COMPUTATION COMPLEXITY OF DEEP RELU NEURAL NETWORKS IN HIGH-DIMENSIONAL APPROXIMATION

The purpose of the present paper is to study the computation complexity of deep ReLU neural networks to approximate functions in H\"older-Nikol'skii spaces of mixed smoothness $H_\infty^\alpha(\mathbb{I}^d)$ on the unit cube $\mathbb{I}^d:=[0,1]^d$. In this context, for any function $f\in H_\infty^\alpha(\mathbb{I}^d)$, we explicitly construct nonadaptive and adaptive deep ReLU neural networks having an output that approximates $f$ with a prescribed accuracy $\varepsilon$, and prove dimension-dependent bounds for the computation complexity of this approximation, characterized by the size and the depth of this deep ReLU neural network, explicitly in $d$ and $\varepsilon$. Our results show the advantage of the adaptive method of approximation by deep ReLU neural networks over nonadaptive one.

The KSBA compactification of the moduli space of $$D_{1,6}$$-polarized Enriques surfaces

We describe a compactification by stable pairs (also known as KSBA compactification) of the 4-dimensional family of Enriques surfaces which arise as the $${\mathbb {Z}}_2^2$$ Z 2 2 -covers of the blow up of $${\mathbb {P}}^2$$ P 2 at three general points branched along a configuration of three pairs of lines. Up to a finite group action, we show that this compactification is isomorphic to the toric variety associated to the secondary polytope of the unit cube. We relate the KSBA compactification considered to the Baily–Borel compactification of the same family of Enriques surfaces. Part of the KSBA boundary has a toroidal behavior, another part is isomorphic to the Baily–Borel compactification, and what remains is a mixture of these two. We relate the stable pair compactification studied here with Looijenga’s semitoric compactifications.

Discrepancy of stratified samples from partitions of the unit cube

We extend the notion of jittered sampling to arbitrary partitions and study the discrepancy of the related point sets. Let $${\varvec{\Omega }}=(\Omega _1,\ldots ,\Omega _N)$$ Ω = ( Ω 1 , … , Ω N ) be a partition of $$[0,1]^d$$ [ 0 , 1 ] d and let the ith point in $${{\mathcal {P}}}$$ P be chosen uniformly in the ith set of the partition (and stochastically independent of the other points), $$i=1,\ldots ,N$$ i = 1 , … , N . For the study of such sets we introduce the concept of a uniformly distributed triangular array and compare this notion to related notions in the literature. We prove that the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy, $${{\mathbb {E}}}{{{\mathcal {L}}}_p}({{\mathcal {P}}}_{\varvec{\Omega }})^p$$ E L p ( P Ω ) p , of a point set $${{\mathcal {P}}}_{\varvec{\Omega }}$$ P Ω generated from any equivolume partition $${\varvec{\Omega }}$$ Ω is always strictly smaller than the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy of a set of N uniform random samples for $$p>1$$ p > 1 . For fixed N we consider classes of stratified samples based on equivolume partitions of the unit cube into convex sets or into sets with a uniform positive lower bound on their reach. It is shown that these classes contain at least one minimizer of the expected $${{{\mathcal {L}}}_p}$$ L p -discrepancy. We illustrate our results with explicit constructions for small N. In addition, we present a family of partitions that seems to improve the expected discrepancy of Monte Carlo sampling by a factor of 2 for every N.

Systematic biases in the on-board navigation solution of a CubeSat GNSS payload board

