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With various jet orders $k$ and weights $n$, let $E_{k,n}^{\rm GG}$ be the Green-Griffiths bundles over the projective space $\mathbb{P}^N (\mathbb{C})$. Denote by $\mathcal{O} (d)$ the tautological line bundle over $\mathbb{P}^N (\mathbb{C})$. Although only negative twists are of interest for applications to complex hyperbolicity (above general type projective submanifolds $Y \subset \mathbb{P}^N (\mathbb{C})$), it is known that the positive twists $E_{k,n}^{\rm GG} \otimes \mathcal{O} (d)$ enjoy nontrivial global sections. In this article, we establish that for every $d \geqslant 1$ and for every jet order $k \geqslant d-1$: \[ \dim\, H^0 \bigg( \mathbb{P}^N,\,\, \bigoplus_{n=1}^{\infty} E_{k, n}^{\text{GG}} \otimes \mathcal{O}(d) \bigg) = (N+1)^d. \] This theorem is actually a corollary of a recent work of Etesse, devoted to a proof, from the point of view of differentially homogeneous polynomials, of the so-called Schmidt-Kolchin-Reinhart conjecture, by means of (advanced) Representation Theory. As Etesse discovered a (simple) tight link with the Green-Griffiths bundles, both statements are in fact equivalent. Our objective is to set up an alternative proof of the above precise dimension estimate, from the Green-Griffiths point of view (only). More precisely, we find an explicit description of all concerned global sections. Our arguments are elementary, and use only determinants, linear algebra, monomial orderings. One old hope is to discover some explicit formulas for global sections of negatively twisted Green-Griffiths bundles over projective general type submanifolds $Y \subset \mathbb{P}^N (\mathbb{C})$, a problem still open., Comment: 32 pages

A graph class $\mathcal{G}$ admits product structure if there exists a constant $k$ such that every $G \in \mathcal{G}$ is a subgraph of $H \boxtimes P$ for a path $P$ and some graph $H$ of treewidth $k$. Famously, the class of planar graphs, as well as many beyond-planar graph classes are known to admit product structure. However, we have only few tools to prove the absence of product structure, and hence know of only a few interesting examples of classes. Motivated by the transition between product structure and no product structure, we investigate subclasses of intersection graphs in the plane (e.g., disk intersection graphs) and present necessary and sufficient conditions for these to admit product structure. Specifically, for a set $S \subset \mathbb{R}^2$ (e.g., a disk) and a real number $\alpha \in [0,1]$, we consider intersection graphs of $\alpha$-free homothetic copies of $S$. That is, each vertex $v$ is a homothetic copy of $S$ of which at least an $\alpha$-portion is not covered by other vertices, and there is an edge between $u$ and $v$ if and only if $u \cap v \neq \emptyset$. For $\alpha = 1$ we have contact graphs, which are in most cases planar, and hence admit product structure. For $\alpha = 0$ we have (among others) all complete graphs, and hence no product structure. In general, there is a threshold value $\alpha^*(S) \in [0,1]$ such that $\alpha$-free homothetic copies of $S$ admit product structure for all $\alpha > \alpha^*(S)$ and do not admit product structure for all $\alpha < \alpha^*(S)$. We show for a large family of sets $S$, including all triangles and all trapezoids, that it holds $\alpha^*(S) = 1$, i.e., we have no product structure, except for the contact graphs (when $\alpha= 1$). For other sets $S$, including regular $n$-gons for infinitely many values of $n$, we show that $0 < \alpha^*(S) < 1$ by proving upper and lower bounds., Comment: An extended abstract of this paper appears in the proceedings of the 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

We determine all affinely homogeneous hypersurfaces S^3 in R^4 whose Hessian is (invariantly) of constant rank 2, including the simply transitive ones. We find 34 inequivalent terminal branches yielding each to a nonempty moduli space of homogeneous models of hypersurfaces S^3 in R^4, sometimes parametrized by a certain complicated algebraic variety, especially for the 15 (over 34) families of models which are simply transitive. We employ the power series method of equivalence, which captures invariants at the origin, creates branches, and infinitesimalizes calculations. In Lie's original classification spirit, we describe the found homogeneous models by listing explicit Lie algebras of infinitesimal transformations, sometimes parametrized by absolute invariants satisfying certain algebraic equations., Comment: 49 pages

