Generalized Heawood Numbers Kühnel, Wolfgang
The Electronic journal of combinatorics,
11/2023, Letnik:
30, Številka:
4
Journal Article
Recenzirano
Odprti dostop
This survey explains the origin and the further development of the Heawood inequalities, the Heawood number, and generalizations to higher dimensions with results and further conjectures.
Color Atlas of Human Anatomy, Volume 2: Internal Organs For over 45 years, the three-volume Color Atlas of Human Anatomy has provided readers with a compact review of the human body and its ...structures. It is ideal for studying, preparing for exams, and as a reference. The new, 7th edition of Volume 2: Internal Organs builds on a robust foundation of scientific knowledge, summarizing in its compactness the macroscopic and topographic anatomy and the functions of the internal organs. Key highlights: Proven concept of concise texts paired with more than 200 color plates of outstanding anatomical illustrations Microscopic anatomy--if necessary for understanding the respective organ Organ functions are explained in connection with the embryological development of the organs, so many anatomical relationships can be better understood For numerous cross-sectional anatomical illustrations, corresponding CT and MRI images are provided, which helps with the application of anatomical knowledge in clinical practice Volume 2: Internal Organsis accompanied byVolume 1: Locomotor System(ISBN 978-3-13-242443-3) andVolume 3: Nervous System and Sensory Organs (ISBN 978-3-13-242451-7).
There are two parallelohedra in dimension 2 and five in dimension 3. We study smallest polyhedral quotients of the tilings defined by them on tori. For 2-tori the situation is well understood, and ...the three minimal models are well known. Here we investigate the five 3-dimensional cases with the result: Each of the five parallelohedra admits a minimal (and weakly neighborly) model, moreover in three cases this is unique up to natural equivalence. Altogether there are 14 inequivalent minimal models.
PL Morse theory in low dimensions Grunert, Romain; Kühnel, Wolfgang; Rote, Günter
Advances in geometry,
01/2023, Letnik:
23, Številka:
1
Journal Article
Recenzirano
Odprti dostop
We discuss a PL analog of Morse theory for PL manifolds. There are several notions of regular and critical points. A point is homologically regular if the homology does not change when passing ...through its level; it is strongly regular if the function can serve as one coordinate in a chart. Several criteria for strong regularity are presented. In particular, we show that in dimensions
≤ 4 a homologically regular point on a PL
-manifold is always strongly regular. Examples show that this fails in higher dimensions
≥ 5. One of our constructions involves an embedding of the dunce hat into 4-space and Mazur’s contractible 4-manifold. Finally, decidability questions in this context are discussed.
Conformally Einstein product spaces Kühnel, Wolfgang; Rademacher, Hans-Bert
Differential geometry and its applications,
December 2016, 2016-12-00, Letnik:
49
Journal Article
Recenzirano
Odprti dostop
We study pseudo-Riemannian Einstein manifolds which are conformally equivalent with a metric product of two pseudo-Riemannian manifolds. Particularly interesting is the case where one of these ...manifolds is 1-dimensional and the case where the conformal factor depends on both manifolds simultaneously. If both factors are at least 3-dimensional then the latter case reduces to the product of two Einstein spaces, each of the special type admitting a non-trivial conformal gradient field. These are completely classified. If each factor is 2-dimensional, there is a special family of examples of non-constant curvature (called extremal metrics by Calabi), where in each factor the gradient of the Gaussian curvature is a conformal vector field. Then the metric of the 2-manifold is a warped product where the warping function is the first derivative of the Gaussian curvature. Moreover we find explicit examples of Einstein warped products with a 1-dimensional fibre and such with a 2-dimensional base. Therefore in the 4-dimensional case our Main Theorem points towards a local classification of conformally Einstein products. Finally we prove an assertion in the book by A. Besse on complete Einstein warped products with a 2-dimensional base. All solutions can be explicitly written in terms of integrals of elementary functions.
