Chromatic numbers of simplicial manifolds

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

Published On 2020/9

Higher chromatic numbers of simplicial complexes naturally generalize the chromatic number of a graph. In any fixed dimension d, the s-chromatic number of d-complexes can become arbitrarily large for (Bing in The geometric topology of 3-manifolds, Colloquium Publications, vol 40, American Mathematical Society, Providence, 1983; Heise et al. in Discrete Comput Geom 52:663–679, 2014). In contrast, , and only little is known on for . A particular class of d-complexes are triangulations of d-manifolds. As a consequence of the Map Color Theorem for surfaces (Ringel in Map color theorem, Grundlehren der mathematischen Wissenschaften, vol 209, Springer, Berlin, 1974), the 2-chromatic number of any fixed surface is finite. However, by combining results from the literature, we will see that for surfaces becomes arbitrarily large with growing genus. The proof for this is via Steiner triple …

Journal

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

Published On

2020/9

Volume

61

Issue

3

Page

419-453

Authors

Jesper Møller

Jesper Møller

Aalborg Universitet

Position

Professor in Statistics

H-Index(all)

46

H-Index(since 2020)

23

I-10 Index(all)

0

I-10 Index(since 2020)

0

Citation(all)

0

Citation(since 2020)

0

Cited By

0

Research Interests

Mathematical Statistics

Probability Theory

University Profile Page

Other Articles from authors

Jesper Møller

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2404.09525

Coupling results and Markovian structures for number representations of continuous random variables

A general setting for nested subdivisions of a bounded real set into intervals defining the digits of a random variable with a probability density function is considered. Under the weak condition that is almost everywhere lower semi-continuous, a coupling between and a non-negative integer-valued random variable is established so that have an interpretation as the ``sufficient digits'', since the distribution of conditioned on does not depend on . Adding a condition about a Markovian structure of the lengths of the intervals in the nested subdivisions, becomes a Markov chain of a certain order . If then are IID with a known distribution. When and the Markov chain is uniformly geometric ergodic, a coupling is established between and a random time so that the chain after time is stationary and follows a simple known distribution. The results are related to several examples of number representations generated by a dynamical system, including base- expansions, generalized L\"uroth series, -expansions, and continued fraction representations. The importance of the results and some suggestions and open problems for future research are discussed.

Jesper Møller

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2404.08387

The asymptotic distribution of the scaled remainder for pseudo golden ratio expansions of a continuous random variable

Let be the base- expansion of a continuous random variable on the unit interval where is the positive solution to for an integer (i.e., is a generalization of the golden mean for which ). We study the asymptotic distribution and convergence rate of the scaled remainder when tends to infinity.

Jesper Møller

Jesper Møller

Aalborg Universitet

Methodology and Computing in Applied Probability

How many digits are needed?

Let be the digits in the base-q expansion of a random variable X defined on [0, 1) where is an integer. For , we study the probability distribution of the (scaled) remainder : If X has an absolutely continuous CDF then converges in the total variation metric to the Lebesgue measure on the unit interval. Under weak smoothness conditions we establish first a coupling between X and a non-negative integer valued random variable N so that follows and is independent of , and second exponentially fast convergence of and its PDF . We discuss how many digits are needed and show examples of our results.

Jesper Møller

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2312.09652

The asymptotic distribution of the remainder in a certain base- expansion

Let be the base- expansion of a continuous random variable on the unit interval where is the golden ratio. We study the asymptotic distribution and convergence rate of the scaled remainder when tends to infinity.

2023/12/15

Article Details
Jesper Møller

Jesper Møller

Aalborg Universitet

Proceedings of the London Mathematical Society

Realizability and tameness of fusion systems

A saturated fusion system over a finite p$p$‐group S$S$ is a category whose objects are the subgroups of S$S$ and whose morphisms are injective homomorphisms between the subgroups satisfying certain axioms. A fusion system over S$S$ is realized by a finite group G$G$ if S$S$ is a Sylow p$p$‐subgroup of G$G$ and morphisms in the category are those induced by conjugation in G$G$. One recurrent question in this subject is to find criteria as to whether a given saturated fusion system is realizable or not. One main result in this paper is that a saturated fusion system is realizable if all of its components (in the sense of Aschbacher) are realizable. Another result is that all realizable fusion systems are tame: a finer condition on realizable fusion systems that involves describing automorphisms of a fusion system in terms of those of some group that realizes it. Stated in this way, these results depend on the …

