scholarly journals Facets of the Generalized Cluster Complex and Regions in the Extended Catalan Arrangement of Type $A$

10.37236/3169 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Susanna Fishel ◽  
Myrto Kallipoliti ◽  
Eleni Tzanaki
Keyword(s):  
Simple Root ◽  
Type A ◽  
Cluster Complex ◽  
Integer Partitions ◽  
Positive Part

In this paper we present a bijection between two well known families of Catalan objects: the set of facets of the $m$-generalized cluster complex $\Delta^m(A_n)$ and that of dominant regions in the $m$-Catalan arrangement ${\rm Cat}^m(A_n)$, where $m\in\mathbb{N}_{>0}$. In particular, the map which we define bijects facets containing the negative simple root $-\alpha$ to dominant regions having the hyperplane $\{v\in V\mid\left\langle v,\alpha \right\rangle=m\}$ as separating wall. As a result, it restricts to a bijection between the set of facets of the positive part of $\Delta^m(A_n)$ and the set of bounded dominant regions in ${\rm Cat}^m(A_n)$. Our map is a composition of two bijections in which integer partitions in an $m$-dilated $n$-staircase shape come into play.

scholarly journals Multi-cluster complexes

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Cesar Ceballos ◽  
Jean-Philippe Labbé ◽  
Christian Stump
Keyword(s):  
Coxeter Groups ◽  
Type A ◽  
Simplicial Complexes ◽  
Cluster Complex ◽  
Geometric Properties ◽  
Cluster Complexes ◽  
Combinatorial Description ◽  
Compatibility Relation ◽  
Positive Roots ◽  
International Audience

International audience We present a family of simplicial complexes called \emphmulti-cluster complexes. These complexes generalize the concept of cluster complexes, and extend the notion of multi-associahedra of types ${A}$ and ${B}$ to general finite Coxeter groups. We study combinatorial and geometric properties of these objects and, in particular, provide a simple combinatorial description of the compatibility relation among the set of almost positive roots in the cluster complex. Nous présentons une famille de complexes simpliciaux appelés \emphcomplexes des multi-amas. Ces complexes généralisent le concept de complexes des amas et étendent la notion de multi-associaèdre de type ${A}$ et ${B}$ aux groupes de Coxeter finis. Nous étudions des propriétés combinatoires et géométriques de ces objets et, en particulier nous fournissons une description combinatoire simple de la relation de compatibilité sur l'ensemble des racines presque positives du complexe des amas.

scholarly journals Index of Seaweed Algebras and Integer Partitions

10.37236/9054 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Seunghyun Seo ◽  
Ae Ja Yee
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Generating Function ◽  
Generating Functions ◽  
Type A ◽  
Integer Partitions ◽  
Algebraic Invariant ◽  
Index Weight

The index of a Lie algebra is an important algebraic invariant.  In 2000, Vladimir Dergachev and Alexandre Kirillov  defined seaweed subalgebras of $\mathfrak{gl}_n$ (or $\mathfrak{sl}_n$) and provided a formula for the index of a seaweed algebra using a certain graph, a so called meander. In a recent paper, Vincent Coll, Andrew Mayers, and Nick Mayers defined a new statistic for partitions, namely  the index of a partition, which arises from seaweed Lie algebras of type A. At the end of their paper, they presented an interesting conjecture, which involves integer partitions into odd parts. Motivated by their work, in this paper, we exploit various index statistics and the index weight generating functions for partitions.  In particular, we examine their conjecture by considering the generating function for partitions into odd parts.  We will also reprove another result  from their paper using generating functions.

Biological and Morphological Studies on Low-Leukemia-Strain Mice with Hodgkin's-Like Neoplasia Induced by Serial Cell-Free Transmission of Leukemic Organs of SJL/J Strain Mice

1968 ◽  
Vol 26 ◽  
pp. 210-211
Author(s):  
S. Fujinaga ◽  
K. Maruyama ◽  
C.W. Williams ◽  
K. Sekhri ◽  
L. Dmochowski
Keyword(s):  
Tissue Culture ◽  
Type A ◽  
Reticulum Cell ◽  
Type B ◽  
Viral Origin ◽  
Serial Transfer ◽  
Strain Mouse ◽  
Morphological Studies ◽  
Cell Neoplasm ◽  
Free Material