Equivelar maps on the torus Brehm, Ulrich; Kühnel, Wolfgang
European journal of combinatorics,
11/2008, Letnik:
29, Številka:
8
Journal Article
Recenzirano
Odprti dostop
We give a classification of all equivelar polyhedral maps on the torus. In particular, we classify all triangulations and quadrangulations of the torus admitting a vertex transitive automorphism ...group. These are precisely the ones which are quotients of the regular tessellations {3,6}, {6,3} or {4,4} by a pure translation group. An explicit formula for the number of combinatorial types of equivelar maps (polyhedral and non-polyhedral) with
n
vertices is obtained in terms of arithmetic functions in elementary number theory, such as the number of integer divisors of
n
. The asymptotic behaviour for
n
→
∞
is also discussed, and an example is given for
n
such that the number of distinct equivelar triangulations of the torus with
n
vertices is larger than
n
itself. The numbers of regular and chiral maps are determined separately, as well as the ones for all other kinds of symmetry. Furthermore, arithmetic properties of the integers of type
p
2
+
p
q
+
q
2
(or
p
2
+
q
2
, resp.) can be interpreted and visualized by the hierarchy of covering maps between regular and chiral equivelar maps or type {3,6} (or {4,4}, resp.).
This paper gives answers to a few questions concerning tilings of Euclidean spaces where the tiles are topological simplices with curvilinear edges. We investigate
lattice triangulations
of Euclidean ...3-space in the sense that the vertices form a lattice of rank 3 and such that the triangulation is invariant under all translations of that lattice. This is the dual concept of a primitive lattice tiling where the tiles are not assumed to be Euclidean polyhedra but only topological polyhedra. In 3-space there is a unique standard lattice triangulation by Euclidean tetrahedra (and with straight edges) but there are infinitely many non-standard lattice triangulations where the tetrahedra necessarily have certain curvilinear edges. From the view-point of Discrete Differential Geometry this tells us that there are such triangulations of 3-space which do not carry any flat discrete metric which is equivariant under the lattice. Furthermore, we investigate lattice triangulations of the 3-dimensional torus as quotients by a sublattice. The standard triangulation admits such quotients with any number
n
≥ 15 of vertices. The unique one with 15 vertices is neighborly, i.e., any two vertices are joined by an edge. It turns out that for any odd
n
≥ 17 there is an
n
-vertex neighborly triangulation of the 3-torus as a quotient of a certain non-standard lattice triangulation. Combinatorially, one can obtain these neighborly 3-tori as slight modifications of the boundary complexes of the cyclic 4-polytopes. As a kind of combinatorial surgery, this is an interesting construction by itself.
We investigate polyhedral 2
k
-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex
k
-Hamiltonian
if it contains the full
k
-skeleton of the polytope. ...Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called
super-neighborly triangulations
), we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the
d
-dimensional cross polytope. These are the “regular cases” satisfying equality in Sparla’s inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of seven copies of
S
2
×
S
2
. By this example all regular cases of
n
vertices with
n
<20 or, equivalently, all cases of regular
d
-polytopes with
d
≤9 are now decided.
The sixth edition of this classic work makes mastering a vast amount of information on internal organs much less daunting. It offers a vivid review of the human body and its structure, and it is an ...ideal study companion as well as an excellent basic reference text. These are some of the many user-friendly features of this book New color plates on embryology and histology More than 200 outstanding full-color illustrations and 130 clinical correlations Side-by-side images with explanatory text An overview of anatomical terms in each section Emphasizing clinical anatomy, this text integrates current information from a wide range of medical disciplines into discussions of the internal organs, including: Cross-sectional anatomy as a basis for working with modern imaging modalities Detailed explanations of organ topography and function Physiological and biochemical information included where appropriate An entire chapter devoted to pregnancy and human development Volume 2: Internal Organs and its companions Volume 1: Locomotor System and Volume 3: Nervous System and Sensory Organs comprise a must-have resource for students of medicine, dentistry, and all allied health fields.
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