Properties and linear representations of Chevalley groups

1970 ◽  
pp. 1-55 ◽  
Author(s):  
Armand Borel
Keyword(s):  
Chevalley Groups ◽  
Linear Representations

Improving Struggling Fifth-Grade Students’ Understanding of Fractions: A Randomized Controlled Trial of an Intervention That Stresses Both Concepts and Procedures

2021 ◽  
pp. 001440292110088
Author(s):  
Madhavi Jayanthi ◽  
Russell Gersten ◽  
Robin F. Schumacher ◽  
Joseph Dimino ◽  
Keith Smolkowski ◽  
...  
Keyword(s):  
Randomized Controlled Trial ◽  
Explicit Instruction ◽  
Controlled Trial ◽  
Number Line ◽  
Randomized Controlled ◽  
Fifth Grade Students ◽  
Linear Representations ◽  
Mathematical Explanations ◽  
Grade 5 ◽  
Number Line Estimation

Using a randomized controlled trial, we examined the effect of a fractions intervention for students experiencing mathematical difficulties in Grade 5. Students who were eligible for the study ( n = 205) were randomly assigned to intervention and comparison conditions, blocked by teacher. The intervention used systematic, explicit instruction and relied on linear representations (e.g., Cuisenaire Rods and number lines) to demonstrate key fractions concepts. Enhancing students’ mathematical explanations was also a focus. Results indicated that intervention students significantly outperformed students from the comparison condition on measures of fractions proficiency and understanding ( g = 0.66–0.78), number line estimation ( g = 0.80–1.08), fractions procedures ( g = 1.07), and explanation tasks ( g = 0.68–1.23). Findings suggest that interventions designed to include explicit instruction, along with consistent use of the number line and opportunities to explain reasoning, can promote students’ proficiency and understanding of fractions.

scholarly journals COMPLETELY REDUCIBLE SETS

2013 ◽  
Vol 23 (04) ◽  
pp. 915-941 ◽  
Author(s):  
DOMINIQUE PERRIN
Keyword(s):  
Closure Properties ◽  
Linear Representations ◽  
Complete Reducibility ◽  
The Family ◽  
Syntactic Representation ◽  
Completely Reducible ◽  
Rational Sets ◽  
Cyclic Sets

We study the family of rational sets of words, called completely reducible and which are such that the syntactic representation of their characteristic series is completely reducible. This family contains, by a result of Reutenauer, the submonoids generated by bifix codes and, by a result of Berstel and Reutenauer, the cyclic sets. We study the closure properties of this family. We prove a result on linear representations of monoids which gives a generalization of the result concerning the complete reducibility of the submonoid generated by a bifix code to sets called birecurrent. We also give a new proof of the result concerning cyclic sets.

Non-linear representations of inhomogeneous groups

1978 ◽  
Vol 2 (6) ◽  
pp. 499-504 ◽  
Author(s):  
Georges Pinczon ◽  
Jacques Simon
Keyword(s):  
Linear Representations ◽  
Non Linear

scholarly journals 1-Cohomology of Chevalley groups

1989 ◽  
Vol 127 (2) ◽  
pp. 353-372 ◽  
Author(s):  
Helmut Völklein
Keyword(s):  
Chevalley Groups

Elements of Order Coxeter Number +1 in Chevalley Groups

1982 ◽  
Vol 34 (4) ◽  
pp. 945-951 ◽  
Author(s):  
Bomshik Chang
Keyword(s):  
Weyl Group ◽  
Chevalley Group ◽  
Trivial Solution ◽  
Affirmative Answer ◽  
Chevalley Groups ◽  
General Solutions ◽  
Coxeter Number ◽  
Image Position

Following the notation and the definitions in [1], let L(K) be the Chevalley group of type L over a field K, W the Weyl group of L and h the Coxeter number, i.e., the order of Coxeter elements of W. In a letter to the author, John McKay asked the following question: If h + 1 is a prime, is there an element of order h + 1 in L(C)? In this note we give an affirmative answer to this question by constructing an element of order h + 1 (prime or otherwise) in the subgroup Lz = 〈xτ(1)|r ∈ Φ〉 of L(K), for any K.Our problem has an immediate solution when L = An. In this case h = n + 1 and the (n + l) × (n + l) matrixhas order 2(h + 1) in SLn+1(K). This seemingly trivial solution turns out to be a prototype of general solutions in the following sense.

On the Generation of Linear Representations of Spatial Configuration

1998 ◽  
Vol 25 (4) ◽  
pp. 559-576 ◽  
Author(s):  
J Peponis ◽  
J Wineman ◽  
S Bafna ◽  
M Rashid ◽  
S H Kim
Keyword(s):  
Spatial Configuration ◽  
Linear Representations
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