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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]

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]

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]

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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]

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]

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]

This paper discusses the neo-logicist approach to the foundations of mathematics by highlighting an issue that arises from looking at the Bad Company objection from an epistemological perspective. For the most part, our issue is independent of the details of any resolution of the Bad Company objection and, as we will show, it concerns other foundational approaches in the philosophy of mathematics. In the first two sections, we give a brief overview of the “Scottish” neo-logicist school, present a generic form of the Bad Company objection and introduce an epistemic issue connected to this general problem that will be the focus of the rest of the paper. In the third section, we present an alternative approach within philosophy of mathematics, a view that emerges from Hilbert’s Grundlagen der Geometrie (1899, Leipzig: Teubner; Foundations of geometry (trans.: Townsend, E.). La Salle, Illinois: Open Court, 1959.). We will argue that Bad Company-style worries, and our concomitant epistemic issue, also affects this conception and other foundationalist approaches. In the following sections, we then offer various ways to address our epistemic concern, arguing, in the end, that none resolves the issue. The final section offers our own resolution which, however, runs against the foundationalist spirit of the Scottish neo-logicist program. [ABSTRACT FROM AUTHOR]

In this paper, we deepen the theoretical study of the geometric structure of a balanced complex polytope (b.c.p.), which is the generalization of a real centrally symmetric polytope to the complex space. We also propose a constructive algorithm for the representation of its facets in terms of their associated linear functionals. The b.c.p.s are used, for example, as a tool for the computation of the joint spectral radius of families of matrices. For the representation of real polytopes, there exist well-known algorithms such as, for example, the Beneath–Beyond method. Our purpose is to modify and adapt this method to the complex case by exploiting the geometric features of the b.c.p. However, due to the significant increase in the difficulty of the problem when passing from the real to the complex case, in this paper, we confine ourselves to examine the two-dimensional case. We also propose an algorithm for the computation of the norm the unit ball of which is a b.c.p. [ABSTRACT FROM AUTHOR]

Abstract  The concept of a hemisystem of a generalised quadrangle has its roots in the work of B. Segre, and this term is used here to denote a set of points $${\mathcal{H}}$$ such that every line ℓ meets $${\mathcal{H}}$$ in half of the points of ℓ. If one takes the point-line geometry on the points of the hemisystem, then one obtains a partial quadrangle and hence a strongly regular point graph. The only previously known hemisystems of generalised quadrangles of order (q, q 2) were those of the elliptic quadric $${\mathsf{Q}^-(5,q)}$$ , q odd. We show in this paper that there exists a hemisystem of the Fisher–Thas–Walker–Kantor generalised quadrangle of order (5, 52), which leads to a new partial quadrangle. Moreover, we can construct from our hemisystem the 3· A 7-hemisystem of $${\mathsf{Q}^-(5,5)}$$ , first constructed by Cossidente and Penttila. [ABSTRACT FROM AUTHOR]

An optimization problem is defined by an objective function to be maximized with respect to a set of constraints. To overcome some theoretical and practical difficulties, the constraint-set is sometimes relaxed and “easier” problems are solved. This led us to study relaxations providing exactly the same set of optimal solutions. We give a complete characterization of these relaxations and present several examples. While the relaxations introduced in this paper are not always easy to solve, they may help to prove that some mathematical programs are equivalent in terms of optimal solutions. An example is given where some of the constraints of a linear program can be relaxed within a certain limit. [ABSTRACT FROM AUTHOR]

In the present paper a general theorem that links characterizations of discrete life distributions based on relationship between failure rate and conditional expectations with those in terms of Chernoff-type inequalities is proposed. Exact expression for lower bounds to the variance is calculated for distributions belonging to the modified power series family, Ord family and mixture geometric models. It is shown that the bounds obtained here contain the Cramer–Rao and Chapman–Robbins inequalities as special cases. An application of the results to real data is also provided. [ABSTRACT FROM AUTHOR]