Jesper Møller

Jesper Møller

Aalborg Universitet

ACM Transactions on Spatial Algorithms and Systems

Stochastic Routing with Arrival Windows

Arriving at a destination within a specific time window is important in many transportation settings. For example, trucks may be penalized for early or late arrivals at compact terminals, and early and late arrivals at general practitioners, dentists, and so on, are also discouraged, in part due to COVID. We propose foundations for routing with arrival-window constraints. In a setting where the travel time of a road segment is modeled by a probability distribution, we define two problems where the aim is to find a route from a source to a destination that optimizes or yields a high probability of arriving within a time window while departing as late as possible. In this setting, a core challenge is to enable comparison between paths that may potentially be part of a result path with the goal of determining whether a path is uninteresting and can be disregarded given the existence of another path. We show that existing solutions …

2023/11/21

Article Details
Jesper Møller

Jesper Møller

Aalborg Universitet

Spatial Statistics

Fitting the grain orientation distribution of a polycrystalline material conditioned on a Laguerre tessellation

The description of distributions related to grain microstructure helps physicists to understand the processes in materials and their properties. This paper presents a general statistical methodology for the analysis of crystallographic orientations of grains in a 3D Laguerre tessellation dataset which represents the microstructure of a polycrystalline material. We introduce complex stochastic models which may substitute expensive laboratory experiments: conditional on the Laguerre tessellation, we suggest interaction models for the distribution of cubic crystal lattice orientations, where the interaction is between pairs of orientations for neighbouring grains in the tessellation. We discuss parameter estimation and model comparison methods based on maximum pseudolikelihood as well as graphical procedures for model checking using simulations. Our methodology is applied for analysing a dataset representing a nickel …

Jesper Møller

Jesper Møller

Aalborg Universitet

Methodology and Computing in Applied Probability

Singular distribution functions for random variables with stationary digits

Let F be the cumulative distribution function (CDF) of the base-q expansion , where is an integer and is a stationary stochastic process with state space . In a previous paper we characterized the absolutely continuous and the discrete components of F. In this paper we study special cases of models, including stationary Markov chains of any order and stationary renewal point processes, where we establish a law of pure types: F is then either a uniform or a singular CDF on [0, 1]. Moreover, we study mixtures of such models. In most cases expressions and plots of F are given.

Jesper Møller

Jesper Møller

Aalborg Universitet

arXiv preprint arXiv:2212.08402

Cox processes driven by transformed Gaussian processes on linear networks

There is a lack of point process models on linear networks. For an arbitrary linear network, we use isotropic covariance functions with respect to the geodesic metric or the resistance metric to construct new models for isotropic Gaussian processes and hence new models for various Cox processes with isotropic pair correlation functions. In particular we introduce three model classes given by log Gaussian, interrupted, and permanental Cox processes on linear networks, and consider for the first time statistical procedures and applications for parametric families of such models. Moreover, we construct new simulation algorithms for Gaussian processes on linear networks and discuss whether the geodesic metric or the resistance metric should be used for the kind of Cox processes studied in this paper.

2022/12/16

Article Details
Jesper Møller

Jesper Møller

Aalborg Universitet

Stat

Determinantal shot noise Cox processes

We present a new class of cluster point process models, which we call determinantal shot noise Cox processes (DSNCP), with repulsion between cluster centres. They are the special case of generalized shot noise Cox processes where the cluster centres are determinantal point processes. We establish various moment results and describe how these can be used to easily estimate unknown parameters in two particularly tractable cases, namely, when the offspring density is isotropic Gaussian and the kernel of the determinantal point process of cluster centres is Gaussian or like in a scaled Ginibre point process. Through a simulation study and the analysis of a real point pattern data set, we see that when modelling clustered point patterns, a much lower intensity of cluster centres may be needed in DSNCP models as compared to shot noise Cox processes.

Jesper Møller

Jesper Møller

Aalborg Universitet

International Statistical Review

Should we condition on the number of points when modelling spatial point patterns?

