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This letter from the editor presents Vol. 25(2) of the Nexus Network Journal. This issue has two connected themes, the first is about the role of regular modules, patterns, or proportions in architecture, and the second, the ways rules or systems underpin architectural form. The papers in this issue address either regularity, rules, or in most cases, both, because spatial and formal patterns are typically derived from rules. Collectively, the eleven research contributions in this issue have topics that span chronologically from the thirteenth century to the present day, and geographically from Asia to Africa and Europe. [ABSTRACT FROM AUTHOR]

Presents several Origami-based mathematical problems. Application of the activity of paper folding in solving math puzzles using the angles, geometric shapes and creases of the paper; Construction of paper shapes based on mathematical rules and parameters provided by the puzzle.

Nexus Network Journal guest editor Roberta Spallone introduces papers selected from the Nexus Conference 2023 in Turin for detailed review, expansion and development. These papers identify close links between architecture, mathematics, and specific areas of interest related to historical periods, architectural cultures, typological elements, and analysis tools. [ABSTRACT FROM AUTHOR]

In Kim et al. (Nagoya Math J 178: 37-53, 2005), Lee and Lee (J Geometry Phys 56(10): 2011-2023, 2006), the authors developed a nice formula to compute the Nielsen number of a self-map on an infra-nilmanifold. For the case of nilmanifolds this formula was extended to n-valued maps in Deconinck and Dekimpe (J Fixed Point Theory Appl 25(4): Paper No. 84, 29, 2023). In this paper, we extend these results further and establish the averaging formula to compute the Nielsen number of any n-valued affine map on an infra-nilmanifold. [ABSTRACT FROM AUTHOR]

Poincaré's conventionalism has been interpreted in many writings as a philosophical position emerged by reflection on certain scientific problems, such as the applicability of geometry to physical space or the status of certain scientific principles. In this paper I would like to consider conventionalism as a philosophical position that emerged from Poincaré's scientific practice. But not so much from dealing with scientific problems, as from the use of two specific methodologies proper to modern mathematics and the modern natural sciences: methodological structuralism and the hypothetical-deductive method—thus, as a philosophical position which emerged from a way (or rather, two ways) of doing science. With this approach, I try to deepen the analysis of connections between Poincaré's scientific practice and his philosophy. [ABSTRACT FROM AUTHOR]

Modern color science finds its birth in the middle of the nineteenth century. Among the chief architects of the new color theory, the name of the polymath Hermann von Helmholtz stands out. A keen experimenter and profound expert of the latest developments of the fields of physiological optics, psychophysics, and geometry, he exploited his transdisciplinary knowledge to define the first non-Euclidean line element in color space, i.e., a three-dimensional mathematical model used to describe color differences in terms of color distances. Considered as the first step toward a metrically significant model of color space, his work inaugurated researches on higher color metrics, which describes how distance in the color space translates into perceptual difference. This paper focuses on the development of Helmholtz's mathematical derivation of the line element. Starting from the first experimental evidence which opened the door to his reflections about the geometry of color space, it will be highlighted the pivotal role played by the studies conducted by his assistants in Berlin, which provided precious material for the elaboration of the final model proposed by Helmholtz in three papers published between 1891 and 1892. Although fallen into oblivion for about three decades, Helmholtz's masterful work was rediscovered by Schrödinger and, since the 1920s, it has provided the basis for all subsequent studies on the geometry of color spaces up to the present time. [ABSTRACT FROM AUTHOR]

In this paper, a joint study of the behavior of solutions of the Heavy Ball ODE and Heavy Ball type algorithms is given. Since the pioneering work of Polyak (USSR Comput Math Math Phys 4(5):1–17, 1964), it is well known that such a scheme is very efficient for C 2 strongly convex functions with Lipschitz gradient. But much less is known when only growth conditions are considered. Depending on the geometry of the function to minimize, convergence rates for convex functions, with some additional regularity such as quasi-strong convexity, or strong convexity, were recently obtained in Aujol et al. (Convergence rates of the Heavy-Ball method for quasi-strongly convex optimization, 2020). Convergence results with much weaker assumptions are given in the present paper: namely, linear convergence rates when assuming a growth condition (which amounts to a Łojasiewicz property in the convex case). This analysis is firstly performed in continuous time for the ODE, and then transposed for discrete optimization schemes. In particular, a variant of the Heavy Ball algorithm is proposed, which converges geometrically whatever the parameters choice, and which has the best state of the art convergence rate for first order methods to minimize composite non smooth convex functions satisfying a Łojasiewicz property. [ABSTRACT FROM AUTHOR]