Students mainly perceive the transition to secondary school as an opportunity to start their school career anew. Reality often proves them wrong, especially in the case of mathematics. In our paper, we briefly discuss children’s transition to secondary school as both an opportunity and a problem, with reference to the Greek context. In discussions about the transition to secondary school and its effect on the teaching and learning of mathematics, primary and secondary school teachers in Greece often depict school mathematics as a “chain” of concepts and procedures. With this metaphor as our reference point, we discuss how ideologies about the teaching of mathematics are enacted in both school levels. We will base our discussion on the analysis of extracts taken from dialogues in primary and secondary school mathematics classrooms in Greece. In our analysis, we employ a Peircean view of semiotics in an attempt to conceptualize students’ rushed introduction to rigor in justifying mathematical statements in secondary school. These extracts are part of a longitudinal study that aimed, on the one hand, to pinpoint discontinuities and continuities in the teaching of geometry between primary and secondary schools and, on the other, to investigate whether a set of curriculum material that we designed could serve as a link in the teaching of geometry between the two school levels. [ABSTRACT FROM AUTHOR]

Abstract In the literature, there are but a few incidence geometries on which the McLaughlin sporadic group $${\mathsf{McL}}$$ acts as a flag-transitive automorphism group. Their highest rank is four. In the present paper, we construct a geometry of rank six on which $${\mathsf{McL}}$$ acts flag-transitively and which has the following diagram. [ABSTRACT FROM AUTHOR]

Although, natural channels are rarely rectangular or trapezoidal in cross section, these cross sections are assumed for the computation of steady, gradually varied flow in open channel networks. The accuracy of the computed results, therefore, becomes questionable due to differences in the hydraulic and geometric characteristics of the main channel and floodplains. To overcome these limitations, an algorithm is presented in this paper to compute steady, gradually varied flow in an open-channel network with compound cross sections. As compared to the presently available methods, the methodology is more general and suitable for application to compound and trapezoidal channel cross sections in series channels, tree-type or looped networks. In this method, the energy and continuity equations are solved for steady, gradually varied flow by the Newton-Raphson method and the proposed methodology is applied to tree-type and looped-channel networks. An algorithm is presented to determine multiple critical depths in a compound channel. Modifications in channel geometry are presented to avoid the occurrence of multiple critical depths. The occurrence of only one critical depth in a compound cross section with modified geometry is demonstrated for a tree-type channel network. [ABSTRACT FROM AUTHOR]

In this paper, we prove that every metric line in the Poincaré ball model of hyperbolic geometry is exactly a classical line of it. We also prove nonexistence of periodic lines in the Poincaré ball model of hyperbolic geometry. [ABSTRACT FROM AUTHOR]

The geometry of a system of second order differential equations is the geometry of a semispray, which is a globally defined vector field on TM. The metric compatibility of a given semispray is of special importance. In this paper, the metric associated with the semispray S is applied in order to study some types of foliations on the tangent bundle which are compatible with SODE structure. Indeed, sufficient conditions for the metric associated with the semispray S are obtained to extend to a bundle-like metric for the lifted foliation on TM. Thus, the lifted foliation converts to a Riemanian foliation on the tangent space which is adapted to the SODE structure. Particularly, the metric compatibility property of the semispray S is applied in order to induce SODE structure on transversals. Finally, some equivalent conditions are presented for the transversals to be totally geodesic. [ABSTRACT FROM AUTHOR]

A proper vertex coloring of a simple graph is κ-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than κ. A graph is κ-forested q-choosable if for a given list of q colors associated with each vertex v, there exists a κ-forested coloring of G such that each vertex receives a color from its own list. In this paper, we prove that the κ-forested choosability of a graph with maximum degree Δ ≥ κ ≥ 4 is at most [Δ/κ-1] + 1, [Δ/κ-1] + 2 or [Δ/κ-1] + 3 if its maximum average degree is less than 12/5, 8/3 or 3, respectively. [ABSTRACT FROM AUTHOR]

Recently, we introduced a new coefficient as a generalization of the modulus of smoothness and Pythagorean modulus such as J , ( t). In this paper, We can compute the constant J , (1) under the absolute normalized norms on ℝ by means of their corresponding continuous convex functions on [0, 1]. Moreover, some sufficient conditions which imply uniform normal structure are presented. 2000 Mathematics Subject Classification: 46B20. [ABSTRACT FROM AUTHOR]

In this paper, the effect of geometrical imperfections due to joint angle deviations of a simple rectangular frame is thoroughly discussed. It has been found that the deviation from the right angle of the joint with respect to the center-lines of the two members of the frame affects significantly the buckling strength of the frame. It is also shown that such geometrical imperfections do not always decrease the strength of such structures but sometimes, even though loading imperfections exist, they act on the favorable side regarding its strength. The influence of various geometric parameters such as slenderness ratio as well as length and moment of inertia ratio of the members on the critical load of the imperfect frame is thoroughly discussed and representative case studies are presented. [ABSTRACT FROM AUTHOR]

