scholarly journals Cutting Convex Polytopes by Hyperplanes

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 381 ◽  
Author(s):  
Takayuki Hibi ◽  
Nan Li
Keyword(s):  
Convex Polytopes ◽  
Unit Cube ◽  
Single Chain ◽  
Combinatorial Properties ◽  
Partial Classification ◽  
Separating Hyperplane ◽  
Birkhoff Polytopes ◽  
Natural Way ◽  
Finite Posets

Cutting a polytope is a very natural way to produce new classes of interesting polytopes. Moreover, it has been very enlightening to explore which algebraic and combinatorial properties of the original polytope are hereditary to its subpolytopes obtained by a cut. In this work, we devote our attention to all the separating hyperplanes for some given polytope (integral and convex) and study the existence and classification of such hyperplanes. We prove the existence of separating hyperplanes for the order and chain polytopes for any finite posets that are not a single chain, and prove there are no such hyperplanes for any Birkhoff polytopes. Moreover, we give a complete separating hyperplane classification for the unit cube and its subpolytopes obtained by one cut, together with some partial classification results for order and chain polytopes.

Partial classification of modules for the algebra of skew-derivations over the d-dimensional torus

2016 ◽  
Vol 57 (10) ◽  
pp. 101702 ◽  
Author(s):  
Chengkang Xu
Keyword(s):  
Dimensional Torus ◽  
Partial Classification

scholarly journals Molecular basis for Glanzmann's thrombasthenia (GT) in a compound heterozygote with glycoprotein IIb gene: a proposal for the classification of GT based on the biosynthetic pathway of glycoprotein IIb-IIIa complex

Blood ◽  
1992 ◽  
Vol 79 (12) ◽  
pp. 3212-3218 ◽  
Author(s):  
A Kato ◽  
K Yamamoto ◽  
S Miyazaki ◽  
SM Jung ◽  
M Moroi ◽  
...  
Keyword(s):  
Molecular Basis ◽  
Biosynthetic Pathway ◽  
Control Level ◽  
Exon Skipping ◽  
Compound Heterozygote ◽  
Acceptor Site ◽  
Single Chain ◽  
Glanzmann’S Thrombasthenia ◽  
Glanzmann's Thrombasthenia

Abstract The genetic basis for Glanzmann's thrombasthenia (GT) was elucidated on a compound heterozygote with glycoprotein (GP)IIb gene: an opal mutation at the end of exon 17 (CGA----TGA) results in only a trace amount of GPIIb mRNA, and a splicing mutation at the acceptor site of exon 26 (CAG----GAG) causes an in-frame, exon skipping process from exon 25 to 27. This aberrant transcript encodes a single-chain polypeptide characterized by a 42-amino acid deletion, which includes the proteolytic cleavage site(s) and a unique, proline-rich region at the location corresponding to the carboxyl-terminal of the normal GPIIb alpha-chain. These characteristics are shared by a previously reported defective GPIIb molecule, which is neither assembled with GPIIIa nor transported to the cellular surface. Despite its normal transcription level, expression of the present defective GPIIb molecule was significantly decreased (approximately 6% of the control level). Because the precursor GPIIb molecule is assembled with GPIIIa in the endoplasmic reticulum (ER) and its processing, as well as stability, is dependent on the GPIIIa subunit, the defective GPIIb molecule may be rapidly degraded by the intrinsic quality control system of the ER due to its inability to form a stable heterodimer complex as a consequence of its misfolded structure. Although we did not confirm that the GPIIIa genes of this individual were normal, GPIIIa may be secondarily decreased (approximately 11% of control), because a large part of it could not be complexed, making it vulnerable to proteolysis. To elucidate the molecular basis for GT, we propose here a classification of GT based on the biosynthetic pathway of the GPIIb-IIIa complex.

FREE INVERSE MONOIDS AND GRAPH IMMERSIONS

1993 ◽  
Vol 03 (01) ◽  
pp. 79-99 ◽  
Author(s):  
STUART W. MARGOLIS ◽  
JOHN C. MEAKIN
Keyword(s):  
Formal Language ◽  
Inverse Monoid ◽  
Finitely Generated ◽  
Inverse Monoids ◽  
Rational Subset ◽  
The Relationship ◽  
Subgroup Theorem ◽  
Free Inverse Monoid ◽  
Natural Way

The relationship between covering spaces of graphs and subgroups of the free group leads to a rapid proof of the Nielsen-Schreier subgroup theorem. We show here that a similar relationship holds between immersions of graphs and closed inverse submonoids of free inverse monoids. We prove using these methods, that a closed inverse submonoid of a free inverse monoid is finitely generated if and only if it has finite index if and only if it is a rational subset of the free inverse monoid in the sense of formal language theory. We solve the word problem for the free inverse category over a graph Γ. We show that immersions over Γ may be classified via conjugacy classes of loop monoids of the free inverse category over Γ. In the case that Γ is a bouquet of X circles, we prove that the category of (connected) immersions over Γ is equivalent to the category of (transitive) representations of the free inverse monoid FIM(X). Such representations are coded by closed inverse submonoids of FIM(X). These monoids will be constructed in a natural way from groups acting freely on trees and they admit an idempotent pure retract onto a free inverse monoid. Applications to the classification of finitely generated subgroups of free groups via finite inverse monoids are developed.