<p>In December 2018 and April 2019, two 3-unit cube satellites of the company Astrocast were launched into orbit. Both satellites are equipped with our low-cost single-frequency multi-GNSS payload board, which provides almost continuous on-board receiver solutions containing the position from GNSS code observations and the velocity from Doppler measurements. We make use of these independent observation types (positions and velocities) to identify and analyse systematic biases in the receiver solution. Therefore, we estimate the parameters of a dynamic orbit model using three different approaches: fitting the orbit model (1) to the positions only, (2) to the velocities only and (3) to both, positions and velocities.</p><p>After removing outliers, the position residuals from the position-only approach are at a level of about 5 m, the velocity residuals from the velocity-only approach at about 15 cm/s. When computing the positions with the velocity-only approach, however, the residuals are much larger and show a once-per revolution periodicity with amplitudes of up to 40 m. Besides that, we identify two offsets in the residuals which are independent of the observation type: a radial position bias of -3 m and an along-track velocity bias of -1.2 cm/s. Additionally, we observe two offsets which are dependent on the observation type: an along-track offset of 13 m in the position residuals when using the velocity-only approach and a radial offset of 1.3 cm/s in radial velocities when using the position-only approach.</p><p>The periodicity in radial and along-track direction is related to the orbit eccentricity and may be due to a general deficiency, when using velocities to estimate geometric orbit parameters. When comparing the orbits from the position-only and the velocity-only approach, we find an offset in the right ascension of the ascending node, which corresponds to a maximum cross-track position difference of 40 m at the equator. We show that this effect is caused by a periodic bias in the velocity solutions with a maximum at the poles. A possible cause for such a periodicity in the velocity solutions may be dynamic effects in the receiver tracking loops related to the LEO satellite velocity relative to the GNSS constellation, which can vary strongly within one revolution.</p><p>Our results show that both, the radial position offset and the along-track velocity offset are dependent on the altitude of the satellite and are likely to be caused by ionospheric refraction. The explanation for the along-track position offset and the along-track velocity offset, however, is not that obvious. We found that these two offsets are geometrically related and, thus, must have the same physical cause. Based on the combined position-and-velocity approach we demonstrate that they originate from a velocity bias rather than from a position bias. To explain the physical cause of such a radial velocity offset, we will study the ionospheric effects on GNSS code and Doppler measurements in more detail, where we use a 3D-ionosphere model and take also the altitude of the two satellites into account.</p>

EVALUATION OF THE PURSUIT TIME IN DIFFERENTIAL GAMES OF MANY PLAYERS ON A CONVEX COMPACT

The paper is devoted to the study of the problem of constructing a pursuit strategy in simple differential games of many persons with phase constraints in the state of the players, in the sense of getting into a certain neighborhood of the evader. The game takes place in -dimensional Euclidean space on a convex compact set. The pursuit problem is considered when the number of pursuing players is , that is, less than , in the sense of — captures. A structure for constructing pursuit controls is proposed, which will ensure the completion of the game in a finite time. An upper bound is obtained for the game time for the completion of the pursuit. An auxiliary problem of simple pursuit on a unit cube in the first orthant is considered, and strategies of pursuing players are constructed to complete the game with special initial positions. The results obtained are used to solve differential games with arbitrary initial positions. For this task, a structure for constructing a pursuit strategy is proposed that will ensure the completion of the game in a finite time. The generalization of the problem in the sense of complicating the obstacle is also considered. A more general problem of simple pursuit on a cube of arbitrary size in the first orthant is considered. With the help of the proposed strategies, the possibilities of completing the pursuit are proved and an estimate of the time is obtained. As a consequence of this result, lower and upper bounds are obtained for the pursuit time in a game with ball-type obstacles. Estimates are obtained for the pursuit time when the compact set is an arbitrarily convex set. The concept of a convex set in a direction relative to a section, which is not necessarily convex, is defined. And in it the problem of simple pursuit in a differential game of many players is studied and the possibilities of completing the pursuit using the proposed strategy are shown. The time of completion of the pursuit of the given game is estimated from above.

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.

EKSPLORASI KETERAMPILAN KERUANGAN MENGENAI KUBUS MELALUI WAWANCARA KLINIS BERBASIS MASALAH

Students’ spatial skills in solving problems related to the arrangement of cubes were relatively low. This research was conducted to describe students' spatial skills regarding cube arrangement through problem-based clinical interviews. The design of this study was descriptive exploratory using one class of students as the research sample. Student tests were used to see students' initial abilities regarding cube arrangements. While problem-based clinical interviews were used to dig deeper into spatial skills and student problem solving regarding cube arrangements. The data obtained was processed using descriptive statistics such as average, cumulative frequency and percent score. Furthermore, the data was analyzed and interpreted using qualitative data analysis techniques to explain the phenomena that occur in the field. The results of the study show that the students' spatial skills regarding the cube were still relatively low; students' spatial skills were not in accordance with the concepts related to solving the problem of unit cube arrangements; students' spatial skills did not show skills related to solving problems in unit cube arrangements; to transform students' skills into spatial skills related to the problem of cube arrangement it is necessary to re-emphasize spatial skills which include spatial visual, spatial perception, spatial relations, and spatial orientation.Keywords: Spatial Ability, Problem Based Clinical Interview

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.