We discuss the practice of directly or indirectly assuming a model for the number of points when modelling spatial point patterns even though it is rarely possible to validate such a model in practice because most point pattern data consist of only one pattern. We therefore explore the possibility to condition on the number of points instead when fitting and validating spatial point process models. In a simulation study with different popular spatial point process models, we consider model validation using global envelope tests based on functional summary statistics. We find that conditioning on the number of points will for some functional summary statistics lead to more narrow envelopes and thus stronger tests and that it can also be useful for correcting for some conservativeness in the tests when testing composite hypothesis. However, for other functional summary statistics, it makes little or no difference to condition …

Jesper Møller

Jesper Møller

Aalborg Universitet

Journal of Applied Probability

Characterization of random variables with stationary digits

Let be an integer, a stochastic process with state space , and F the cumulative distribution function (CDF) of . We show that stationarity of is equivalent to a functional equation obeyed by F, and use this to characterize the characteristic function of X and the structure of F in terms of its Lebesgue decomposition. More precisely, while the absolutely continuous component of F can only be the uniform distribution on the unit interval, its discrete component can only be a countable convex combination of certain explicitly computable CDFs for probability distributions with finite support. We also show that is a Rajchman measure if and only if F is the uniform CDF on [0, 1].

Jesper Møller

Jesper Møller

Aalborg Universitet

Spatial Statistics

Fitting three-dimensional Laguerre tessellations by hierarchical marked point process models

We present a general statistical methodology for analysing a Laguerre tessellation data set viewed as a realization of a marked point process model. In the first step, for the points, we use a nested sequence of multiscale processes which constitute a flexible parametric class of pairwise interaction point process models. In the second step, for the marks/radii conditioned on the points, we consider various exponential family models where the canonical sufficient statistic is based on tessellation characteristics. For each step, parameter estimation based on maximum pseudolikelihood methods is tractable. For model selection, we consider maximized log pseudolikelihood functions for models of the radii conditioned on the points. Model checking is performed using global envelopes and corresponding tests in both steps and moreover by comparing observed and simulated tessellation characteristics in the second step …

Jesper Møller

Jesper Møller

Aalborg Universitet

Translational psychiatry

Layer III pyramidal cells in the prefrontal cortex reveal morphological changes in subjects with depression, schizophrenia, and suicide

Brodmann Area 46 (BA46) has long been regarded as a hotspot of disease pathology in individuals with schizophrenia (SCH) and major depressive disorder (MDD). Pyramidal neurons in layer III of the Brodmann Area 46 (BA46) project to other cortical regions and play a fundamental role in corticocortical and thalamocortical circuits. The AutoCUTS-LM pipeline was used to study the 3-dimensional structural morphology and spatial organization of pyramidal cells. Using quantitative light microscopy, we used stereology to calculate the entire volume of layer III in BA46 and the total number and density of pyramidal cells. Volume tensors estimated by the planar rotator quantified the volume, shape, and nucleus displacement of pyramidal cells. All of these assessments were carried out in four groups of subjects: controls (C, n = 10), SCH (n = 10), MDD (n = 8), and suicide subjects with a history of depression (SU …

Jesper Møller

Jesper Møller

Aalborg Universitet

Graphs and Combinatorics

Equivariant Euler characteristics of symplectic buildings

We compute the equivariant Euler characteristics of the buildings for the symplectic groups over finite fields.

Jesper Møller

Jesper Møller

Aalborg Universitet

Journal of Computational and Graphical Statistics

MCMC computations for Bayesian mixture models using repulsive point processes

Repulsive mixture models have recently gained popularity for Bayesian cluster detection. Compared to more traditional mixture models, repulsive mixture models produce a smaller number of well-separated clusters. The most commonly used methods for posterior inference either require to fix a priori the number of components or are based on reversible jump MCMC computation. We present a general framework for mixture models, when the prior of the “cluster centers” is a finite repulsive point process depending on a hyperparameter, specified by a density which may depend on an intractable normalizing constant. By investigating the posterior characterization of this class of mixture models, we derive a MCMC algorithm which avoids the well-known difficulties associated to reversible jump MCMC computation. In particular, we use an ancillary variable method, which eliminates the problem of having intractable …

Jesper Møller

Jesper Møller

Aalborg Universitet

Scandinavian Journal of Statistics

Approximate Bayesian inference for a spatial point process model exhibiting regularity and random aggregation

In this article, we propose a doubly stochastic spatial point process model with both aggregation and repulsion. This model combines the ideas behind Strauss processes and log Gaussian Cox processes. The likelihood for this model is not expressible in closed form but it is easy to simulate realizations under the model. We therefore explain how to use approximate Bayesian computation (ABC) to carry out statistical inference for this model. We suggest a method for model validation based on posterior predictions and global envelopes. We illustrate the ABC procedure and model validation approach using both simulated point patterns and a real data example.

