scholarly journals Place-Difference-Value Patterns: A Generalization of Generalized Permutation and Word Patterns

Integers ◽  
2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Sergey Kitaev ◽  
Jeffrey Remmel
Keyword(s):  
Partially Ordered Sets ◽  
General Definition ◽  
Permutation Patterns ◽  
Ordered Sets ◽  
Combinatorial Objects ◽  
Permutation Pattern ◽  
New Type ◽  
Number Of Publications ◽  
Definition Of ◽  
Patterns In Permutations

AbstractMotivated by the study of Mahonian statistics, in 2000, Babson and Steingrímsson [Sém. Lothar. Comb] introduced the notion of a “generalized permutation pattern” (GP) which generalizes the concept of “classical” permutation pattern introduced by Knuth in 1969. The invention of GPs led to a large number of publications related to properties of these patterns in permutations and words. Since the work of Babson and Steingrímsson, several further generalizations of permutation patterns have appeared in the literature, each bringing a new set of permutation or word pattern problems and often new connections with other combinatorial objects and disciplines. For example, Bousquet-Mélou et al. [J. Comb. Theory A] introduced a new type of permutation pattern that allowed them to relate permutation patterns theory to the theory of partially ordered sets.In this paper we introduce yet another, more general definition of a pattern, called place-difference-value patterns (PDVP) that covers all of the most common definitions of permutation and/or word patterns that have occurred in the literature. PDVPs provide many new ways to develop the theory of patterns in permutations and words. We shall give several examples of PDVPs in both permutations and words that cannot be described in terms of any other pattern conditions that have been introduced previously. Finally, we discuss several bijective questions linking our patterns to other combinatorial objects.

scholarly journals Mesh Patterns and the Expansion of Permutation Statistics as Sums of Permutation Patterns

10.37236/2001 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Petter Brändén ◽  
Anders Claesson
Keyword(s):  
Fixed Points ◽  
Relative Position ◽  
Permutation Statistics ◽  
Permutation Patterns ◽  
Alternating Permutations ◽  
Major Index ◽  
Permutation Pattern ◽  
New Type ◽  
Permutation Statistic ◽  
Infinite Linear

Any permutation statistic $f:{\mathfrak{S}}\to{\mathbb C}$ may be represented uniquely as a, possibly infinite, linear combination of (classical) permutation patterns: $f= \Sigma_\tau\lambda_f(\tau)\tau$. To provide explicit expansions for certain statistics, we introduce a new type of permutation patterns that we call mesh patterns. Intuitively, an occurrence of the mesh pattern $p=(\pi,R)$ is an occurrence of the permutation pattern $\pi$ with additional restrictions specified by $R$ on the relative position of the entries of the occurrence. We show that, for any mesh pattern $p=(\pi,R)$, we have $\lambda_p(\tau) = (-1)^{|\tau|-|\pi|}{p}^{\star}(\tau)$ where ${p}^{\star}=(\pi,R^c)$ is the mesh pattern with the same underlying permutation as $p$ but with complementary restrictions. We use this result to expand some well known permutation statistics, such as the number of left-to-right maxima, descents, excedances, fixed points, strong fixed points, and the major index. We also show that alternating permutations, André permutations of the first kind and simsun permutations occur naturally as permutations avoiding certain mesh patterns. Finally, we provide new natural Mahonian statistics.

scholarly journals Flag-symmetric and Locally Rank-symmetric Partially Ordered Sets

10.37236/1264 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Richard P. Stanley
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Partially Ordered Sets ◽  
Schur Function ◽  
Ordered Sets ◽  
Graded Poset ◽  
Kostka Polynomials ◽  
Relative Version ◽  
Definition Of ◽  
Formal Power

For every finite graded poset $P$ with $\hat{0}$ and $\hat{1}$ we associate a certain formal power series $F_P(x) = F_P(x_1,x_2,\dots)$ which encodes the flag $f$-vector (or flag $h$-vector) of $P$. A relative version $F_{P/\Gamma}$ is also defined, where $\Gamma$ is a subcomplex of the order complex of $P$. We are interested in the situation where $F_P$ or $F_{P/\Gamma}$ is a symmetric function of $x_1,x_2,\dots$. When $F_P$ or $F_{P/\Gamma}$ is symmetric we consider its expansion in terms of various symmetric function bases, especially the Schur functions. For a class of lattices called $q$-primary lattices the Schur function coefficients are just values of Kostka polynomials at the prime power $q$, thus giving in effect a simple new definition of Kostka polynomials in terms of symmetric functions. We extend the theory of lexicographically shellable posets to the relative case in order to show that some examples $(P,\Gamma)$ are relative Cohen-Macaulay complexes. Some connections with the representation theory of the symmetric group and its Hecke algebra are also discussed.