In this paper, an approach to the analysis of arbitrary thin to moderately thick plates and shells by the edge-based smoothed finite element method (ES-FEM) is presented. The formulation is based on the first order shear deformation theory, and Discrete Shear Gap (DSG) method is employed to mitigate the shear locking. Triangular meshes are used as they can be generated automatically for complicated geometries. The discretized system equations are obtained using the smoothed Galerkin weak form, and the numerical integration is applied based on the edge-based smoothing domains. The smoothing operation can provide a much needed softening effect to the FEM model to reduce the well-known “overly stiff” behavior caused by the fully compatible implementation of the displacement approach based on the Galerkin weakform, and hence improve significantly the solution accuracy. A number of benchmark problems have been studied and the results confirm that the present method can provide accurate results for both plate and shell using triangular mesh. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of the geometry of Norden manifolds. Some properties of Riemannian curvature tensors and curvature scalars of Kähler-Norden manifolds using the theory of Tachibana operators is presented. [ABSTRACT FROM AUTHOR]

Abstract  Minihypers are substructures of projective spaces introduced to study linear codes meeting the Griesmer bound. Recently, many results in finite geometry were obtained by applying characterization results on minihypers (De Beule et al. 16:342–349, 2008; Govaerts and Storme 4:279–286, 2004; Govaerts et al. 28:659–672, 2002). In this paper, using characterization results on certain minihypers, we present new results on tight sets in classical finite polar spaces and weighted m-covers, and on weighted m-ovoids of classical finite generalized quadrangles. The link with minihypers gives us characterization results of i-tight sets in terms of generators and Baer subgeometries contained in the Hermitian and symplectic polar spaces, and in terms of generators for the quadratic polar spaces. We also present extendability results on partial weighted m-ovoids and partial weighted m-covers, having small deficiency, to weighted m-covers and weighted m-ovoids of classical finite generalized quadrangles. As a particular application, we prove in an alternative way the extendability of 53-, 54-, and 55-caps of PG(5,3), contained in a non-singular elliptic quadric Q−(5,3), to 56-caps contained in this elliptic quadric Q−(5,3). [ABSTRACT FROM AUTHOR]

This paper reports the outcomes of an empirical study undertaken to investigate the effect of students’ cognitive styles on achievement in measurement tasks in a dynamic geometry learning environment, and to explore the ability of dynamic geometry learning in accommodating different cognitive styles and enhancing students’ learning. A total of 49 6th grade students were tested using the VICS and the extended CSA-WA tests (Peterson, Verbal imagery cognitive styles and extended cognitive style analysis-wholistic analytic test—Administration guide. New Zealand: Peterson, ) for cognitive styles. The same students were also administered a pre-test and a post-test involving 20 measurement tasks. All students were taught a unit in measurement (area of triangles and parallelograms) with the use of dynamic geometry, after a pre-test. As expected, the dynamic geometry software seems to accommodate different cognitive styles and enhances students’ learning. However, contrary to expectations, verbalisers and wholist/verbalisers gained more in their measurement achievement in the environment of dynamic geometry than students who had a tendency towards other cognitive styles. The results are discussed in terms of the nature of the measurement tasks administered to the students. [ABSTRACT FROM AUTHOR]

Abstract In this paper we outline a construction method which has been used for minimal blocking sets in PG(2, q) and maximal partial line spreads in PG(n, q) and which must have a lot of more applications. We also give a survey on what is known about the spectrum of sizes of maximal partial line spreads in PG(n, q). At the end we list some more elaborate random techniques used in finite geometry. [ABSTRACT FROM AUTHOR]

In the present paper, we classify the local singularities of chord sets, i.e., of the envelopes of two-parameter families of straight lines connecting pairs of points on two smooth curves in ℝ3; we also present geometric criteria for the chord set to have a given local singularity. [ABSTRACT FROM AUTHOR]

This paper presents a new contact searching algorithm for large deformation mortar-based contact formulations. In this algorithm, a bounding volume hierarchy, defined in the context of a binary tree, is built for each contact surface based on the geometry of the surface. A global contact searching procedure based on these bounding volume trees is first performed to find all candidate contact element pairs, and then a local searching procedure is done to find all the mortar segments having contributions to the mortar integrals that define the contact formulation. The searching algorithm is shown to be very efficient and readily applicable to a variety of large sliding contact problems. [ABSTRACT FROM AUTHOR]

The article provides an overview of the study which examines the locally uniform nonsquare in generalized Cesàro sequence spaces. It found that the generalized Cesàro sequence spaces possessed the locally uniform nonsquare and have the fixed point property, but they are not uniformly nonsquare. It specified that the result is related to the result of the paper J. False and colleagues in 2006. It was done through giving examples and the motivation to find the geometric properties that are weaker than uniformly nonsquare, but still possessed the fixed point property in any Banach spaces .