Jesper Møller

Jesper Møller

Aalborg Universitet

Journal of Algebraic Combinatorics

Equivariant Euler characteristics of unitary buildings

The (p-primary) equivariant Euler characteristics of the buildings for the general unitary groups over finite fields are determined.

Jesper Møller

Jesper Møller

Aalborg Universitet

Communications Biology

Cellular 3D-reconstruction and analysis in the human cerebral cortex using automatic serial sections

Techniques involving three-dimensional (3D) tissue structure reconstruction and analysis provide a better understanding of changes in molecules and function. We have developed AutoCUTS-LM, an automated system that allows the latest advances in 3D tissue reconstruction and cellular analysis developments using light microscopy on various tissues, including archived tissue. The workflow in this paper involved advanced tissue sampling methods of the human cerebral cortex, an automated serial section collection system, digital tissue library, cell detection using convolution neural network, 3D cell reconstruction, and advanced analysis. Our results demonstrated the detailed structure of pyramidal cells (number, volume, diameter, sphericity and orientation) and their 3D spatial organization are arranged in a columnar structure. The pipeline of these combined techniques provides a detailed analysis of tissues …

Jesper Møller

Jesper Møller

Aalborg Universitet

AMERICAN MATHEMATICAL SOCIETY

THE NUMBER OF p-ELEMENTS IN FINITE GROUPS OF LIE TYPE OF CHARACTERISTIC p

The combinatorics of the poset of p-radical p-subgroups of a finite group is used to count the number of p-elements.

Other articles from Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry journal

Martin Kreuzer

Martin Kreuzer

Universität Passau

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

Re-embeddings of affine algebras via Gröbner fans of linear ideals

Given an affine algebra over a field K, where I is an ideal in the polynomial ring, we examine the task of effectively calculating re-embeddings of I, ie, of presentations such that has fewer indeterminates. For cases when the number of indeterminates n is large and Gröbner basis computations are infeasible, we have introduced the method of Z-separating re-embeddings in Kreuzer et al.(J Algebra Appl 21, 2022) and Kreuzer, et al.(São Paulo J Math Sci, 2022). This method tries to detect polynomials of a special shape in I which allow us to eliminate the indeterminates in the tuple Z by a simple substitution process. Here we improve this approach by showing that suitable candidate tuples Z can be found using the Gröbner fan of the linear part of I. Then we describe a method to compute the Gröbner fan of a linear ideal, and we improve this computation in the …

Suat Koç

Suat Koç

Marmara Üniversitesi

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

On morphic modules over commutative rings

A commutative ring R is said to be a morphic ring if for each there exists such that and . In this paper, we extend the notion of morphic rings to modules and we study the introduced concept by comparing it with some related notions.

Frank Kutzschebauch

Frank Kutzschebauch

Universität Bern

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

Algebraic overshear density property

We introduce the notion of the algebraic overshear density property which implies both the algebraic notion of flexibility and the holomorphic notion of the density property. We investigate basic consequences of this stronger property, and propose further research directions in this borderland between affine algebraic geometry and elliptic holomorphic geometry. As an application, we show that any smoothly bordered Riemann surface with finitely many boundary components that is embedded in a complex affine surface with the algebraic overshear density property admits a proper holomorphic embedding.

ünsal tekir

ünsal tekir

Marmara Üniversitesi

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

On morphic modules over commutative rings

A commutative ring R is said to be a morphic ring if for each there exists such that and . In this paper, we extend the notion of morphic rings to modules and we study the introduced concept by comparing it with some related notions.