scholarly journals New Type Direction Curves in 3-Dimensional Compact Lie Group

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 387 ◽  
Author(s):  
Ali Çakmak
Keyword(s):  
Lie Group ◽  
Three Dimensional ◽  
Normal Direction ◽  
General Definition ◽  
Compact Lie Group ◽  
3 Dimensional ◽  
New Type ◽  
Definition Of

In this paper, new types of associated curves, which are defined as rectifying-direction, osculating-direction, and normal-direction, in a three-dimensional Lie group G are achieved by using the general definition of the associated curve, and some characterizations for these curves are obtained. Additionally, connections between the new types of associated curves and the curves, such as helices, general helices, Bertrand, and Mannheim, are given.

To the spectral theory of partially ordered sets

2018 ◽  
Vol 60 (3) ◽  
pp. 578-598
Author(s):  
Yu. L. Ershov ◽  
M. V. Schwidefsky
Keyword(s):  
Spectral Theory ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered

scholarly journals A remark on local fractional calculus and ordinary derivatives

2016 ◽  
Vol 14 (1) ◽  
pp. 1122-1124 ◽  
Author(s):  
Ricardo Almeida ◽  
Małgorzata Guzowska ◽  
Tatiana Odzijewicz
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Short Note ◽  
General Definition ◽  
Local Fractional Derivative ◽  
Fundamental Properties ◽  
Local Fractional Calculus ◽  
Definition Of

AbstractIn this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.

scholarly journals AB0677 REVISING THE DEFINITION OF REACTIVE ARTHRITIS AND DIFFERENTIATION FROM UNDIFFERENTIATED SPONDYLOARTHRITIS

2020 ◽  
Vol 79 (Suppl 1) ◽  
pp. 1633.2-1634
Author(s):  
F. Cosan ◽  
O. M. Gedar
Keyword(s):  
Causative Agent ◽  
Reactive Arthritis ◽  
Case Reports ◽  
Experimental Studies ◽  
Classification Criteria ◽  
General Definition ◽  
Parasitic Infections ◽  
Undifferentiated Spondyloarthritis ◽  
Classical Definition ◽  
Definition Of

Background:Reactive arthritis (ReA) is defined by 1999 ACR criteria as arthritis preceding a bacterial genitourinary (GUS) or gastrointestinal (GIS) infection in 3 days-6 weeks and evidence of triggering infection. Recently, ReA is classified as SpA and patients who do not fulfill SpA criteria are classified as undifferentiated spondyloarthritis (USpA) according to ASAS/EULAR SpA classification criteria.Objectives:In several case reports which are associated with other infective agents are reported and the definition is extended for some clinicians so that SpA which is occurred after any infection is called as ReA. On the other hand, some researchers still accept the classical definition of ReA. The problem with the heterogeneity of opinions and unstandardized definition of ReA hinders studies about pathogenesis and standardization of treatments. In this study, we aimed to determine the spectrum of the use of the definition of reactive arthritis in publications in PubMed between 2009-2019.Methods:The ReA keyword is searched in PubMed for the years between 2009-2019. 248 different publications have been identified and included in this research. 89 articles, 47 reviews, 108 case reports, 2 guidelines, and 2 editorials reviewed for the definition of ReA.Results:Only 42.7% (106 patients) of these publications meet the classical definition which suggests ReA after only GIS and GUS infections. In 4 (1.6%) of the publications ReA was defined after GIS, GUS and oropharyngeal infections; in 3 (1,2%) of the publications after any bacterial infection; in 9 (3.6%) of the publications after any infection. In 8 (3.2%) of the publications, ReA and USPA was used correspondingly. In 39 (15,7%) of the publications the term agent related, ReA was used without making a general definition for ReA. 79 publications (31,9%) have not defined ReA.According to causative agent and ReA relationship, in 64 (24,6%) general infective agents, in 75 (30,2%) classical agents, in 22 (8,9%) other bacterial agents, in 23 (9,3%) streptococcus, in 10(4%) intravesical BCG, in 6 (2.4%) HIV, in 6 (2.4%) tuberculosis, in 12 (4,8%) clostrudium difficle, in 2 (0.8%) parasites were reported. In 31 (12,5%) of the publications the causative agent for the ReA was unknown, the diagnosis was made clinically.Conclusion:In this study, it is aimed to draw attention terminology intricacy and the need for the standardization of the definition of ReA and USpA. It is clear that to standardize the definition of Rea and USpA is necessary. Between 2009-2019 there are reported cases diagnosed as ReA associated with bacterial infections (especially with Clostridium difficile, streptococcus and tuberculosis infections), and viral infections (by a majority with HIV), and parasitic infections. It is not clear if we need to define them classically or define them as USPA. Another important consideration is the necessity of extended laboratory investigations to find out the real causative agent even if the patient is clinically diagnosed with ReA. The requirement of the differentiation between ReA and USpA must be revealed for therapeutic researches.References:[1]A proposal for the classification of patients for clinical and experimental studies on reactive arthritis. Pacheco-Tena C, Burgos-Vargas R, Vázquez-Mellado J, Cazarín J, Pérez-Díaz JA. J Rheumatol. 1999 Jun;26(6):1338-46.[2]The Assessment of SpondyloArthritis International Society classification criteria for peripheral spondyloarthritis and for spondyloarthritis in general. Rudwaleit M, van der Heijde D, Landewé R, Akkoc N, Brandt J, Chou CT, Dougados M, Huang F, Gu J, Kirazli Y, et al. Ann Rheum Dis. 2011;70:25–31.Disclosure of Interests:None declared