Yumoto and Dmochowski (Cancer Res.27, 2098 (1967)) reported the presence of mature and immature type C leukemia virus particles in leukemic organs and tissues such as lymph nodes, spleen, thymus, liver, and kidneys of SJL/J strain mice with Hodgki's-like disease or reticulum cell neoplasm (type B). In an attempt to ascertain the possibility that this neoplasia may be of viral origin, experiments with induction and transmission of this neoplasm were carried out using cell-free extracts of leukemic organs from an SJL/J strain mouse with spontaneous disease.It has been possible to induce the disease in low-leukemia BALB/c and C3HZB strain mice and serially transfer the neoplasia by cell-free extracts of leukemic organs of these mice. Histological examination revealed the neoplasia to be of either reticulum cell-type A or type B. Serial transfer is now in its fifth passage. In addition leukemic spleen from another SJL/J strain mouse with spontaneous reticulum cell neoplasm (type A) was set up in tissue culture and is now in its 141st serial passage in vitro. Preliminary results indicate that cell-free material of 39th tissue culture passage can reproduce neoplasia in BALB/c mice.

Hepatitis A Antigen in Liver, Bile and Stool

1975 ◽  
Vol 33 ◽  
pp. 340-341
Author(s):  
D.R. Jackson ◽  
J.H. Hoofnagle ◽  
A.N. Schulman ◽  
J.L. Dienstag ◽  
R.H. Purcell ◽  
...  
Keyword(s):  
Hepatitis A ◽  
Liver Enzymes ◽  
Hepatitis A Virus ◽  
Type A ◽  
Initial Study ◽  
Virus Like Particle ◽  
Elevated Liver Enzymes ◽  
Ag Particles ◽  
Ag Particle ◽  
Human Stool

Using immune electron microscopy Feinstone et. al. demonstrated the presence of a 27 nm virus-like particle in acute-phase stools of patients with viral hepatitis, type A, These hepatitis A antigen (HA Ag) particles were aggregated by convalescent serum from patients with type A hepatitis but not by pre-infection serum. Subsequently Dienstag et. al. and Maynard et. al. produced acute hepatitis in chimpanzees by inoculation with human stool containing HA Ag. During the early acute disease, virus like particles antigenically, morphologically and biophysically identical to the human HA Ag particle were found in chimpanzee stool. Recently Hilleman et. al. have described similar particles in liver and serum of marmosets infected with hepatitis A virus (HAV). We have investigated liver, bile and stool from chimpanzees and marmosets experimentally infected with HAV. In an initial study, a chimpanzee (no.785) inoculated with HA Ag-containing stool developed elevated liver enzymes 21 days after exposure.

Detection of Viral Antigens in Mouse and Rat Tumor Cells by Immunoelectron Microscopy Following Periodate-Lysine-Paraformaldehyde (PLP) Fixation

1976 ◽  
Vol 34 ◽  
pp. 252-253
Author(s):  
Y. Ohtsuki ◽  
G. Seman ◽  
J. M. Bowen ◽  
M. Scanlon ◽  
L. Dmochowski
Keyword(s):  
Immunoelectron Microscopy ◽  
Mammary Tumor ◽  
Type A ◽  
Rabbit Serum ◽  
Virus Particles ◽  
Viral Antigens ◽  
Culture Cells ◽  
Mouse Mammary Tumor ◽  
Tumor Virus ◽  
Immunoperoxidase Labeling

Recently, periodate-lysine-paraformaldehyde (PLP) fixation was reported for immunoelectron microscopy (1). In PLP fixation, carbohydrates are oxidized by periodate and cross-linked by lysine; paraformaldehyde stabilizes proteins and lipids. By using PLP fixation, intracytoplasmic type A viral antigens have been previously demonstrated by immunoperoxidase labeling (2). In the present study, PLP fixation has been applied for the detection of the same antigens in mouse mammary tumor culture cells by both immunoferritin and immunoperoxidase methods. Rabbit anti-intracytoplasmic type A virus serum (anti-A), kindly provided by Dr. M. Muller (3), rabbit anti-strain A mouse mammary tumor virus (anti-MMTV) and preimmune rabbit serum as control were used to detect viral antigens in cells of C3H/HeJ strain mouse mammary tumor culture. Attempts have been also made to demonstrate peroxidase labeling of type C virus particles in frozen sections of an SD-MSV-induced NZB rat bone tumor tissue by rabbit anti-MuLV serum.

Type A and Heart Disease: Past, Present, and Future

1995 ◽  
Vol 40 (3) ◽  
pp. 272-272
Author(s):  
Lori A. Ingram ◽  
Gail M. Williamson
Keyword(s):  
Heart Disease ◽  
Type A
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