Ali Jaballah

Ali Jaballah

University of Sharjah

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

The dimension-overrings equation and maximal ideals of integral domains

We investigate integral domains with only finitely many overrings and establish several new sharp inequalities relating the cardinality of the set of all overrings, the Krull dimension, and the number of maximal ideals. In fact it is shown that knowing any two of these parameters will induce sharp lower and upper bounds for the third. Similar results for the length of an integral domain are also obtained. In particular, we show that if the number of overrings of an integrally closed domain R is a prime number, then the field of fractions of R has a unique maximal subring.

john r owen

john r owen

University of Southampton

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

Mobility of geometric constraint systems with extrusion symmetry

If we take a (bar-joint) framework, prepare an identical copy of this framework, translate it by some vector , and finally join corresponding points of the two copies, then we obtain a framework with ‘extrusion’ symmetry in the direction of . This process may be repeated t times to obtain a framework whose underlying graph has as a subgroup of its automorphism group and which has ‘t-fold extrusion’ symmetry. Extruding a framework is a widely used technique in CAD for generating a 3D model from an initial 2D sketch, and hence it is important to understand the flexibility of extrusion-symmetric frameworks. Using group representation theory, we show that while t-fold extrusion symmetry is not a point-group symmetry, the rigidity matrix of a framework with t-fold extrusion symmetry can still be transformed into a block-decomposed form in the analogous way as for point-group symmetric frameworks. This allows us to …

Vladimir Lazić

Vladimir Lazić

Universität des Saarlandes

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

Programming the Minimal Model Program: a proposal

The aim of this paper is to propose a strategy to implement the Minimal Model Program in modern computer algebra systems.

Gordon Ian Williams

Gordon Ian Williams

University of Alaska Anchorage

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

On infinite 2-orbit polyhedra in classes and

We describe all 2-orbit skeletal polyhedra in classes and in . Those in class are continuous deformations of the three Petrie–Coxeter polyhedra and therefore are arranged in three families. Those in class are the Petrie duals of the chiral polyhedra in and are classified in six families.

Götz Pfeiffer

Götz Pfeiffer

National University of Ireland, Galway

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

Centers of Hecke algebras of complex reflection groups

We provide a dual version of the Geck–Rouquier Theorem (Geck and Rouquier in Finite Reductive Groups (Luminy, 1994), Progr. Math., vol. 141, Birkhäuser Boston, Boston, pp. 251–272, 1997) on the center of an Iwahori–Hecke algebra, which also covers the complex case. For the eight complex reflection groups of rank 2, for which the symmetrising trace conjecture is known to be true, we provide a new faithful matrix model for their Hecke algebra H. These models enable concrete calculations inside H. For each of the eight groups, we compute an explicit integral basis of the center of H.

Yahya Talebi

Yahya Talebi

University of Mazandaran

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

Exploring virtual versions of H-supplemented and NS-modules

In this work we introduce virtual versions of H-supplemented modules and NS-modules. These modules are defined by replacing the condition of being a “direct summand” with being “isomorphic to a direct summand”. The paper explores various equivalent conditions for a module to be virtually H-supplemented and investigates their fundamental properties. It is discovered that over a right V-ring for a module, the concepts virtually H-supplemented, virtually semisimple and VNS, coincide. Additionally, it is proven that each right R-module is VNS if and only if every noncosingular right R-module is injective.

2023/10/24

Article Details
Claudia He Yun

Claudia He Yun

Brown University

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

Some thoughts and experiments on Bergman’s compact amalgamation problem

We study the question whether copies of in can be amalgamated in a compact group. This is the simplest instance of a fundamental open problem in the theory of compact groups raised by George Bergman in 1987. Considerable computational experiments suggest that the answer is positive in this case. We obtain a positive answer for a relaxed problem using theoretical considerations.

Abdeslam Mimouni

Abdeslam Mimouni

King Fahd University of Petroleum and Minerals

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

Nil-ideals, J-ideals and their generalizations in commutative rings

Let R be a commutative ring with identity element, J(R) its Jacobson radical and Nil(R) its subset of all nilpotent elements. Recall that an ideal I is said to be an n-ideal (resp. a J-ideal) if whenever and such that (resp. ), then . A quasi J-ideal is an ideal I such that is a J-ideal. In this paper, we unify the notions of prime ideals, primary ideals, n-ideals and J-ideals in the so-called Q-ideals as follows: given ideals in R, I is a Q-ideal if whenever and , then (equivalently, given ideals A and B of R such that and , then ). Clearly, if , respectively , respectively , respectively , then I is a Q-ideal if and only if I is prime, respectively a primary ideal, respectively a J-ideal, respectively an n-ideal. We investigate the properties of these notions in different contexts of commutative rings. Precisely, in trivial ring extensions …