Modelling Nondeterministic Concurrent Processes with Event Structures

1991 ◽  
Vol 14 (1) ◽  
pp. 39-73
Author(s):  
Rita Loogen ◽  
Ursula Goltz
Keyword(s):  
Dynamic Behaviour ◽  
Partially Ordered Sets ◽  
Parallel Composition ◽  
Ordered Sets ◽  
Communicating Sequential Processes ◽  
Partially Ordered ◽  
Event Structures ◽  
Transition Relation ◽  
Concurrent Processes ◽  
Sequential Processes

We present a non-interleaving model for non deterministic concurrent processes that is based on labelled event structures. We define operators on labelled event structures like parallel composition, nondeterministic combination, choice, prefixing and hiding. These operators correspond to the operations of the “Theory of Communicating Sequential Processes” (TCSP). Infinite processes are defined using the metric approach. The dynamic behaviour of event structures is defined by a transition relation which describes the execution of partially ordered sets of actions, abstracting from internal events.

Estimating the area under a receiver operating characteristic curve using partially ordered sets

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Ehsan Zamanzade ◽  
Xinlei Wang
Keyword(s):  
Roc Curve ◽  
Characteristic Curve ◽  
Partially Ordered Sets ◽  
Sampling Technique ◽  
Real Data ◽  
Cost Effective ◽  
Ordered Sets ◽  
Operating Characteristics ◽  
Partially Ordered ◽  
Receiver Operating

AbstractRanked set sampling (RSS), known as a cost-effective sampling technique, requires that the ranker gives a complete ranking of the units in each set. Frey (2012) proposed a modification of RSS based on partially ordered sets, referred to as RSS-t in this paper, to allow the ranker to declare ties as much as he/she wishes. We consider the problem of estimating the area under a receiver operating characteristics (ROC) curve using RSS-t samples. The area under the ROC curve (AUC) is commonly used as a measure for the effectiveness of diagnostic markers. We develop six nonparametric estimators of the AUC with/without utilizing tie information based on different approaches. We then compare the estimators using a Monte Carlo simulation and an empirical study with real data from the National Health and Nutrition Examination Survey. The results show that utilizing tie information increases the efficiency of estimating the AUC. Suggestions about when to choose which estimator are also made available to practitioners.

scholarly journals A lattice-theoretic characterization of pure subgroups of Abelian groups

2021 ◽  
Author(s):  
M. Ferrara ◽  
M. Trombetti
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Subgroup Lattice ◽  
General Definition ◽  
Short Paper ◽  
Theoretic Characterization ◽  
Definition Of

AbstractLet G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice $$\mathcal {L}(H)$$ L ( H ) embeds in $$\mathcal {L}(G)$$ L ( G ) . It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.

scholarly journals Fixed Point of Interpolative Rus–Reich–Ćirić Contraction Mapping on Rectangular Quasi-Partial b-Metric Space

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 32
Author(s):  
Pragati Gautam ◽  
Luis Manuel Sánchez Ruiz ◽  
Swapnil Verma
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Real Number ◽  
Metric Space ◽  
Contraction Mapping ◽  
Metric Spaces ◽  
Rectangular Metric Space ◽  
New Type ◽  
Symmetry Requirement ◽  
Definition Of

The purpose of this study is to introduce a new type of extended metric space, i.e., the rectangular quasi-partial b-metric space, which means a relaxation of the symmetry requirement of metric spaces, by including a real number s in the definition of the rectangular metric space defined by Branciari. Here, we obtain a fixed point theorem for interpolative Rus–Reich–Ćirić contraction mappings in the realm of rectangular quasi-partial b-metric spaces. Furthermore, an example is also illustrated to present the applicability of our result.

Sign in / Sign up

Export Citation Format

Share Document