Abstract In this paper, we prove that the Hermitian quadrangle $${\mathsf{H}}(4, q^2)$$ is the unique generalized quadrangle Γ of order (q 2, q 3) containing some subquadrangle of order (q 2, q) isomorphic to $${\mathsf{H}}(3, q^2)$$ such that every central elation of the subquadrangle is induced by a collineation of Γ. [ABSTRACT FROM AUTHOR]

Abstract  In this paper, we consider the action of (2, q) on the finite projective line $${\mathbb{F}}_q\cup\{\infty\}$$ for q ≡ 1 (mod 4) and construct several infinite families of simple 3-designs which admit PSL(2, q) as an automorphism group. Some of the designs are also minimal. We also indicate a general outline to obtain some other algebraic constructions of simple 3-designs. [ABSTRACT FROM AUTHOR]

Abstract  In the symplectic polar space W 5(q) every 1-system which satisfies the BLT-property (and then q is odd) defines a generalized quadrangle (GQ) of order (q 2,q 3). In this paper, we show that this 1-system is unique, so that the only GQ arising in this way is isomorphic to the classical GQ H(4,q 2), q odd. [ABSTRACT FROM AUTHOR]

Abstract  It is well known that associated with a translation plane π there is a family of equivalent spreads. In this paper, we prove that if one of these spreads is symplectic and π is finite, then all the associated spreads are symplectic. Also, using the geometric intepretation of the Knuth’s cubical array, we prove that a symplectic semifield spread of dimension n over its left nucleus is associated via a Knuth operation to a commutative semifield of dimension n over its middle nucleus. [ABSTRACT FROM AUTHOR]

In this paper, we propose a complete framework for 3D geometry modeling and processing that uses only fast geodesic computations. The basic building block for these techniques is a novel greedy algorithm to perform a uniform or adaptive remeshing of a triangulated surface. Our other contributions include a parameterization scheme based on barycentric coordinates, an intrinsic algorithm for computing geodesic centroidal tessellations, and a fast and robust method to flatten a genus-0 surface patch. On large meshes (more than 500,000 vertices), our techniques speed up computation by over one order of magnitude in comparison to classical remeshing and parameterization methods. Our methods are easy to implement and do not need multilevel solvers to handle complex models that may contain poorly shaped triangles. [ABSTRACT FROM AUTHOR]

Diophantos' solutions to the problems of Arithmetica have been the object of extensive reading and interpretation in modern times, especially from the point of view of identifying ``hidden steps'' or ``general methods''. In this paper, after examining the relevance of various interpretations given for the famous problem II 8 in the context of modern algebra or geometry, we focus on a close reading of the ancient text of some problems of Arithmetica in order to investigate Diophantos' solving practices. This inquiry reveals certain pointers, which enable us to create a framework for defining the generality of Diophantos' methods. [ABSTRACT FROM AUTHOR]

Linear scale space methodology uses Gaussian probes at scale s to observe the differential structure. In observing the differential image structure through the Gaussian derivative probes at scale s we implicitly construct the Taylor series expansion of the smoothed image. The Gaussian facet model, as a generalization of the classic Haralick facet model, constructs a polynomial approximation of the unsmoothed image. The measured differential structure therefore is closer to the ‘real’ structure than the differential structure measured using Gaussian derivatives. At the points in an image where the differential structure changes abruptly (because of discontinuities in the imaging conditions, e.g. a material change, or a depth discontinuity) both the Gaussian derivatives and the Gaussian facet model diffuse the information from both sides of the discontinuity (smoothing across the edge). Robust estimators that are classically meant to deal with statistical outliers can also be used to deal with these ‘mixed model distributions’. In this paper we introduce the robust estimators of local image structure. Starting with the Gaussian facet model where we replace the quadratic error norm with a robust (Gaussian) error norm leads to a robust Gaussian facet model. We will show examples of using the robust differential structure estimators for luminance and color images, for zero and higher order differential structure. Furthermore we look at a ‘robustified’ structure tensor that forms the basis of robust orientation estimation. [ABSTRACT FROM AUTHOR]