Matthias Beck

Matthias Beck

San Francisco State University

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

Lattice zonotopes of degree 2

The Ehrhart polynomial of a lattice polytope P gives the number of integer lattice points in the n-th dilate of P for all integers . The degree of P is defined as the degree of its -polynomial, a particular transformation of the Ehrhart polynomial with many useful properties which serves as an important tool for classification questions in Ehrhart theory. A zonotope is the Minkowski (pointwise) sum of line segments. We classify all Ehrhart polynomials of lattice zonotopes of degree 2 thereby complementing results of Scott (Bull Aust Math Soc 15(3), 395–399, 1976), Treutlein (J Combin Theory Ser A 117(3), 354–360, 2010), and Henk and Tagami (Eur J Combin 30(1), 70–83, 2009). Our proof is constructive: by considering solid-angles and the lattice width, we provide a characterization of all 3-dimensional zonotopes of degree 2.

Georgios Tsapogas

Georgios Tsapogas

Agricultural University of Athens

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

A specific model of Hilbert geometry on the unit disc

A new metric on the open 2-dimensional unit disk is defined making it a geodesically complete metric space whose geodesic lines are precisely the Euclidean straight lines. Moreover, it is shown that the unit disk with this new metric is not isometric to any hyperbolic model of constant negative curvature, nor to any convex domain in equipped with its Hilbert metric.

2023/11/17

Article Details
ünsal tekir

ünsal tekir

Marmara Üniversitesi

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

On -1-absorbing prime ideals

In this paper, we introduce -1-absorbing prime ideals in commutative rings. Let R be a commutative ring with a nonzero identity and be a function where is the set of all ideals of R. A proper ideal I of R is called a -1-absorbing prime ideal if for each nonunits with , then either or . In addition to give many properties and characterizations of -1-absorbing prime ideals, we also determine rings in which every proper ideal is -1-absorbing prime.

Fuensanta Aroca

Fuensanta Aroca

Universidad Nacional Autónoma de México

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

Groebner fans and embedded resolutions of ideals on toric varieties

We consider the notions of Groebner fan and Newton non-degeneracy for an ideal on a toric variety, extending the two existing notions for ideals on affine spaces. We prove, without assumptions on the characteristic of the base fields, that the “Groebner fan” of such an ideal is actually a polyhedral fan and that a subvariety defined by a Newton non-degenerate ideal on a toric variety admits a toric embedded resolution of singularities

Miklos Laczkovich

Miklos Laczkovich

Eötvös Loránd Tudományegyetem

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

Quadrilateral reptiles

A polygon P is called a reptile, if it can be decomposed into nonoverlapping and congruent polygons similar to P. We prove that if a cyclic quadrilateral is a reptile, then it is a trapezoid. Comparing with results of Betke and Osburg we find that every convex reptile is a triangle or a trapezoid.

Markus Johannes Stroppel

Markus Johannes Stroppel

Universität Stuttgart

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

Semiaffine stable planes

A locally compact stable plane of positive topological dimension will be called semiaffine if for every line L and every point p not in L there is at most one line passing through p and disjoint from L. We show that then the plane is either an affine or projective plane or a punctured projective plane (i.e., a projective plane with one point deleted). We also compare this with the situation in general linear spaces (without topology), where P. Dembowski showed that the analogue of our main result is true for finite spaces but fails in general.

Michael Beeson

Michael Beeson

San José State University

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

On the notion of equal figures in Euclid

Euclid uses an undefined notion of “equal figures”, to which he applies the common notions about equals added to equals or subtracted from equals. This notion does not occur in modern geometrical theories such as those of Hilbert or Tarski. Therefore to account for Euclid in modern geometry, one must somehow replace Euclid’s “equal figures” with a defined notion. In this paper we present a new solution to this problem, and moreover we argue that “Euclid could have done it”. That is, it is based on mathematics that was available in Euclid’s time, including ideas related to Euclid’s Proposition I.44. The proof uses the theory of proportions. Hence we also discuss the “early theory of proportions”, which has a long history.

Ahmed Abdelwanis

Ahmed Abdelwanis

Cairo University

Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry

On Hom-Jordan algebras and their type derivations

In this paper we generalize the results in Huang et al. (Commun Algebra 46:2600–2614, 2018). The current paper studies -type derivations of Hom-Jordan algebras. First, we give some properties of Hom-Jordan algebra and homomorphisms of Hom-Jordan algebras. Second, we get on some properties of -centroids and -quasicentroids of Hom-Jordan algebras. Finally, we study quasiderivations and -quasiderivations of Hom-Jordan algebras.