C-Nodal Surfaces of Order Three

1983 ◽  
Vol 35 (1) ◽  
pp. 68-100
Author(s):  
Tibor Bisztriczky
Keyword(s):  
Nineteenth Century ◽  
Singular Point ◽  
Algebraic Surface ◽  
Partial Classification ◽  
Differentiability Condition ◽  
Nodal Surfaces

The problem of describing a surface of order three can be said to originate in the mid-nineteenth century when A. Cayley discovered that a non-ruled cubic (algebraic surface of order three) may contain up to twenty-seven lines. Besides a classification of cubics, not much progress was made on the problem until A. Marchaud introduced his theory of synthetic surfaces of order three in [9]. While his theory resulted in a partial classification of a now larger class of surfaces, it was too general to permit a global description. In [1], we added a differentiability condition to Marchaud's definition. This resulted in a partial classification and description of surfaces of order three with exactly one singular point in [2]-[5]. In the present paper, we examine C-nodal surfaces and thus complete this survey.

scholarly journals The algebraic structure of relativistic wave equations

1978 ◽  
Vol 20 (4) ◽  
pp. 446-486 ◽  
Author(s):  
A. Cant ◽  
C. A. Hurst
Keyword(s):  
Lie Algebras ◽  
Algebraic Structure ◽  
Mass Spectra ◽  
Wave Equations ◽  
Relativistic Wave Equations ◽  
Relativistic Wave ◽  
Partial Classification ◽  
The Family ◽  
Image Position

The algebraic structure of relativistic wave equations of the formis considered. This leads to the problem of finding all Lie algebrasLwhich contain the Lorentz Lie algebraso(3, 1) and also contain a “four-vector” αμa such anLgives rise to a family of wave equations. The simplest possibility is the Bhabha equations whereL≅so(5). Some authors have claimed that this is theonlyone, but it is shown that there are many other possibilities still in accord with physical requirements. Known facts about representations, along with Dynkin's theory of the embeddings of Lie algebras, are used to obtain a partial classification of wave equations. The discrete transformationsC, P, Tare also discussed, along with reality properties. Finally, a simple example of a family of wave equations based onL=sp(12) is considered in detail. Theso(3, 1) content and mass spectra are given for the low order members of the family, and the problem of causality is briefly discussed.

scholarly journals Partial classification of heteroclinic behaviour associated with the perturbation of hexagonal planforms

2008 ◽  
Vol 23 (2) ◽  
pp. 137-162 ◽  
Author(s):  
M.J. Parker ◽  
I.N. Stewart ◽  
M.G.M. Gomes
Keyword(s):  
Partial Classification

NEUTRAL EVOLUTION AND MUTATION RATES OF SEQUENTIAL DYNAMICAL SYSTEMS

2004 ◽  
Vol 07 (03n04) ◽  
pp. 395-418
Author(s):  
H. S. MORTVEIT ◽  
C. M. REIDYS
Keyword(s):  
Dynamical Systems ◽  
Mutation Rate ◽  
Distance Measure ◽  
Neutral Evolution ◽  
Mutation Rates ◽  
Combinatorial Properties ◽  
Labeled Graph ◽  
Sequential Dynamical Systems ◽  
Natural Way ◽  
Independent Probability

In this paper we study the evolution of sequential dynamical systems [Formula: see text] as a result of the erroneous replication of the SDS words. An [Formula: see text] consists of (a) a finite, labeled graph Y in which each vertex has a state, (b) a vertex labeled sequence of functions (Fvi,Y), and (c) a word w, i.e. a sequence (w1,…,wk), where each wi is a Y-vertex. The function Fwi,Y updates the state of vertex wi as a function of the states of wi and its Y-neighbors and leaves the states of all other vertices fixed. The [Formula: see text] over the word w and Y is the composed map: [Formula: see text]. The word w represents the genotype of the [Formula: see text] in a natural way. We will randomly flip consecutive letters of w with independent probability q and study the resulting evolution of the [Formula: see text]. We introduce combinatorial properties of [Formula: see text] which allow us to construct a new distance measure [Formula: see text] for words. We show that [Formula: see text] captures the similarity of corresponding [Formula: see text]. We will use the distance measure [Formula: see text] to study neutrality and mutation rates in the evolution of words. We analyze the structure of neutral networks of words and the transition of word populations between them. Furthermore, we prove the existence of a critical mutation rate beyond which a population of words becomes essentially randomly distributed, and the existence of an optimal mutation rate at which a population maximizes its mutant offspring.

scholarly journals Partial classification of modules for Lie-algebra of diffeomorphisms of d-dimensional torus

2004 ◽  
Vol 45 (8) ◽  
pp. 3322-3333 ◽  
Author(s):  
S. Eswara Rao
Keyword(s):  
Lie Algebra ◽  
Dimensional Torus ◽  
Partial Classification

scholarly journals Antiautomorphisms and Biantiautomorphisms of Some Finite Abelian Groups

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 294
Author(s):  
Daniel López-Aguayo ◽  
Servando López Aguayo
Keyword(s):  
Lower Bound ◽  
Abelian Groups ◽  
Additive Group ◽  
Exact Formula ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Partial Classification ◽  
Open Questions ◽  
Odd Order

We extend the concepts of antimorphism and antiautomorphism of the additive group of integers modulo n, given by Gaitanas Konstantinos, to abelian groups. We give a lower bound for the number of antiautomorphisms of cyclic groups of odd order and give an exact formula for the number of linear antiautomorphisms of cyclic groups of odd order. Finally, we give a partial classification of the finite abelian groups which admit antiautomorphisms and state some open questions.

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