Proof and reasoning are fundamental aspects of mathematics. Yet, how to help students develop the skills they need to engage in this type of higher-order thinking remains elusive. In order to contribute to the dialogue on this subject, we share results from a classroom-based interpretive study of teaching and learning proof in geometry. The goal of this research was to identify factors that may be related to the development of proof understanding. In this paper, we identify and interpret students' actions, teacher's actions, and social aspects that are evident in a classroom in which students discuss mathematical conjectures, justification processes and student-generated proofs. We conclude that pedagogical choices made by the teacher, as manifested in the teacher's actions, are key to the type of classroom environment that is established and, hence, to students' opportunities to hone their proof and reasoning skills. More specifically, the teacher's choice to pose open-ended tasks (tasks which are not limited to one specific solution or solution strategy), engage in dialogue that places responsibility for reasoning on the students, analyze student arguments, and coach students as they reason, creates an environment in which participating students make conjectures, provide justifications, and build chains of reasoning. In this environment, students who actively participate in the classroom discourse are supported as they engage in proof development activities. By examining connections between teacher and student actions within a social context, we offer a first step in linking teachers' practice to students' understanding of proof. [ABSTRACT FROM AUTHOR]

In this paper we investigate holomorphically projective mappings of generalized Kählerian spaces. In the case of equitorsion holomorphically projective mappings of generalized Kählerian spaces we obtain five invariant geometric objects for these mappings. [ABSTRACT FROM AUTHOR]

In the present paper we consider algebras satisfying both the congruence extension property (briefly the CEP) and the weak congruence intersection property (WCIP for short). We prove that subalgebras of such algebras have these properties. We deduce that a lattice has the CEP and the WCIP if and only if it is a two-element chain. We also show that the class of all congruence modular algebras with the WCIP is closed under the formation of homomorphic images. [ABSTRACT FROM AUTHOR]

Assuming an odd number of voters, E. S. Maskin recently provided a characterization of majority rule based on full transitivity. This paper characterizes majority rule with a set of axioms that includes two of Maskin's, dispenses with another, and contains weak versions of his other two axioms. It allows the number of voters to be odd or even. [ABSTRACT FROM AUTHOR]

We explore extensions of tropical methods to arithmetic enumerative problems such as A 1 -enumeration with values in the Grothendieck–Witt ring and rationality over Henselian valued fields, using bitangents to plane quartics as a test case. We consider quartic curves over valued fields whose tropicalizations are smooth and satisfy a mild genericity condition. We then express obstructions to rationality of bitangents and their points of tangency in terms of twisting of edges of the tropicalization; the latter depends only on the tropicalization and the initial coefficients of the defining equation modulo squares. We also show that the GW -multiplicity of a tropical bitangent, i.e. the multiplicity with which its lifts contribute to the A 1 -enumeration of bitangents as defined by Larson and Vogt (Res Math Sci 8(26):1–21, 2021), can be computed from the tropicalization of the quartic together with the initial coefficients of the defining equation. As an application, we show that the four lifts of most tropical bitangent classes contribute 2 H , twice the class of the hyperbolic plane, to the A 1 -enumeration. These results rely on a degeneration theorem relating the Grothendieck–Witt ring of a Henselian valued field to the Grothendieck–Witt ring of its residue field, in residue characteristic not equal to two. [ABSTRACT FROM AUTHOR]

David Hilbert’s early foundational views, especially those corresponding to the 1890s, are analysed here. I consider strong evidence for the fact that Hilbert was a logicist at that time, following upon Dedekind’s footsteps in his understanding of pure mathematics. This insight makes it possible to throw new light on the evolution of Hilbert’s foundational ideas, including his early contributions to the foundations of geometry and the real number system. The context of Dedekind-style logicism makes it possible to offer a new analysis of the emergence of Hilbert’s famous ideas on mathematical existence, now seen as a revision of basic principles of the “naive logic” of sets. At the same time, careful scrutiny of his published and unpublished work around the turn of the century uncovers deep differences between his ideas about consistency proofs before and after 1904. Along the way, we cover topics such as the role of sets and of the dichotomic conception of set theory in Hilbert’s early axiomatics, and offer detailed analyses of Hilbert’s paradox and of his completeness axiom (Vollständigkeitsaxiom). [ABSTRACT FROM AUTHOR]