We determine all affinely homogeneous models for surfaces $S^2 \subset \mathbb{R}^4$, including the simply transitive models. We employ an improved power series method of equivalence, which captures invariants at the origin, creates branches, and infinitesimalizes calculations. We find several inequivalent terminal branches yielding each to some nonempty moduli space of homogeneous models, sometimes parametrized by a certain invariant algebraic variety. Three main features may be emphasized: 1) Iterated single-pointed jet bundles; 2) Cartan-enhanced power series method of equivalence; 3) Constant ping-pong between normal forms (nf) and vector fields (vf)., Comment: 135 pages, 22 figures

In a joint effort, MeteoSwiss and Deutscher Wetterdienst (DWD) address the need for improving the initial state of the atmospheric boundary layer (ABL) by exploiting ground-based profiling observations that aim to fill the existing observational gap in the ABL. We implemented the brightness temperature observations from ground-based microwave radiometers (MWRs) into our data assimilation systems using a local ensemble transform Kalman filter (LETKF) with RTTOV-gb (Radiative Transfer for TOVS, ground-based) as a forward operator. We were able to obtain a positive impact on the brightness temperature first guess and analysis as well as a slight impact on the ABL humidity using two MWRs at MeteoSwiss. These results led to a subsequent operational implementation of the observing system at MeteoSwiss. Furthermore, we performed an extensive set of assimilation experiments at DWD to further investigate various aspects such as the vertical localisation of selected single channels. We obtained a positive impact on the 6h-forecast of ABL temperature and humidity by assimilating two channels employing a dynamical localisation based on the sensitivity functions of RTTOV-gb but also with a static localisation in a single-channel setup. Our experiments indicate the importance of the vertical localisation when using more than one channel, although reliable improvements are challenging to obtain without a larger number of observations for both assimilation and verification.

A linear layout of a graph $ G $ consists of a linear order $\prec$ of the vertices and a partition of the edges. A part is called a queue (stack) if no two edges nest (cross), that is, two edges $ (v,w) $ and $ (x,y) $ with $ v \prec x \prec y \prec w $ ($ v \prec x \prec w \prec y $) may not be in the same queue (stack). The best known lower and upper bounds for the number of queues needed for planar graphs are 4 [Alam et al., Algorithmica 2020] and 42 [Bekos et al., Algorithmica 2022], respectively. While queue layouts of special classes of planar graphs have received increased attention following the breakthrough result of [Dujmovi\'c et al., J. ACM 2020], the meaningful class of bipartite planar graphs has remained elusive so far, explicitly asked for by Bekos et al. In this paper we investigate bipartite planar graphs and give an improved upper bound of 28 by refining existing techniques. In contrast, we show that two queues or one queue together with one stack do not suffice; the latter answers an open question by Pupyrev [GD 2018]. We further investigate subclasses of bipartite planar graphs and give improved upper bounds; in particular we construct 5-queue layouts for 2-degenerate quadrangulations., Comment: An extended abstract of this paper appears in the Proceedings of the 18th International Symposium on Algorithms and Data Structures, WADS 2023

In this companion paper to our article {\em Accidental CR structures} (arxiv.org, January 2023), thought of as an appendix not submitted for publication, we provide complete explicit lists of infinitesimal CR automorphisms for the concerned CR models having respective Lie algebra structures: $${\bf E}_{II}, \qquad\ {\bf E}_{III}, \qquad\ \mathfrak{so}(\ell-1,\ell+1), \qquad\ \mathfrak{su}(p,q).$$ We start from our lists of {\em quadric} CR submanifolds $M^{2n+c} \subset \mathbb{C}^{n+c}$ of codimension $c >1$ which are shown to be {\em accidental}, in the sense that their CR symmetry groups are {\em equal to} (and not smaller than) the symmetry groups of the underlying real distribution structures -- after forgetting the complex structure. Thanks to intensive symbolic computer explorations, we then determine embedded vector field generators of these CR symmetries Lie algebras, and we express them in {\em extrinsic} holomorphic coordinates, because intrinsic formulas would be too extended to be shown.