Consider a point particle moving through a Poisson distributed array of cubes all oriented along the axes—the random wind-tree model introduced in Ehrenfest–Ehrenfest (1912) as reported by Ehrenfest, Ehrenfest (Begriffliche Grundlagen der statistischen Auffassung in der Mechanik Encykl. d. Math. Wissensch. IV 2 II, Heft 6, 90 S (1912) (Translated:) The conceptual foundations of the statistical approach in mechanics. Dover Books on Physics, 1912). We show that in the joint Boltzmann–Grad and diffusive limit this process satisfies an invariance principle. That is, the process converges in distribution to Brownian motion in a particular scaling limit. In a previous paper (2020) (Lutsko, Tóth in Commun. Math. Phys. 379:589–632, 2020) the authors used a novel coupling method to prove the same statement for the random Lorentz gas with spherical scatterers. In this paper we show that, despite the change in dynamics, a similar strategy with some modification can be used to prove the invariance principle for the random wind-tree model. The key differences from our previous work are that the individual path segments of the underlying Markov process are no longer fully independent and the geometry of recollisions is simpler. [ABSTRACT FROM AUTHOR]

This paper develops an improved tri-coloured rooted-tree theory for the order conditions for ERKN methods solving general multi-frequency and multidimensional second-order oscillatory systems. The bottleneck of the original tricoloured rooted-tree theory is the existence of numerous redundant trees. In light of the fact that the sum of the products of the symmetries and the elementary differentials is meaningful, this paper naturally introduces the so-called extended elementary differential mappings. Then, the new improved tri-coloured rooted tree theory is established based on a subset of the original tri-coloured rooted-tree set. This new theory makes all redundant trees disappear, and thus, the order conditions of ERKN methods for general multi-frequency and multidimensional second-order oscillatory systems are reduced greatly. Furthermore, with this new theory, we present some new ERKN methods of order up to four. Numerical experiments are implemented and the results show that ERKN methods can be competitive with other existing methods in the scientific literature, especially when comparatively large stepsizes are used. [ABSTRACT FROM AUTHOR]

Between 1873 and 1876, Felix Klein published a series of papers that he later placed under the rubric anschauliche Geometrie in the second volume of his collected works (1922). The present study attempts not only to follow the course of this work, but also to place it in a larger historical context. Methodologically, Klein's approach had roots in Poncelet's principle of continuity, though the more immediate influences on him came from his teachers, Plücker and Clebsch. In the 1860s, Clebsch reworked some of the central ideas in Riemann's theory of Abelian functions to obtain complicated results for systems of algebraic curves, most published earlier by Hesse and Steiner. These findings played a major role in enumerative geometry, whereas Plücker's work had a strongly qualitative character that imbued Klein's early studies. A leitmotif in these works can be seen in the interplay between real curves and surfaces as reflected by their transformational properties. During the early 1870s, Klein and Zeuthen began to explore the possibility of deriving all possible forms for real cubic surfaces as well as quartic curves. They did so using continuity methods reminiscent of Poncelet's earlier approach. Both authors also relied on visual arguments, which Klein would later advance under the banner of intuitive geometry (anschauliche Geometrie). [ABSTRACT FROM AUTHOR]

In view of a better understanding of the geometry of scalar flat Kähler metrics, this paper studies two families of scalar flat Kähler metrics constructed by Hwang and Singer (Trans Am Math Soc 354(6):2285–2325, 2002) on C n + 1 and on O (- k) . For the metrics in both the families, we prove the existence of an asymptotic expansion for their ϵ -functions and we show that they can be approximated by a sequence of projectively induced Kähler metrics. Further, we show that the metrics on C n + 1 are not projectively induced, and that the Burns–Simanca metric is characterized among the scalar flat metrics on O (- k) to be the only projectively induced one as well as the only one whose second coefficient in the asymptotic expansion of the ϵ -function vanishes. [ABSTRACT FROM AUTHOR]

