Abstract
In this paper, we extend the result of 1 by calculating some examples in detail, including the inscribed ellipses in triangles, quadrilaterals, and pentagons. We also improve the original ...proof and reduce the requirements through projective geometry methods in the quadrilateral and pentagon cases. Furthermore, we see the inscribed ellipse problems from the perspective of two projective planes simultaneously, which offers a new way to determine the inscribed ellipses in triangles. Also, we use python to realize the method provided in this paper of drawing inscribed ellipse.
In this paper, we propose a family of quantum synchronizable codes from repeated-root cyclic codes and constacyclic codes. This family of quantum synchronizable codes are based on (λ(u + v)|u - v) ...construction which is constructed from constacyclic codes. Under this construction, we enrich the varieties of valid quantum synchronizable codes. We also prove that the obtained quantum synchronizable codes can achieve maximum synchronization error tolerance. Furthermore, quantum synchronizable codes based on (λ(u + v)|u - v) construction are shown to be able to have a better capability in correcting bit errors than those from projective geometry codes.
Comprehensive descriptions of animal behavior require precise three-dimensional (3D) measurements of whole-body movements. Although two-dimensional approaches can track visible landmarks in ...restrictive environments, performance drops in freely moving animals, due to occlusions and appearance changes. Therefore, we designed DANNCE to robustly track anatomical landmarks in 3D across species and behaviors. DANNCE uses projective geometry to construct inputs to a convolutional neural network that leverages learned 3D geometric reasoning. We trained and benchmarked DANNCE using a dataset of nearly seven million frames that relates color videos and rodent 3D poses. In rats and mice, DANNCE robustly tracked dozens of landmarks on the head, trunk, and limbs of freely moving animals in naturalistic settings. We extended DANNCE to datasets from rat pups, marmosets, and chickadees, and demonstrate quantitative profiling of behavioral lineage during development.
The PSL(4,R) Hitchin component of a closed surface group π1(S) consists of holonomies of properly convex foliated projective structures on the unit tangent bundle of S. We prove that the leaves of ...the codimension-1 foliation of any such projective structure are all projectively equivalent if and only if its holonomy is Fuchsian. This implies constraints on the symmetries and shapes of these leaves.
We also give an application to the topology of the non-T0 space C(RPn) of projective classes of properly convex domains in RPn. Namely, Benzécri asked in 1960 if every closed subset of C(RPn) that contains no proper nonempty closed subset is a point. Our results imply a negative resolution for n≥2.
The Fundamental Theorem of Projective Geometry states that, in a vector space, a permutation of vector lines preserving triples that span a vector plane is induced by a semi-linear automorphism. We ...consider a generalisation to triples of subspaces, not necessarily of the same dimension, spanning, or being contained in a subspace of fixed dimension. We determine all cases in which the permutation is necessarily induced by a semi-linear automorphism.
A linear code with parameters <inline-formula> <tex-math notation="LaTeX">n, k, n-k+1 </tex-math></inline-formula> is called a maximum distance separable (MDS for short) code. A linear code with ...parameters <inline-formula> <tex-math notation="LaTeX">n, k, n-k </tex-math></inline-formula> is said to be almost maximum distance separable (AMDS for short). A linear code is said to be near maximum distance separable (NMDS for short) if both the code and its dual are AMDS. MDS codes are very important in both theory and practice. There is a classical construction of a <inline-formula> <tex-math notation="LaTeX">q+1, 2u-1, q-2u+3 </tex-math></inline-formula> MDS code for each <inline-formula> <tex-math notation="LaTeX">u </tex-math></inline-formula> with <inline-formula> <tex-math notation="LaTeX">1 \leq u \leq \lfloor \frac {q+1}2\rfloor </tex-math></inline-formula>, which is a reversible and cyclic code. The objective of this paper is to study the extended codes of this family of MDS codes. Two families of MDS codes and several families of NMDS codes are obtained. The NMDS codes have applications in finite geometry, cryptography and distributed and cloud data storage systems. The weight distributions of some of the extended codes are determined.
Anosov representations of word hyperbolic groups into higher-rank semisimple Lie groups are representations with finite kernel and discrete image that have strong analogies with convex cocompact ...representations into rank-one Lie groups. However, the most naive analogy fails: generically, Anosov representations do not act properly and cocompactly on a convex set in the associated Riemannian symmetric space. We study representations into projective indefinite orthogonal groups
PO
(
p
,
q
)
by considering their action on the associated pseudo-Riemannian hyperbolic space
H
p
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q
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1
in place of the Riemannian symmetric space. Following work of Barbot and Mérigot in anti-de Sitter geometry, we find an intimate connection between Anosov representations and a natural notion of convex cocompactness in this setting.
Let E be a set of solids (hyperplanes) in PG(4,q), q even, q>2, such that every point of PG(4,q) lies in either 0, 12(q3−q2) or 12q3 solids of E, and every plane of PG(4,q) lies in either 0, 12q or q ...solids of E. This article shows that E is either the set of solids that are disjoint from a hyperoval, or the set of solids that meet a non-singular quadric Q(4,q) in an elliptic quadric.
Biological visual systems rely on pose estimation of 3D objects to navigate and interact with their environment, but the neural mechanisms and computations for inferring 3D poses from 2D retinal ...images are only partially understood, especially where stereo information is missing. We previously presented evidence that humans infer the poses of 3D objects lying centered on the ground by using the geometrical back-transform from retinal images to viewer-centered world coordinates. This model explained the almost veridical estimation of poses in real scenes and the illusory rotation of poses in obliquely viewed pictures, which includes the “pointing out of the picture” phenomenon. Here we test this model for more varied configurations and find that it needs to be augmented. Five observers estimated poses of sloped, elevated, or off-center 3D sticks in each of 16 different poses displayed on a monitor in frontal and oblique views. Pose estimates in scenes and pictures showed remarkable accuracy and agreement between observers, but with a systematic fronto-parallel bias for oblique poses similar to the ground condition. The retinal projection of the pose of an object sloped wrt the ground depends on the slope. We show that observers’ estimates can be explained by the back-transform derived for close to the correct slope. The back-transform explanation also applies to obliquely viewed pictures and to off-center objects and elevated objects, making it more likely that observers use internalized perspective geometry to make 3D pose inferences while actively incorporating inferences about other aspects of object placement.
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