We noticed a discrepancy between \'Elie Cartan and Sigurdur Helgason about the lowest possible dimension in which the simple exceptional Lie group ${\bf E}_8$ can be realized. This raised the question about the lowest dimensions in which various real forms of the exceptional groups ${\bf E}_\ell$ can be realized. Cartan claims that ${\bf E}_6$ can be realized in dimension 16. However Cartan refers to the complex group ${\bf E}_6$, or its split real form $E_I$. His claim is also valid in the case of the real form denoted by $E_{IV}$. We find however that the real forms $E_{II}$ and $E_{III}$ of ${\bf E}_6$ can not be realized in dimension 16 \`a la Cartan. In this paper we realize them in dimension 24 as groups of CR automorphisms of certain CR structures of higher codimension. As a byproduct of these two realizations, we provide a full list of CR structures $(M,H,J)$ and their CR embeddings in an appropriate ${\bf C}^N$, which satisfy the following conditions: (1) they have real codimension $k>1$, (2) the real vector distribution $H$ proper for the action of the complex structure $J$ is such that $[H,H]+H=TM$, (3) the local group $G_J$ of CR automorphisms of the structure $(M,H,J)$ is simple, acts transitively on $M$ and has isotropy $P$ being a parabolic subgroup in $G_J$, (4) the local symmetry group $G$ of the vector distribution $H$ on $M$ coincides with the group $G_J$ of CR automorphisms of $(M,H,J)$. Because all the CR structures from our list satisfy the last property we call them accidental. Our CR structures of higher codimension with the exceptional symmetries $E_{II}$ and $E_{III}$ are particular entries in this list.

The stack number of a directed acyclic graph $G$ is the minimum $k$ for which there is a topological ordering of $G$ and a $k$-coloring of the edges such that no two edges of the same color cross, i.e., have alternating endpoints along the topological ordering. We prove that the stack number of directed acyclic outerplanar graphs is bounded by a constant, which gives a positive answer to a conjecture by Heath, Pemmaraju and Trenk [SIAM J. Computing, 1999]. As an immediate consequence, this shows that all upward outerplanar graphs have constant stack number, answering a question by Bhore et al. [Eur. J. Comb., 2023] and thereby making significant progress towards the problem for general upward planar graphs originating from Nowakowski and Parker [Order, 1989]. As our main tool we develop the novel technique of directed $H$-partitions, which might be of independent interest. We complement the bounded stack number for directed acyclic outerplanar graphs by constructing a family of directed acyclic 2-trees that have unbounded stack number, thereby refuting a conjecture by N\"ollenburg and Pupyrev [GD 2023].

We study the dielectric function of CuBr$_\mathrm{x}$I$_{1-\mathrm{x}}$ thin film alloys using spectroscopic ellipsometry in the spectral range between 0.7 eV to 6.4 eV, in combination with first-principles calculations based on density functional theory. Through the comparison of theory and experiment, we attribute features in the dielectric function to electronic transitions at specific k-points in the Brillouin zone. The observed bandgap bowing as a function of alloy composition is discussed in terms of different physical and chemical contributions. The band splitting at the top of the valence band due to spin-orbit coupling is found to decrease with increasing Br-concentration, from a value of 660 meV for CuI to 150 meV for CuBr. This result can be understood considering the contribution of copper d-orbitals to the valence band maximum as a function of the alloy composition., Comment: Michael Seifert and Evgeny Kr\"{u}ger contributed equally. Some typos were corrected and references were added

Nowadays cross-industry ranging challenges include the reduction of greenhouse gas emission and enabling a circular economy. However, the production of paper from waste paper is still a highly resource intensive task, especially in terms of energy consumption. While paper machines produce a lot of data, we have identified a lack of utilization of it and implement a concept using an operator assistance system and state-of-the-art machine learning techniques, e.g., classification, forecasting and alarm flood handling algorithms, to support daily operator tasks. Our main objective is to provide situation-specific knowledge to machine operators utilizing available data. We expect this will result in better adjusted parameters and therefore a lower footprint of the paper machines.