Mine blasting produces excavation geometry changes which induce stress change that can be observed in the seismic source parameter apparent stress calculated for local seismic events. Using high apparent stress as a proxy for increasing stress within a rock mass, areas experiencing increases in the local stress conditions can be determined. This paper presents the use of apparent stress of seismic events to identify areas within a rock mass experiencing local stress change. Examples from a deep Canadian mine, operating in excess of 2900 m below surface, are provided. [ABSTRACT FROM AUTHOR]

The teaching of Mathematics – in particular, Geometry, through conventional methods has been a challenging task for tutors. Augmented Reality (AR) based applications available in commercial space, have not followed any structured pedagogical approach in the designing process, and also do not ensure that the learning time of students is spent prolifically. In this paper, we explore the use of AR in mathematics for geometry education, to aid visualization of multidimensional objects and long-term retention of concepts by the learners. For designing an appropriate AR application it is necessary to identify some principles which support better memory retention of the students. The application has been specifically designed on the basis of identified principles affecting memory retention. We further explain the development of an AR-based Geometry Learning Assistant (AR-GLA), using a structural approach to pedagogical-design for teaching 3-dimensional geometry to higher school students through improved visualization and enhance their memory retention for related concepts. A sample of 54 K-12 students and 2 teachers with expertise in mathematics were part of the experiment. The students were divided into two different groups; one of the groups was taught with AR-based content whereas the other group was given Interactive Simulation (IS) based learning. The results illustrated that AR-based learning provides better retention of memory as compared to IS-based learning were tested over a period of two months. [ABSTRACT FROM AUTHOR]

This paper addresses an article by Felix Klein of 1886, in which he generalized his theory of polynomial equations of degree 5—comprehensively discussed in his Lectures on the Icosahedron two years earlier—to equations of degree 6 and 7. To do so, Klein used results previously established in line geometry. I review Klein's 1886 article, its diverse mathematical background, and its place within the broader history of mathematics. I argue that the program advanced by this article, although historically overlooked due to its eventual failure, offers a valuable insight into a time of crucial evolution of the subject. [ABSTRACT FROM AUTHOR]

NNJ guest editor Marco Giorgio Bevilacqua introduces the papers in the special issue of the Nexus Network Journal dedicated to Military Architecture and Mathematics. [ABSTRACT FROM AUTHOR]

Subgroups of the spinor group Spin(2l + 1,K) with l ≥ 3 over a field K such that 2 ∈ K* and |K| ≥ 9, which contain a split maximal torus, are described. We prove that the description of these subgroups is standard in two cases: (1) l is even; (2) l is odd and -1 ∈ K*. It is shown that, as in the papers of N. A. Vavilov and V. Hołubovsky devoted to subgroups of the orthogonal group, one can reduce the odd case to the case of even n = 2l. However, here the calculations are somewhat more involved, since we can only use diagonal elements of Spin(2l + 1,K). Moreover, the results of N. A. Vavilov pertaining to the even case are strengthened by relaxing the condition on the field K to |K| ≥ 9. Bibliography: 17 titles. [ABSTRACT FROM AUTHOR]

In this note we introduce a new technique to answer an issue posed in Fávaro et al. (Bull Braz Math Soc 51:27–46, 2020) concerning geometric properties of the set of non-surjective linear operators. We also extend and improve a related result from the same paper. [ABSTRACT FROM AUTHOR]

Topology Control is one of the most important techniques used in wireless multi-hop networks to obtain the desired network property. The most conventional MST-based topology control schemes both achieve a very good performance in terms of transmission power and implicitly consider network capacity, but they hardly have flexibility in determining whether more importance is placed on network capacity or not. Therefore, we jointly consider network capacity and energy efficiency in this paper. A new localized topology control scheme is proposed to meet the above two competing objectives by finding a balance between them through using Get_min- cost_for_link_(). The simulation results show that, when compared with some typical MST-based schemes, the proposed scheme can achieve the desired performance in terms of network capacity under the appropriate combination of weight values though it has a slightly worse performance in terms of transmission power. Moreover, the proposed scheme achieves the most balanced distribution with respect to path capacity among the compared schemes, especially when both k-hop neighborhood size and network node density increase. More importantly, the proposed scheme can flexibly determine that either network capacity or energy consumption should be given more attention according to the demand of the network application. [ABSTRACT FROM AUTHOR]

