On a remarkable geometric-mechanical synergism based on a novel linear eigenvalue problem

The vertices of two specific eigenvectors, obtained from a novel linear eigenvalue problem, describe two curves on the surface of an N-dimensional unit hypersphere. N denotes the number of degrees of freedom in the framework of structural analysis by the Finite Element Method. The radii of curvature of these two curves are 0 and 1. They correlate with pure stretching and pure bending, respectively, of structures. The two coefficient matrices of the eigenvalue problem are the tangent stiffness matrix at the load level considered and the one at the onset of loading. The goals of this paper are to report on the numerical verification of the aforesaid geometric-mechanical synergism and to summarize current attempts of its extension to combinations of stretching and bending of structures.

Tetrahedral cages for unit discs

A cage is the 1-skeleton of a convex polytope in ℝ3. A cage is said to hold a set if the set cannot be continuously moved to a distant location, remaining congruent to itself and disjoint from the cage. In how many positions can (compact 2-dimensional) unit discs be held by a tetrahedral cage? We completely answer this question for all tetrahedra.

The Typical Cell of a Voronoi Tessellation on the Sphere

The typical cell of a Voronoi tessellation generated by $$n+1$$ n + 1 uniformly distributed random points on the d-dimensional unit sphere $$\mathbb {S}^d$$ S d is studied. Its f-vector is identified in distribution with the f-vector of a beta’ polytope generated by n random points in $$\mathbb {R}^d$$ R d . Explicit formulas for the expected f-vector are provided for any d and the low-dimensional cases $$d\in \{2,3,4\}$$ d ∈ { 2 , 3 , 4 } are studied separately. This implies an explicit formula for the total number of k-dimensional faces in the spherical Voronoi tessellation as well.

Bose–Einstein Condensation for Two Dimensional Bosons in the Gross–Pitaevskii Regime

We consider systems of N bosons trapped on the two-dimensional unit torus, in the Gross-Pitaevskii regime, where the scattering length of the repulsive interaction is exponentially small in the number of particles. We show that low-energy states exhibit complete Bose–Einstein condensation, with almost optimal bounds on the number of orthogonal excitations.

Estimation of 6D Object Pose Using a 2D Bounding Box

This paper provides an efficient way of addressing the problem of detecting or estimating the 6-Dimensional (6D) pose of objects from an RGB image. A quaternion is used to define an object′s three-dimensional pose, but the pose represented by q and the pose represented by -q are equivalent, and the L2 loss between them is very large. Therefore, we define a new quaternion pose loss function to solve this problem. Based on this, we designed a new convolutional neural network named Q-Net to estimate an object’s pose. Considering that the quaternion′s output is a unit vector, a normalization layer is added in Q-Net to hold the output of pose on a four-dimensional unit sphere. We propose a new algorithm, called the Bounding Box Equation, to obtain 3D translation quickly and effectively from 2D bounding boxes. The algorithm uses an entirely new way of assessing the 3D rotation (R) and 3D translation rotation (t) in only one RGB image. This method can upgrade any traditional 2D-box prediction algorithm to a 3D prediction model. We evaluated our model using the LineMod dataset, and experiments have shown that our methodology is more acceptable and efficient in terms of L2 loss and computational time.

Asymptotics of $L_p$-error for adaptive approximation of $n$-variable functions by harmonic splines

For a twice continuously differentiable function, defined on $n$-dimensional unit cube, we obtain sharp asymptotics of $L_p$-error for approximation by harmonic splines, and construct the asymptotically optimal sequence of partitions.

Realizing arbitrary $d$-dimensional dynamics by renormalization of $C^d$-perturbations of identity

<p style='text-indent:20px;'>Any <inline-formula><tex-math id="M3">\begin{document}$ C^d $\end{document}</tex-math></inline-formula> conservative map <inline-formula><tex-math id="M4">\begin{document}$ f $\end{document}</tex-math></inline-formula> of the <inline-formula><tex-math id="M5">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimensional unit ball <inline-formula><tex-math id="M6">\begin{document}$ {\mathbb B}^d $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ d\geq 2 $\end{document}</tex-math></inline-formula>, can be realized by renormalized iteration of a <inline-formula><tex-math id="M8">\begin{document}$ C^d $\end{document}</tex-math></inline-formula> perturbation of identity: there exists a conservative diffeomorphism of <inline-formula><tex-math id="M9">\begin{document}$ {\mathbb B}^d $\end{document}</tex-math></inline-formula>, arbitrarily close to identity in the <inline-formula><tex-math id="M10">\begin{document}$ C^d $\end{document}</tex-math></inline-formula> topology, that has a periodic disc on which the return dynamics after a <inline-formula><tex-math id="M11">\begin{document}$ C^d $\end{document}</tex-math></inline-formula> change of coordinates is exactly <inline-formula><tex-math id="M12">\begin{document}$ f $\end{document}</tex-math></inline-formula>.</p>

The symplectic geometry of higher Auslander algebras: Symmetric products of disks

We show that the perfect derived categories of Iyama’s d-dimensional Auslander algebras of type ${\mathbb {A}}$ are equivalent to the partially wrapped Fukaya categories of the d-fold symmetric product of the $2$ -dimensional unit disk with finitely many stops on its boundary. Furthermore, we observe that Koszul duality provides an equivalence between the partially wrapped Fukaya categories associated to the d-fold symmetric product of the disk and those of its $(n-d)$ -fold symmetric product; this observation leads to a symplectic proof of a theorem of Beckert concerning the derived Morita equivalence between the corresponding higher Auslander algebras of type ${\mathbb {A}}$ . As a by-product of our results, we deduce that the partially wrapped Fukaya categories associated to the d-fold symmetric product of the disk organise into a paracyclic object equivalent to the d-dimensional Waldhausen $\text {S}_{\bullet }$ -construction, a simplicial space whose geometric realisation provides the d-fold delooping of the connective algebraic K-theory space of the ring of coefficients.

On Proinov’s Lower Bound for the Diaphony

In 1986, Proinov published an explicit lower bound for the diaphony of finite and infinite sequences of points contained in the d−dimensional unit cube [Proinov, P. D.:On irregularities of distribution, C. R. Acad. Bulgare Sci. 39 (1986), no. 9, 31–34]. However, his widely cited paper does not contain the proof of this result but simply states that this will appear elsewhere. To the best of our knowledge, this proof was so far only available in a monograph of Proinov written in Bulgarian [Proinov, P. D.: Quantitative Theory of Uniform Distribution and Integral Approximation, University of Plovdiv, Bulgaria (2000)]. The first contribution of our paper is to give a self contained version of Proinov’s proof in English. Along the way, we improve the explicit asymptotic constants implementing recent, and corrected results of [Hinrichs, A.—Markhasin, L.: On lower bounds for the ℒ2-discrepancy, J. Complexity 27 (2011), 127–132.] and [Hinrichs, A.—Larcher, G.: An improved lower bound for the ℒ2-discrepancy, J. Complexity 34 (2016), 68–77]. (The corrections are due to a note in [Hinrichs, A.—Larcher, G. An improved lower bound for the ℒ2-discrepancy, J. Complexity 34 (2016), 68–77].) Finally, as a main result, we use the method of Proinov to derive an explicit lower bound for the dyadic diaphony of finite and infinite sequences in a similar fashion.