scholarly journals On Relative Ranks of the Semigroup of Orientation-Preserving Transformations on Infinite Chains

2020 ◽  
pp. 2150146
Author(s):  
Ilinka Dimitrova ◽  
Jörg Koppitz
Keyword(s):  
Relative Rank

In this paper, we determine the relative rank of the semigroup [Formula: see text] of all orientation-preserving transformations on infinite chains modulo the semigroup [Formula: see text] of all order-preserving transformations.

Tits Indices

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Coxeter Group ◽  
Coxeter System ◽  
Relative Rank ◽  
Coxeter Diagram ◽  
The Absolute ◽  
Relative Type

This chapter introduces the notion of a Tits index and the notion of the relative Coxeter diagram of a Tits index. It first defines a Tits index, which can be anisotropic or isotropic, quasi-split or split, before considering a number of propositions regarding compatible representations. It then gives a proof of the theorem that includes two assumptions about a Coxeter system, focusing on the absolute Coxeter system, the relative Coxeter system, and the relative Coxeter group of the Tits index, as well as the absolute Coxeter diagram (or absolute type), the relative Coxeter diagram (or relative type), and the absolute rank and the relative rank of the Tits index. The chapter concludes with some observations about the case that (W, S) is spherical, irreducible or affine.

Why Does Construct Validity Matter in Measuring Implementation Fidelity? A Methodological Case Study

2021 ◽  
pp. 153450842199877
Author(s):  
Wilhelmina van Dijk ◽  
A. Corinne Huggins-Manley ◽  
Nicholas A. Gage ◽  
Holly B. Lane ◽  
Michael Coyne
Keyword(s):  
Construct Validity ◽  
Reading Instruction ◽  
Model Building ◽  
Implementation Fidelity ◽  
Modeling Framework ◽  
Relative Rank ◽  
Rank Ordering ◽  
Adherence Data ◽  
Funded Project

In reading intervention research, implementation fidelity is assumed to be positively related to student outcomes, but the methods used to measure fidelity are often treated as an afterthought. Fidelity has been conceptualized and measured in many different ways, suggesting a lack of construct validity. One aspect of construct validity is the fidelity index of a measure. This methodological case study examined how different decisions in fidelity indices influence relative rank ordering of individuals on the construct of interest and influence our perception of the relation between the construct and intervention outcomes. Data for this study came from a large State-funded project to implement multi-tiered systems of support for early reading instruction. Analyses were conducted to determine whether the different fidelity indices are stable in relative rank ordering participants and if fidelity indices of dosage and adherence data influence researcher decisions on model building within a multilevel modeling framework. Results indicated that the fidelity indices resulted in different relations to outcomes with the most commonly used fidelity indices for both dosage and adherence being the worst performing. The choice of index to use should receive considerable thought during the design phase of an intervention study.

Tits Triangles

2018 ◽  
Vol 62 (3) ◽  
pp. 583-601
Author(s):  
Bernhard Mühlherr ◽  
Richard M. Weiss
Keyword(s):  
Bipartite Graph ◽  
Mild Condition ◽  
Stable Rank ◽  
Relative Rank ◽  
Standard Construction ◽  
Coxeter Diagram

AbstractA Tits polygon is a bipartite graph in which the neighborhood of every vertex is endowed with an “opposition relation” satisfying certain properties. Moufang polygons are precisely the Tits polygons in which these opposition relations are all trivial. There is a standard construction that produces a Tits polygon whose opposition relations are not all trivial from an arbitrary pair $(\unicode[STIX]{x1D6E5},T)$, where $\unicode[STIX]{x1D6E5}$ is a building of type $\unicode[STIX]{x1D6F1}$, $\unicode[STIX]{x1D6F1}$ is a spherical, irreducible Coxeter diagram of rank at least $3$, and $T$ is a Tits index of absolute type $\unicode[STIX]{x1D6F1}$ and relative rank $2$. A Tits polygon is called $k$-plump if its opposition relations satisfy a mild condition that is satisfied by all Tits triangles coming from a pair $(\unicode[STIX]{x1D6E5},T)$ such that every panel of $\unicode[STIX]{x1D6E5}$ has at least $k+1$ chambers. We show that a $5$-plump Tits triangle is parametrized and uniquely determined by a ring $R$ that is alternative and of stable rank $2$. We use the connection between Tits triangles and the theory of Veldkamp planes as developed by Veldkamp and Faulkner to show existence.

Optimal selection based on relative ranks of a sequence by ties

1984 ◽  
Vol 16 (01) ◽  
pp. 131-146
Author(s):  
Gregory Campbell
Keyword(s):  
Distribution Function ◽  
Dirichlet Process ◽  
Random Distribution ◽  
Discrete Distribution ◽  
Optimal Selection ◽  
Dirichlet Process Prior ◽  
Limiting Behavior ◽  
Relative Rank ◽  
Atomic Parameter ◽  
Selection Of

The optimal selection of a maximum of a sequence with the possibility of ties is considered. The object is to examine each observation in the sequence of known length n and, based only on the relative rank among predecessors, either to stop and select it as a maximum or to continue without recall. Rules which maximize the probability of correctly selecting a maximum from a sequence with ties are investigated. These include rules which randomly break ties, rules which discard tied observations, and minimax rules based on the atoms of a discrete distribution function. If the sequence is random from F, a random distribution function from a Dirichlet process prior with non-atomic parameter, optimal rules are developed. The limiting behavior of these rules is studied and compared with other rules. The selection of the parameter of the Dirichlet process regulates the ties.

Recognizing the maximum of a random sequence based on relative rank with backward solicitation

1974 ◽  
Vol 11 (03) ◽  
pp. 504-512 ◽  
Author(s):  
Mark C. K. Yang
Keyword(s):  
General Formula ◽  
Optimal Stopping ◽  
Random Sequence ◽  
Geometric Probability ◽  
Secretary Problem ◽  
Stopping Rules ◽  
Relative Rank ◽  
Special Cases ◽  
Optimal Stopping Rules

The classical secretary problem is generalized to admit stochastically successful procurement of previous interviewees, but each has a certain probability of refusing the offer. A general formula for solving this problem is obtained. Two special cases: constant probability of refusing and geometric probability of refusing are discussed in detail. The optimal stopping rules in these two cases turn out to be simple.

scholarly journals Do Student Evaluations of University Reflect Inaccurate Beliefs or Actual Experience? A Relative Rank Model

2014 ◽  
Vol 28 (1) ◽  
pp. 14-26 ◽  
Author(s):  
Gordon D. A. Brown ◽  
Alex M. Wood ◽  
Ruth S. Ogden ◽  
John Maltby
Keyword(s):  
Student Evaluations ◽  
Relative Rank ◽  
Actual Experience
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