In this paper, we deal with issues related to global multiplicity of W loc 1 , p (Ω) -solutions for the very-singular and non-local μ -problem - g ∫ Ω u q Δ p u = μ u - δ + u β in Ω , u > 0 in Ω and u = 0 on ∂ Ω , where Ω ⊂ R N is a smooth bounded domain, δ > 0 , q > 0 , 0 < β ≤ p - 1 and g : [ 0 , ∞) → [ 0 , ∞) is a continuous function that achieves critical values for the class of non-local problems (i.e., the level zero if β < p - 1 and 1 / λ 1 if β = p - 1 , where λ 1 stands for the principal eigenvalue of the p-Laplacian in Ω under homogeneous Dirichlet boundary conditions). To overcome the difficulties arising from the geometry of g and the presence of very-singular term combined with a (p - 1) -sublinear/asymptotically linear ones, we take advantage of a comparison principle for sub-supersolutions in W loc 1 , p (Ω) -sense proved in Santos and Santos (Z Angew Math Phys 69:Art. 145, 2018), together with sub-supersolutions techniques and bifurcation theory. [ABSTRACT FROM AUTHOR]

In this paper, we study variants of the classical geometric shortest path problem inside a simple polygon, where we allow a part of the path to go outside the polygon. Let P be a simple polygon consisting of n vertices and let s, t be a pair of points in P. Let int(P) represent the interior of P and let P¯ represent the exterior of P, i.e. int(P)=P\∂(P) and P¯=R2\int(P). For an integer k≥0, we define a k-violation path from s to t to be a path connecting s and t such that its intersection with P¯ consists of at most k segments. There is no restriction in terms of the number of segments of the path within P. The objective is to compute a path of minimum Euclidean length among all possible (≤k)-violation paths from s to t. In this paper, we study this problem for k=1 and propose an algorithm that computes the shortest one-violation path in O(n3) time. We show that for rectilinear polygons, the minimum length rectilinear one-violation path can be computed in O(nlogn) time. We extend the concept of one-violation path to a one-stretch violation path. In this case, the path between s and t is composed of (a) a path in P from s to a vertex u of P, (b) a path in P¯ between u and a vertex v of P, and (c) a path in P between v and t. We show that a minimum length one-stretch violation path can be computed in O(nlognloglogn) time. Next, we introduce one- and two-violation monotone rectilinear path problems among a set of n disjoint axis-parallel rectangular objects. Let s, t be two points in R2 that are not in the interior of any of the objects. In the case of one-violation monotone path problem, the desired rectilinear path from s to t consists of horizontal edges that are directed towards the right and vertical edges that are directed upwards, except for at most one edge. Similarly, in the case of a two-violation monotone path problem, all horizontal edges are directed towards the right except at most one and all vertical edges are directed upwards except at most one. Our algorithms for both of these problems run in O(nlogn) time. [ABSTRACT FROM AUTHOR]

In this paper, we perform $$L^\infty $$ and $$W^{1,\infty }$$ error estimates for a class of bi- k finite volume schemes on a quadrilateral mesh for elliptic equations, where $$k\ge 2$$ is arbitrary. We show that the errors of the finite volume solution in both the $$L^\infty $$ and $$W^{1,\infty }$$ norms converge to zero with optimal orders, provided the solution $$u\in W^{k+2,\infty }$$ . Our analysis is based mainly on an estimate of the difference between the finite volume and the corresponding finite element bilinear forms, as well as some techniques derived for $$L^\infty $$ and $$W^{1,\infty }$$ estimates of the finite element method. Our theoretical findings are supported by several numerical examples. [ABSTRACT FROM AUTHOR]

PL homotopy methods are effective numerical methods for highly nonlinear problems. It is widely believed that the feasibility of a PL homotopy method depends on the nondegeneracy condition that the zero set (or the fixed point set in the case of finding fixed points instead of zeroes) of the PL approximation of the homotopy does not intersect the triangulation's skeletons of co-dimensions two and above. This paper shows that, although the sections of the PL approximation's zero set tracked by the PL homotopy method are of dimension one (while other sections may have higher dimensions), the paths generated by the pivoting method are potentially and essentially of dimension two. It makes path-crossing a safe thing. Thus, this paper first sets up the without exception feasibility of PL homotopy methods geometrically. [ABSTRACT FROM AUTHOR]

