scholarly journals A Short Approach to Catalan Numbers Modulo $2^r$

10.37236/664 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Guoce Xin ◽  
Jing-Feng Xu
Keyword(s):  
Catalan Numbers ◽  
Binary Expansion ◽  
Dyck Paths ◽  
Combinatorial Interpretations ◽  
Recent Classification ◽  
Motzkin Paths

We notice that two combinatorial interpretations of the well-known Catalan numbers $C_n=(2n)!/n!(n+1)!$ naturally give rise to a recursion for $C_n$. This recursion is ideal for the study of the congruences of $C_n$ modulo $2^r$, which attracted a lot of interest recently. We present short proofs of some known results, and improve Liu and Yeh's recent classification of $C_n$ modulo $2^r$. The equivalence $C_{n}\equiv_{2^r} C_{\bar n}$ is further reduced to $C_{n}\equiv_{2^r} C_{\tilde{n}}$ for simpler $\tilde{n}$. Moreover, by using connections between weighted Dyck paths and Motzkin paths, we find new classes of combinatorial sequences whose $2$-adic order is equal to that of $C_n$, which is one less than the sum of the digits of the binary expansion of $n+1$.

scholarly journals Path Counting and Random Matrix Theory

10.37236/1736 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Ioana Dumitriu ◽  
Etienne Rassart
Keyword(s):  
Random Walks ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Open Problem ◽  
Matrix Theory ◽  
Dyck Paths ◽  
Combinatorial Interpretations ◽  
Motzkin Paths ◽  
Path Counting

We establish three identities involving Dyck paths and alternating Motzkin paths, whose proofs are based on variants of the same bijection. We interpret these identities in terms of closed random walks on the halfline. We explain how these identities arise from combinatorial interpretations of certain properties of the $\beta$-Hermite and $\beta$-Laguerre ensembles of random matrix theory. We conclude by presenting two other identities obtained in the same way, for which finding combinatorial proofs is an open problem.

scholarly journals A57 THE USE AND INTERPRETATION OF HLA-DQ2/DQ8 GENOTYPING BY PEDIATRIC GASTROENTEROLOGISTS

2021 ◽  
Vol 4 (Supplement_1) ◽  
pp. 14-15
Author(s):  
B Moreau ◽  
E Robidoux
Keyword(s):  
Risk Groups ◽  
Risk Classification ◽  
Low Risk ◽  
Risk Alleles ◽  
Seronegative Patient ◽  
Espghan Guidelines ◽  
Recent Classification ◽  
American Society ◽  
Hla Dq2

Abstract Background A recent classification of high and low risk alleles associated with celiac disease (CD) shows that the presence of a single allele (DQA1*05 or DQB1*02; coding together for HLA-DQ2), without a positive genotype (HLA-DQ2 and or HLA-DQ8), represents a risk of developing the disease. Aims The aim of this study is to evaluate the use and interpretation of the HLA-DQ2/DQ8 genotyping by pediatric gastroenterologists, as there is no study on the matter and the latest guidelines do not address this risk classification. Methods A web-based survey was sent by email to all NASPGHAN (North American society of pediatric gastroenterolgy, hepatology and nutrition) members. Results Results 294 pediatric gastroenterologists sent a complete survey. 86,1% use the HLA-DQ2/DQ8 genotyping according mainly to the NASPGHAN and ESPGHAN guidelines. The main indications considered were to exclude CD in a patient on a gluten-free diet with a resolution of his symptoms and in a seronegative patient with equivocal biopsies. A minority would consider the genotyping for screening high risk groups or for making a diagnosis in children with high specific CD antibodies and strong clinical suspicion without performing biopsies, as suggested by the ESPGHAN guidelines. The alleles associated with CD are not well known, but 76,7% the participants are aware of the risk classification. While only 62,8% have access to the complete genotype, 47,8% consider it useful. Nevertheless, 82,6% would still want to know the presence of a low risk allele. Conclusions The risk classification of alleles related to CD warrants a modification of the genotyping result with access to the alleles and an adaptation of the guidelines. Funding Agencies None

scholarly journals On Pattern-Avoiding Partitions

10.37236/763 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Vít Jelínek ◽  
Toufik Mansour
Keyword(s):  
Stirling Numbers ◽  
Equivalence Classes ◽  
Catalan Numbers ◽  
Set Partition ◽  
Combinatorial Interpretations