In a previous memoir 2202.03030, we showed that in every dimension $n \geq 5$, there exists (unexpectedly) no affinely homogeneous hypersurface $H^n \subset \mathbb{R}^{n+1}$ having Hessian of constant rank 1 (and not being affinely equivalent to a product with $\mathbb{R}^{m \geqslant 1}$). The present work is devoted to determine all non-product constant Hessian rank 1 affinely homogeneous hypersurfaces $H^n \subset \mathbb{R}^{n+1}$ in dimensions $n = 2, 3, 4$, the cases $n = 1, 2$ being known. With complete details in the case $n = 2$, we illustrate the main features of what can be termed the "Power Series Method of Equivalence". The gist is to capture invariants at the origin only, to create branches, and to infinitesimalize calculations. In dimension $n = 3$, we find a single homogeneous model: \[ u \,=\, \frac{1}{3\,z^2} \Big\{ \big( 1-2\,y+y^2-2\,xz \big)^{3/2} - (1-y)\, \big( 1-2\,y+y^2-3\,xz \big) \Big\}, \] the singularity $\frac{1}{3z^2}$ being illusory. In dimension $n = 4$, without reaching closed forms, we find two simply homogeneous models, differing by some $\pm$ sign., Comment: 29 pages

A squaregraph is a plane graph in which each internal face is a $4$-cycle and each internal vertex has degree at least 4. This paper proves that every squaregraph is isomorphic to a subgraph of the semi-strong product of an outerplanar graph and a path. We generalise this result for infinite squaregraphs, and show that this is best possible in the sense that "outerplanar graph" cannot be replaced by "forest".

We prove that the stack-number of the strong product of three $n$-vertex paths is $\Theta(n^{1/3})$. The best previously known upper bound was $O(n)$. No non-trivial lower bound was known. This is the first explicit example of a graph family with bounded maximum degree and unbounded stack-number. The main tool used in our proof of the lower bound is the topological overlap theorem of Gromov. We actually prove a stronger result in terms of so-called triangulations of Cartesian products. We conclude that triangulations of three-dimensional Cartesian products of any sufficiently large connected graphs have large stack-number. The upper bound is a special case of a more general construction based on families of permutations derived from Hadamard matrices. The strong product of three paths is also the first example of a bounded degree graph with bounded queue-number and unbounded stack-number. A natural question that follows from our result is to determine the smallest $\Delta_0$ such that there exist a graph family with unbounded stack-number, bounded queue-number and maximum degree $\Delta_0$. We show that $\Delta_0\in \{6,7\}$.

Equivalences under the affine group ${\rm Aff} (\mathbb{R}^3)$ of constant Hessian rank $1$ surfaces $S^2 \subset \mathbb{R}^3$, sometimes called parabolic, were, among other objects, studied by Doubrov, Komrakov, Rabinovich, Eastwood, Ezhov, Olver, Chen, Merker, Arnaldsson, Valiquette. Especially, homogeneous models and algebras of differential invariants in various branches have been fully understood. Then what about higher dimensions? We consider hypersurfaces $H^n \subset \mathbb{R}^{n+1}$ graphed as $\{ u = F(x_1, \dots, x_n) \}$ whose Hessian matrix $(F_{x_i x_j})$, a relative affine invariant, is, similarly, of constant rank $1$. Are there homogeneous models? Complete explorations were done by the author on a computer in dimensions $n = 2, 3, 4, 5, 6, 7$. The first, expected outcome, was to obtain a complete classification of homogeneous models in dimensions $n = 2, 3, 4$ (forthcoming article, case $n = 2$ already known). The second, unexpected outcome, was that in dimensions $n = 5, 6, 7$, there are no affinely homogenous models! (Except those that are affinely equivalent to a product of $\mathbb{R}^m$ with a homogeneous model in dimensions $2, 3, 4$.) The present article establishes such a non-existence result in every dimension $n \geqslant 5$, based on the production of a normal form for $\{ u = F(x_1, \dots, x_n) \}$ under ${\rm Aff} (\mathbb{R}^{n+1})$, up to order $\leqslant n+5$, valid in any dimension $n \geqslant 2$., Comment: 68 pages, 0 figure