Perverse schobers are categorical analogs of perverse sheaves. Examples arise from varieties admitting flops, determined by diagrams of derived categories of coherent sheaves associated to the flop: in this paper we construct mirror partners to such schobers, determined by diagrams of Fukaya categories with stops, for examples in dimensions 2 and 3. Interpreting these schobers as supported on loci in mirror moduli spaces, we prove homological mirror symmetry equivalences between them. Our construction uses the coherent–constructible correspondence and a recent result of Ganatra et al. (Microlocal morse theory of wrapped fukaya categories. arXiv:1809.08807) to relate the schobers to certain categories of constructible sheaves. As an application, we obtain new mirror symmetry proofs for singular varieties associated to our examples, by evaluating the categorified cohomology operators of Bondal et al. (Selecta Math 24(1):85–143, 2018) on our mirror schobers. [ABSTRACT FROM AUTHOR]

The classical way to study a finite poset ( X, ≤ ) using topology is by means of the simplicial complex Δ X of its nonempty chains. There is also an alternative approach, regarding X as a finite topological space. In this article we introduce new constructions for studying X topologically: inspired by a classical paper of Dowker (Ann Math 56:84–95, ), we define the simplicial complexes K X and L X associated to the relation ≤. In many cases these polyhedra have the same homotopy type as the order complex Δ X. We give a complete characterization of the simplicial complexes that are the K or L-complexes of some finite poset and prove that K X and L X are topologically equivalent to the smaller complexes K′ X, L′ X induced by the relation <. More precisely, we prove that K X (resp. L X) simplicially collapses to K′ X (resp. L′ X). The paper concludes with a result that relates the K-complexes of two posets X, Y with closed relations R ⊂ X × Y. [ABSTRACT FROM AUTHOR]

This paper reports on a New Zealand iteration of the Encouraging Persistence, Maintaining Challenge (EPMC) project, which proposes that students learn mathematics best when they build connections between mathematical ideas for themselves. This iteration explores the actions, perceptions and learning of 12 primary teachers and their 281 students during the implementation of a set of challenging tasks related to geometric reasoning. The teachers launched the suggested tasks, ensuring that the challenge was maintained. The students explored these tasks with minimal input from the teacher, and learning was summarised and extended. The teachers were positive about the intervention. The challenging task approach enabled students' thinking became visible and, at times, the teachers' prior perceptions of their students' ability were challenged. A highly significant difference between the students' pre- and post-assessment scores was found. The students were supported to have autonomy in their learning and make mathematical connections themselves. The students became less reliant on their teachers' help and were positive about their involvement in the project. [ABSTRACT FROM AUTHOR]

Supersymmetric microstate geometries were recently conjectured (Eperon et al. in JHEP 10:031, 2016. 10.1007/JHEP10(2016)031) to be nonlinearly unstable due to numerical and heuristic evidence, based on the existence of very slowly decaying solutions to the linear wave equation on these backgrounds. In this paper, we give a thorough mathematical treatment of the linear wave equation on both two- and three-charge supersymmetric microstate geometries, finding a number of surprising results. In both cases, we prove that solutions to the wave equation have uniformly bounded local energy, despite the fact that three-charge microstates possess an ergoregion; these geometries therefore avoid Friedman's "ergosphere instability" (Friedman in Commun Math Phys 63(3):243–255, 1978). In fact, in the three-charge case we are able to construct solutions to the wave equation with local energy that neither grows nor decays, although these data must have non-trivial dependence on the Kaluza–Klein coordinate. In the two-charge case, we construct quasimodes and use these to bound the uniform decay rate, showing that the only possible uniform decay statements on these backgrounds have very slow decay rates. We find that these decay rates are sublogarithmic, verifying the numerical results of Eperon et al. (2016). The same construction can be made in the three-charge case, and in both cases the data for the quasimodes can be chosen to have trivial dependence on the Kaluza–Klein coordinates. [ABSTRACT FROM AUTHOR]

