optimally chosen fixed set of coefficients for the linear combination delivers a better estimation than each of the methods used solely.

scholarly journals Formal Modeling and Analysis of Fairness Characterization of E-Commerce Protocols

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Chengwei Zhang ◽  
Xiaohong Li ◽  
Jing Hu ◽  
Zhiyong Feng ◽  
Jiaojiao Song
Keyword(s):  
Formal Modeling ◽  
Sufficient Condition ◽  
Formal Approach ◽  
Common Assumption ◽  
Modeling And Analysis ◽  
Verification Method ◽  
Space Model ◽  
The Past

In the past, fairness verification of exchanges between the traders in E-commerce was based on a common assumption, so-called nonrepudiation property, which says that if the parties involved can deny that they have received or sent some information, then the exchanging protocol is unfair. So, the nonrepudiation property is not a sufficient condition. In this paper, we formulate a new notion of fairness verification based on the strand space model and propose a method for fairness verification, which can potentially determine whether evidences have been forged in transactions. We first present an innovative formal approach not to depend on nonrepudiation, and then establish a relative trader model and extend the strand space model in accordance with traders’ behaviors of E-commerce. We present a case study to demonstrate the effectiveness of our verification method.

Unique Indexing Scheme of Decagonal Phases Based on a Six Dimensional Model

2003 ◽  
Vol 805 ◽  
Author(s):  
R. K. Mandal ◽  
A. K. Pramanick ◽  
G. V. S. Sastry ◽  
S. Lele
Keyword(s):  
Linear Combination ◽  
Physical Space ◽  
Null Vector ◽  
Structural Description ◽  
Dimensional Model ◽  
Indexing Scheme ◽  
The Past ◽  
Diffraction Patterns ◽  
Parity Condition ◽  
Basis Vectors

ABSTRACTMandal and Lele (1989) have proposed a six dimensional model for the structural description of the decagonal phases. The integral linear combination of six basis vectors for indicating a physical vector in their model, however, leaves the problem of redundancy in indexing. While revisiting their model, we have noted that the condition of a null vector in physical space permits the formulation of unique indexing scheme both in physical reciprocal and direct spaces, we will demonstrate that our scheme, unlike all previously discussed ones, relies only on the information contained in the model. It will also be shown that diffracted spot having equivalent indices possesses identical intensity. This aspect, though equally important, has been totally ignored in the past. We shall substantiate our claim by taking examples from the known decagonal phases. We shall also present parity condition on indices that will be helpful in discussing subtle features of diffraction patterns. The notion of weak and strong diffracting conditions is explained based on the zone rule in physical space in terms of Cartesian components of direct and reciprocal vectors.

scholarly journals Canonical and log canonical thresholds of multiple projective spaces

2019 ◽  
Author(s):  
Aleksandr V. Pukhlikov
Keyword(s):  
Projective Space ◽  
Projective Spaces ◽  
Regularity Conditions ◽  
Large Classes ◽  
Log Canonical Threshold ◽  
Birational Rigidity ◽  
Log Canonical ◽  
Finite Set ◽  
Dimensional Projective Space ◽  
Almost All

AbstractWe show that the global (log) canonical threshold of d-sheeted covers of the M-dimensional projective space of index 1, where $$d\geqslant 4$$d⩾4, is equal to 1 for almost all families (except for a finite set). The varieties are assumed to have at most quadratic singularities, the rank of which is bounded from below, and to satisfy the regularity conditions. This implies birational rigidity of new large classes of Fano–Mori fibre spaces over a base, the dimension of which is bounded from above by a constant that depends (quadratically) on the dimension of the fibre only.

scholarly journals Homography in ℝℙ

2016 ◽  
Vol 24 (4) ◽  
pp. 239-251 ◽  
Author(s):  
Roland Coghetto
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Projective Spaces ◽  
Invertible Matrix ◽  
Real Projective Plane ◽  
The Real ◽  
Mizar Mathematical Library ◽  
Plücker Relation

Summary The real projective plane has been formalized in Isabelle/HOL by Timothy Makarios [13] and in Coq by Nicolas Magaud, Julien Narboux and Pascal Schreck [12]. Some definitions on the real projective spaces were introduced early in the Mizar Mathematical Library by Wojciech Leonczuk [9], Krzysztof Prazmowski [10] and by Wojciech Skaba [18]. In this article, we check with the Mizar system [4], some properties on the determinants and the Grassmann-Plücker relation in rank 3 [2], [1], [7], [16], [17]. Then we show that the projective space induced (in the sense defined in [9]) by ℝ3 is a projective plane (in the sense defined in [10]). Finally, in the real projective plane, we define the homography induced by a 3-by-3 invertible matrix and we show that the images of 3 collinear points are themselves collinear.

scholarly journals Critical hypersurfaces and instability for reconstruction of scenes in high dimensional projective spaces

2020 ◽  
Vol 29 (1) ◽  
pp. 3-20
Author(s):  
Marina Bertolini ◽  
Luca Magri
Keyword(s):  
Projective Space ◽  
Projective Spaces ◽  
High Dimensional ◽  
Projective Reconstruction ◽  
Algebraic Varieties ◽  
Reconstruction Problem ◽  
Complex Projective Spaces ◽  
Simulated Experiment ◽  
Complex Projective ◽  
Practical Implications

In the context of multiple view geometry, images of static scenes are modeled as linear projections from a projective space P^3 to a projective plane P^2 and, similarly, videos or images of suitable dynamic or segmented scenes can be modeled as linear projections from P^k to P^h, with k>h>=2. In those settings, the projective reconstruction of a scene consists in recovering the position of the projected objects and the projections themselves from their images, after identifying many enough correspondences between the images. A critical locus for the reconstruction problem is a configuration of points and of centers of projections, in the ambient space, where the reconstruction of a scene fails. Critical loci turn out to be suitable algebraic varieties. In this paper we investigate those critical loci which are hypersurfaces in high dimension complex projective spaces, and we determine their equations. Moreover, to give evidence of some practical implications of the existence of these critical loci, we perform a simulated experiment to test the instability phenomena for the reconstruction of a scene, near a critical hypersurface.

scholarly journals Cofinitely and co-countably projective spaces

2002 ◽  
Vol 3 (2) ◽  
pp. 185
Author(s):  
Pablo Mendoza Iturralde ◽  
Vladimir V. Tkachuk
Keyword(s):  
Topological Group ◽  
Projective Space ◽  
Infinite Family ◽  
Projective Spaces ◽  
Finite Union ◽  
Paracompact Space ◽  
Finite Set ◽  
Discrete Spaces

<p>We show that X is cofinitely projective if and only if it is a finite union of Alexandroff compactatifications of discrete spaces. We also prove that X is co-countably projective if and only if X admits no disjoint infinite family of uncountable cozero sets. It is shown that a paracompact space X is co-countably projective if and only if there exists a finite set B C X such that B C U ϵ τ (X) implies │X\U│ ≤ ω. In case of existence of such a B we will say that X is concentrated around B. We prove that there exists a space Y which is co-countably projective while there is no finite set B C Y around which Y is concentrated. We show that any metrizable co-countably projective space is countable. An important corollary is that every co-countably projective topological group is countable.</p>

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