While lots of research has been conducted on the architecture of Industrial Internet of Things (IIoT) systems, concepts of structuring their development processes are missing. Therefore, we propose a holistic approach supporting organizations in rapid development of IIoT systems. It includes the structuring of the development process into multiple projects sharing project conventions. Utilizing a single configurable build script for all projects, our goal is to make the integration of code from various projects into broader IIoT systems easy and the systems decomposable with minimal effort.

This article gives an account of a teaching experience carried out from 2008 to 2021 at the university of Franche-Comt{\'e} as an answer to the ministerial command of proposing cross-disciplinary courses in the curricula. The goal of the experience was to develop simultaneously a discourse on mathematics and a discourse on literature, two independent discourses, but each filled with the gap between one domain and the other and mindful of the flashes of thought that are lightening from one to the other, witnesses of unity in their reasoning. -- Cet article rend compte d'une exp{\'e}rience d'enseignement men{\'e}e de 2008 {\`a} 2012 {\`a} l'universit{\'e} de Franche-Comt{\'e} en r{\'e}ponse {\`a} l'injonction minist{\'e}rielle de proposer des unit{\'e}s transversales dans les maquettes de dipl{\^o}me. Le but de cette exp{\'e}rience a {\'e}t{\'e} de d{\'e}velopper {\`a} la fois un discours sur les math{\'e}matiques et un discours sur la litt{\'e}rature, deux discours ind{\'e}pendants, mais chacun rempli de l'{\'e}cart entre l'un des domaines et l'autre, et attentif aux {\'e}clairs de la pens{\'e}e qui jaillissent de l'un vers l'autre, t{\'e}moignages d'unit{\'e} dans la d{\'e}marche intellectuelle., Comment: in French

We study the equivalence problem of classifying second order ordinary differential equations $y_{xx}=J(x,y,y_{x})$ modulo fibre-preserving point transformations $x\longmapsto \varphi(x)$, $y\longmapsto \psi(x,y)$ by using Moser's method of normal forms. We first compute a basis of the Lie algebra ${\frak{g}}_{{\{y_{xx}=0\}}}$ of fibre-preserving symmetries of $y_{xx}=0$. In the formal theory of Moser's method, this Lie algebra is used to give an explicit description of the set of normal forms $\mathcal{N}$, and we show that the set is an ideal in the space of formal power series. We then show the existence of the normal forms by studying flows of suitable vector fields with appropriate corrections by the Cauchy-Kovalevskaya theorem. As an application, we show how normal forms can be used to prove that the identical vanishing of Hsu-Kamran primary invariants directly imply that the second order differential equation is fibre-preserving point equivalent to $y_{xx}=0$., Comment: 27 pages

A page (queue) with respect to a vertex ordering of a graph is a set of edges such that no two edges cross (nest), i.e., have their endpoints ordered in an ABAB-pattern (ABBA-pattern). A union page (union queue) is a vertex-disjoint union of pages (queues). The union page number (union queue number) of a graph is the smallest $ k $ such that there is a vertex ordering and a partition of the edges into $ k $ union pages (union queues). The local page number (local queue number) is the smallest $ k $ for which there is a vertex ordering and a partition of the edges into pages (queues) such that each vertex has incident edges in at most $ k $ pages (queues). We present upper and lower bounds on these four parameters for the complete graph $ K_n $ on $ n $ vertices. In three cases we obtain the exact result up to an additive constant. In particular, the local page number of $ K_n $ is $ n/3 \pm O(1) $, while its local and union queue number is $ (1-1/\sqrt{2})n \pm O(1) $. The union page number of $ K_n $ is between $ n/3 - O(1) $ and $ 4n/9 + O(1) $., Comment: Appears in the Proceedings of the 29th International Symposium on Graph Drawing and Network Visualization (GD 2021)