French mathematician Michel Chasles (1793–1880), a staunch defender of pure geometrical methods, is now mostly remembered as the author of the Aperçu historique (1837). In this book, he retraced the history of geometry in order to expound epistemological theses on what constitutes a virtuous practice of geometry. Amongst these stands out the assertion that the values of generality and simplicity in mathematics are intimately connected. In this paper, we flesh out this claim by analysing Chasles's geometrical solutions to the century-old problem of the attraction of the ellipsoids. We show how these solutions echo Chasles's evaluation of the relative strengths of geometrical and analytical methods, and how they embody a set of normative rules for the geometer's practice whose observance Chasles deemed necessary and sufficient for the development of general methods and theories. [ABSTRACT FROM AUTHOR]

In the study of the collapsed manifolds with bounded sectional curvature, the following two results provide basic tools: a (singular) fibration theorem by K. Fukaya [J. Differential Geom., 1987, 25(1): 139–156] and J. Cheeger, K. Fukaya, and M. Gromov [J. Amer. Math. Soc., 1992, 5(2): 327–372], and the stability for isometric compact Lie group actions on manifolds by R. S. Palais [Bull. Amer. Math. Soc., 1961, 67(4): 362–364] and K. Grove and H. Karcher [Math. Z., 1973, 132: 11–20]. The main results in this paper (partially) generalize the two results to manifolds with local bounded Ricci covering geometry. [ABSTRACT FROM AUTHOR]

In this paper, we consider a reaction-diffusion boundary value problem in a three-dimensional thin domain. The very different length scales in the geometry result in an anisotropy effect. Our study is motivated by a parabolic heat conduction problem in a thin foil leading to such anisotropic reaction-diffusion problems in each time step of an implicit time integration method [7]. The reaction-diffusion problem contains two important parameters, namely ε >0 which parameterizes the thickness of the domain and μ >0 denoting the measure for the size of the reaction term relative to that of the diffusion term. In this paper we analyze the convergence of a multigrid method with a robust (line) smoother. Both, for the W- and the V-cycle method we derive contraction number bounds smaller than one uniform with respect to the mesh size and the parameters ε and μ. [ABSTRACT FROM AUTHOR]

Abstract??The paper addresses the classical problem of optimal truss design where cross-sectional areas and the positions of joints are simultaneously optimized. Se-veral approaches are discussed from a general point of view. In particular, we focus on the difference between simultaneous and alternating optimization of geometry and topology. We recall a rigorously mathematical approach based on the implicit programming technique which considers the classical single load minimum compliance problem subject to a volume constraint. This approach is refined leading to three new problem formulations which can be treated by methods of Mathematical Programming. In particular, these formulations cover the effect of melting end nodes, i.e., vanishing potential bars due to changes in the geometry. In one of these new problem formulations, the objective function is a polynomial of degree three and the constraints are bilinear or just sign constraints. Because heuristics is avoided, certain optimality properties can be proven for resulting structures. The paper closes with two numerical test examples. [ABSTRACT FROM AUTHOR]

In this paper the doubly noncentral beta and F distributions are represented alternatively by using the results on the product of two hypergeometric functions. Their moments and the cumulative distribution functions are also given in terms of hypergeometric functions, which can be easily calculated by the Mathematica package. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to propose a noncommutative generalization of a gauge connection and the free-field Yang–Mills equations. The paper draws upon the techniques proposed by Heller et al. for the noncommutative generalization of the Einstein field equations. [ABSTRACT FROM AUTHOR]

The sequential organization of actions necessary to construct a figure in a dynamic geometry environment (DGE) introduces an explicit order of construction. Such sequential organization produces what is, in effect, a hierarchy of dependences, as elements of the construction depend on something created earlier. This hierarchy of dependences is one of the main factors that determine the dynamic behavior (DB) within a DGE, and the order is often explicitly stated by terms such as parent and child. This article is a part of a larger study that examines various instruments developed by users when they use dragging. It addresses one aspect of dragging: the connection between dynamic behavior and the sequential order of construction. Junior high students and graduate students in mathematics education were asked to predict the DB of points that were part of a geometric construction they had executed using a DGE according to a given procedure, and to explain their predictions. The study reveals that while hierarchy in geometric constructions in a DGE is mirrored by the DB, user actions and perceptions of DB indicate that users often grasp a reverse hierarchy in which dragging a child affects its parent. The study reveals four categories of reverse-order predictions, and suggests that these may be caused by terms and knowledge built into paper-and-pencil geometry, which are part of the resources users bring to dragging. Examining various DGEs1 in detail reveals that in some cases the DB is in fact compatible with the reverse-order predictions. The paper concludes with a brief discussion of some implications for learning activities and software design. [ABSTRACT FROM AUTHOR]