A set partition of size $n$ is a collection of disjoint blocks $B_1,B_2,\ldots$, $B_d$ whose union is the set $[n]=\{1,2,\ldots,n\}$. We choose the ordering of the blocks so that they satisfy $\min B_1 < \min B_2 < \cdots < \min B_d$. We represent such a set partition by a canonical sequence $\pi_1,\pi_2,\ldots,\pi_n$, with $\pi_i=j$ if $i\in B_j$. We say that a partition $\pi$ contains a partition $\sigma$ if the canonical sequence of $\pi$ contains a subsequence that is order-isomorphic to the canonical sequence of $\sigma$. Two partitions $\sigma$ and $\sigma'$ are equivalent, if there is a size-preserving bijection between $\sigma$-avoiding and $\sigma'$-avoiding partitions. We determine all the equivalence classes of partitions of size at most $7$. This extends previous work of Sagan, who described the equivalence classes of partitions of size at most $3$. Our classification is largely based on several new infinite families of pairs of equivalent patterns. For instance, we prove that there is a bijection between $k$-noncrossing and $k$-nonnesting partitions, with a notion of crossing and nesting based on the canonical sequence. Our results also yield new combinatorial interpretations of the Catalan numbers and the Stirling numbers.

scholarly journals Neuropathology of Temporal Lobe Epilepsy

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Fahd Al Sufiani ◽  
Lee Cyn Ang
Keyword(s):  
Temporal Lobe Epilepsy ◽  
Temporal Lobe ◽  
Focal Cortical Dysplasia ◽  
Cortical Dysplasia ◽  
Wide Range ◽  
Pathologic Findings ◽  
Pathologic Lesions ◽  
Recent Classification ◽  
Associated Tumors

Pathologic findings in surgical resections from patients with temporal lobe epilepsy include a wide range of diagnostic possibilities that can be categorized into different groups on the basis of etiology. This paper outlines the various pathologic entities described in temporal lobe epilepsy, including some newly recognized epilepsy-associated tumors, and briefly touch on the recent classification of focal cortical dysplasia. This classification takes into account coexistent pathologic lesions in focal cortical dysplasia.

A Global Phylogenomic Study of the Thelypteridaceae

2021 ◽  
Vol 46 (4) ◽  
pp. 891-915
Author(s):  
Susan Fawcett ◽  
Alan R. Smith ◽  
Michael Sundue ◽  
J. Gordon Burleigh ◽  
Emily B. Sessa ◽  
...  
Keyword(s):  
Morphological Characters ◽  
Morphological Evidence ◽  
Phylogenetic Studies ◽  
The Family ◽  
Likelihood Analysis ◽  
Single Genus ◽  
The Subject ◽  
Recent Classification ◽  
Phylogenomic Study

Abstract— The generic classification of the Thelypteridaceae has been the subject of much controversy. Proposed taxonomic systems have varied from recognizing the approximately 1200 species in the family within the single genus Thelypteris, to systems favoring upwards of 30 genera. Insights on intrafamilial relationships, especially for neotropical taxa, have been gained from recent phylogenetic studies; however, in the most recent classification, 10 of 30 recognized genera are either non-monophyletic or untested. We sequenced 407 nuclear loci for 621 samples, representing all recognized genera and approximately half the known species diversity. These were analyzed using both maximum likelihood analysis of a concatenated matrix and multi-species coalescent methods. Our phylogenomic results, informed by recently published morphological evidence, provide the foundation for a generic classification which recircumscribed 14 genera and recognized seven new genera. The 37 monophyletic genera sampled demonstrate greater geographic coherence than previous taxonomic concepts suggested. Additionally, our results demonstrate that certain morphological characters, such as frond division, are evolutionarily labile and are thus inadequate for defining genera.