Abstract  This article is part of the program described in Bennett et al. (in: Ivanov et al. (ed.) Groups, combinatorics, and geometry, 2003) [3]. We study the Phan amalgams and their universal completions that occur for q = 3 in rank n = 3 for the diagram A 3 = D 3, corresponding to SU(4, 32) ≅ Spin −(6, 3). We show that the associated geometries admit universal 9-fold coverings, by showing that the universal completion of the Phan amalgam is the central extension of SU(4, 32) by its Schur multiplier. This information provides the last missing piece of information in the full classification of Phan amalgams and their universal completions for A n and D n . [ABSTRACT FROM AUTHOR]

The article establishes some geometric inequalities for a new Banach Sequence Space. It denotes the space whose elements are finite sets of distinct positive integers through the operator C. A space has been considered for matrix transformations. Some of its geometric properties are also considered. A Banach space is defined with a norm and a complete p-normed space with the p-norm. It examines the Banach-Saks property of type p, strict convexity and uniform convexity. It describes when a Banach space is said to have the Banach-Saks property of type p or property (BS) if every weakly null-sequence has a subsequence.

This paper generalizes the Theorem of the Maximum (Berge [2]) to allow for discontinuous changes in the domain and the objective function. It also provides a geometrical version of the (generalized) theorem. [ABSTRACT FROM AUTHOR]

Abstract definitions and explanations frequently come together and permeate almost all fields of knowledge. This does not exclude mathematics, even when these definitions hold clear links and close connections with our physical world. Here we propose a rather different perspective. Making operational physical assumptions, we show how it is possible to rigorously reconstruct some features of both geometry and topology. Broadly speaking, assuming this operational and more concrete philosophy we not only are capable of defining primitive concepts like points, straight lines, planes, angles and spaces, but we also go all the way up to more complex constructions. We show that within our method it is also possible to come up with what we call a compass-based topology as well as a normed and a metric space. We hope this operationalist point-of-view can assist other researchers and students to have a clearer understanding of these quite abstract concepts. [ABSTRACT FROM AUTHOR]

In this article, I reminisce about my fond interactions with John Torrence Tate (13 March, 1925-16 October, 2019), my teacher, mentor and PhD advisor. It is hard to trace the influence of Tate's work on other works accurately, as the basic objects, ideas and theorems that he introduced have permeated (and have been generalized by others into widely used theories) throughout mathematics. I briefly describe his many fundamental contributions to number theory and arithmetic geometry, and end with some personal anecdotes. [ABSTRACT FROM AUTHOR]

In this article we introduce the concept of midrange geometric constraints into semantic segmentation. We call these constraints 'midrange' since they are neither global constraints, which take into account all pixels without any spatial limitation, nor are they local constraints, which only regard single pixels or pairwise relations. Instead, the proposed constraints allow to discourage the occurrence of labels in the vicinity of each other, e.g., 'wolf' and 'sheep'. 'Vicinity' encompasses spatial distance as well as specific spatial directions simultaneously, e.g., 'plates' are found directly above 'tables', but do not fly over them. It is up to the user to specifically define the spatial extent of the constraint between each two labels. Such constraints are not only interesting for scene segmentation, but also for part-based articulated or rigid objects. The reason is that object parts such as for example arms, torso and legs usually obey specific spatial rules, which are among the few things that remain valid for articulated objects over many images and which can be expressed in terms of the proposed midrange constraints, i.e. closeness and/or direction. We show, how midrange geometric constraints are formulated within a continuous multi-label optimization framework, and we give a convex relaxation, which allows us to find globally optimal solutions of the relaxed problem independent of the initialization. [ABSTRACT FROM AUTHOR]

A correction to the article "Metric Geometry of Nonregular Weighted Carnot - Caratheodory Spaces" that was published in the February 10, 2015 issue is presented.

The theorem of three circles in real algebraic geometry guarantees the termination and correctness of an algorithm of isolating real roots of a univariate polynomial. The main idea of its proof is to consider polynomials whose roots belong to a certain area of the complex plane delimited by straight lines. After applying a transformation involving inversion this area is mapped to an area delimited by circles. We provide a formalisation of this rather geometric proof in Ssreflect, an extension of the proof assistant Coq, that supports a variety of algebraic tools. They allow us to formalise the proof from an algebraic point of view. [ABSTRACT FROM AUTHOR]