In the context of discrete curve evolution the following problem is of relevance: decompose the boundary of a plane digital object into convex and concave parts. Such a decomposition is very useful for describing the form of an object, e.g. for shape databases. Although the problem is relatively trivial in ordinary plane geometry, in digital geometry its statement becomes a very difficult task due to the fact that in digital geometry there is no simple set-complement duality. The paper is based on results given by Hübler et al. The main new contribution of the paper is the generalization of the concepts introduced by these authors to nonconvex sets. The digital geometric ?low level? segmentation of the boundary of a digital object can be used as a starting basis for further reduction of the boundary by means of discrete evolution. [ABSTRACT FROM AUTHOR]

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In this paper, we present some new refinements of the Erdös-Mordell inequality, Barrow's inequality, and Oppenheim's inequality. Based on verification by computer, several related interesting conjectures are put forward. [ABSTRACT FROM AUTHOR]

Unlike a jigsaw puzzle where a piece must fit in only one way to complete a picture, the seven pieces of a geometric tangram can be arranged in many different ways to make figures of birds, animals, people, and objects. This article presents a Tangram Teaching Guide to provide a five-step developmental sequence for teaching tangramming to young children. Teaching resources are included, and a checklist provides a summary outline of the teaching sequence. Tangram experiences help children develop positive attitudes toward geometry, further their shape identification and classification skills, and foster an understanding of basic geometric concepts and relationships. [ABSTRACT FROM AUTHOR]

In this paper, we present results dealing with properties of well-known geometric random problems in the plane, together with their motivations. The paper specifically concentrates on the traveling salesman and minimum spanning tree problems, even though most of the results apply to other problems such as the Steiner tree problem and the minimum weight matching problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider polytopes P that are contained in the unit hypercube. We provide conditions on the set of 0, 1 vectors not contained in P that guarantee that P has a small Chvátal rank. Our conditions are in terms of the subgraph induced by these infeasible 0, 1 vertices in the skeleton graph of the unit hypercube. In particular, we show that when this subgraph contains no 4-cycle, the Chvátal rank is at most 3; and when it has tree width 2, the Chvátal rank is at most 4. We also give polyhedral decomposition theorems when this graph has a vertex cutset of size one or two. [ABSTRACT FROM AUTHOR]

The notion of degree-constrained spanning hierarchies, also called k-trails, was recently introduced in the context of network routing problems. They describe graphs that are homomorphic images of connected graphs of degree at most k. First results highlight several interesting advantages of k-trails compared to previous routing approaches. However, so far, only little is known regarding computational aspects of k-trails. In this work we aim to fill this gap by presenting how k-trails can be analyzed using techniques from algorithmic matroid theory. Exploiting this connection, we resolve several open questions about k-trails. In particular, we show that one can recognize efficiently whether a graph is a k-trail, and every graph containing a k-trail is a (k+1)-trail. Moreover, further leveraging the connection to matroids, we consider the problem of finding a minimum weight k-trail contained in a graph G. We show that one can efficiently find a (2k-1)-trail contained in G whose weight is no more than the cheapest k-trail contained in G, even when allowing negative weights. The above results settle several open questions raised by Molnár, Newman, and Sebő. [ABSTRACT FROM AUTHOR]