SCALING THEORY AND THE CLASSIFICATION OF PHASE TRANSITIONS

1992 ◽  
Vol 06 (13) ◽  
pp. 773-784 ◽  
Author(s):  
R. HILFER
Keyword(s):  
Phase Transitions ◽  
Statistical Physics ◽  
Thermodynamic Description ◽  
Classification Theory ◽  
Nonequilibrium Phase Transitions ◽  
Critical Limit ◽  
Scaling Hypothesis ◽  
Recent Classification ◽  
Thermodynamics And Statistical Mechanics

The recent classification theory for phase transitions (R. Hilfer, Physica Scripta 44, 321 (1991)) and its relation with the foundations of statistical physics is reviewed. First it is outlined how Ehrenfests classification scheme can be generalized into a general thermodynamic classification theory for phase transitions. The classification theory implies scaling and multiscaling thereby eliminating the need to postulate the scaling hypothesis as a fourth law of thermodynamics. The new classification has also led to the discovery and distinction of nonequilibrium transitions within equilibrium statistical physics. Nonequilibrium phase transitions are distinguished from equilibrium transitions by orders less than unity and by the fact that equilibrium thermodynamics and statistical mechanics become inapplicable at the critical point. The latter fact requires a change in the Gibbs assumption underlying the canonical and grandcanonical ensembles in order to recover the thermodynamic description in the critical limit.

scholarly journals Recent Classification of the Shrews

Nature ◽  
1890 ◽  
Vol 42 (1093) ◽  
pp. 567-568
Author(s):  
R. W. SHUFELDT
Keyword(s):  
Recent Classification

Range-compatible homomorphisms over the field with two elements

2018 ◽  
Vol 34 ◽  
pp. 71-114
Author(s):  
Clément De Seguins Pazzis
Keyword(s):  
Linear Subspace ◽  
Linear Operators ◽  
Vector Spaces ◽  
Operator Spaces ◽  
Affine Subspaces ◽  
Finite Dimensional ◽  
Reflexive Operator ◽  
Recent Classification ◽  
Special Case

Let U and V be finite-dimensional vector spaces over a field K, and S be a linear subspace of the space L(U, V ) of all linear operators from U to V. A map F : S → V is called range-compatible when F(s) ∈ Im s for all s ∈ S. Previous work has classified all the range-compatible group homomorphisms provided that codimL(U,V )S ≤ 2 dim V − 3, except in the special case when K has only two elements and codimL(U,V )S = 2 dim V − 3. This article gives a thorough treatment of that special case. The results are partly based upon the recent classification of vector spaces of matrices with rank at most 2 over F2. As an application, the 2-dimensional non-reflexive operator spaces are classified over any field, and so do the affine subspaces of Mn,p(K) with lower-rank at least 2 and codimension 3.

SOME COMMENTS ON A RECENT CLASSIFICATION OF THE MONIMIACEAE

Taxon ◽  
1988 ◽  
Vol 37 (4) ◽  
pp. 834-837 ◽  
Author(s):  
Juan B. Martínez‐Laborde
Keyword(s):  
Recent Classification

Agent Orange in War Medicine: An Aftermath Myth

1998 ◽  
Vol 28 (4) ◽  
pp. 715-724 ◽  
Author(s):  
Lennart Hardell ◽  
Mikael Eriksson ◽  
Olav Axelson
Keyword(s):  
Malignant Tumors ◽  
Epidemiological Studies ◽  
Agent Orange ◽  
Royal Commission ◽  
International Agency ◽  
Cancer Risks ◽  
Phenoxy Herbicides ◽  
Recent Classification ◽  
War Medicine

Since the late 1970s several epidemiological studies have appeared linking exposure to phenoxy herbicides or chlorophenols to some malignant tumors. Most of these compounds are contaminated with dioxins and dibenzofurans; for example, 2,3,7,8-tetrachlorodibenzo- p-dioxin (TCDD) is a contaminant of 2,4,5-trichlorophenoxyacetic acid (2,4,5-T), a component of Agent Orange which was sprayed in Vietnam during the war. The results of some of the epidemiological studies on cancer risks associated with exposure to these compounds have been manipulated and misinterpreted, particularly by the Australian Royal Commission on the Use and Effects of Chemical Agents on Australian Personnel in Vietnam. Furthermore, a book on Australian war history entitled Medicine at War, commissioned by the Federal Government, reiterates several of these misinterpretations, despite available contrary evaluations from Australian and U.S. authorities. These remarkable and confusing circumstances in the scientific process are considered also in the light of the recent classification of TCDD as carcinogenic to humans, Group 1, by a Working Group at the International Agency for Research on Cancer in Lyon, France.

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