A blocking semioval in a projective plane is a set of points which is both a semioval and a blocking set. In this paper, blocking semiovals in the Desaguesian projective plane $$\mathrm{PG}(2,s^2)$$ admitting an order $$s+1$$ homology group are considered. The geometry of the point-orbits of such a group is studied. Using this geometry two new blocking semiovals are constructed in $$\mathrm{PG}(2,5^2)$$ . Their automorphism group is also discussed. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to put on solid mathematical grounds the so-called non-equilibrium Green's function transport formalism for open systems. In particular, we derive the Jauho-Meir-Wingreen formula for the time-dependent current through an interacting sample coupled to non-interacting leads. Our proof is non-perturbative and uses neither complex-time Keldysh contours nor Langreth rules of 'analytic continuation.' We also discuss other technical identities (Langreth, Keldysh) involving various many-body Green's functions. Finally, we study the Dyson equation for the advanced/retarded interacting Green's function and we rigorously construct its (irreducible) self-energy, using the theory of Volterra operators. [ABSTRACT FROM AUTHOR]

This paper introduces a new algorithm for solving large-scale continuous-time algebraic Riccati equations (CARE). The advantage of the new algorithm is in its immediate and efficient low-rank formulation, which is a generalization of the Cholesky-factored variant of the Lyapunov ADI method. We discuss important implementation aspects of the algorithm, such as reducing the use of complex arithmetic and shift selection strategies. We show that there is a very tight relation between the new algorithm and three other algorithms for CARE previously known in the literature-all of these seemingly different methods in fact produce exactly the same iterates when used with the same parameters: they are algorithmically different descriptions of the same approximation sequence to the Riccati solution. [ABSTRACT FROM AUTHOR]

This paper is concerned with the implementation of efficient solution algorithms for elliptic problems with constraints. We establish theory which shows that including a simple scaling within well-established block diagonal preconditioners for Stokes problems can result in significantly faster convergence when applying the preconditioned MINRES method. The codes used in the numerical studies are available online. [ABSTRACT FROM AUTHOR]

This paper is concerned with a 1D Schrödinger scattering problem involving both oscillatory and evanescent regimes, separated by jump discontinuities in the potential function, to avoid 'turning points'. We derive a non-overlapping domain decomposition method to split the original problem into sub-problems on these regions, both for the continuous and afterwards for the discrete problem. Further, a hybrid WKB-based numerical method is designed for its efficient and accurate solution in the semi-classical limit: a WKB-marching method for the oscillatory regions and a FEM with WKB-basis functions in the evanescent regions. We provide a complete error analysis of this hybrid method and illustrate our convergence results by numerical tests. [ABSTRACT FROM AUTHOR]

For any x 휖 (0, 1], let the series ∑n=1∞1/dnx be the Sylvester expansion of x, where {dj(x), j ≥ 1} is a sequence of positive integers satisfying d1(x) ≥ 2 and dj + 1(x) ≥ dj(x)(dj(x) − 1) + 1 for j ≥ 1. Suppose ϕ : ℕ → ℝ+ is a function satisfying ϕ(n+1) - ϕ (n) → ∞ as n → ∞. In this paper, we consider the setEϕ=x∈01:limn→∞logdnx−∑j=1n−1logdjxϕn=1and quantify the size of the set in the sense of Hausdorff dimension. As applications, for any β > 1 and γ > 0, we get the Hausdorff dimension of the set x∈01:limn→∞logdnx−∑j=1n−1logdjx/nβ=γ, and for any τ > 1 and η > 0, we get a lower bound of the Hausdorff dimension of the set x∈01:limn→∞logdnx−∑j=1n−1logdjx/τn=η. [ABSTRACT FROM AUTHOR]

This paper reports on how the geometry component of the National Curricula for mathematics in Japan and in one selected country of the UK, specifically England, is interpreted in school mathematics textbooks from major publishers sampled from each country. The findings we report identify features of geometry, and approaches to geometry teaching and learning, that are found in a sample of textbooks aimed at students in Grade 8 (aged 13-14). Our analysis raises two issues which are widely recognised as very important in mathematics education: the teaching of mathematical reasoning and proof, and the teaching of problem-solving. In terms of the teaching of mathematical reasoning and proof, our evidence indicates that this is dispersed in the textbook in England while it is concentrated in geometry in the textbook in Japan. In terms of the teaching of mathematical problem-solving and modeling, our analysis shows that it is more concentrated in the textbook from England, and rather more dispersed in the textbook from Japan. These findings indicate how important it is to consider ways in which these issues can be carefully designed in the geometry sections of future textbooks. [ABSTRACT FROM AUTHOR]