scholarly journals COMPLETE RESIDUE SYSTEMS IN THE RING OF MATRICES OVER EUCLIDEAN DOMAINS AND A GREATEST COMMON DIVISOR OF MATRICES

2013 ◽  
Vol 87 (3) ◽  
Author(s):  
S. Damkaew ◽  
S. Prugsapitak
Keyword(s):  
Greatest Common Divisor ◽  
Residue Systems ◽  
Euclidean Domains

Square-Reduced Residue Systems (Mod r) and Related Arithmetical Functions

1979 ◽  
Vol 22 (2) ◽  
pp. 207-220 ◽  
Author(s):  
R. Sivaramakrishnan
Keyword(s):  
Symmetric Functions ◽  
Arithmetic Function ◽  
Exponential Sum ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Class Division ◽  
Arithmetical Functions ◽  
New Class ◽  
Image Position ◽  
Residue Systems

AbstractWe define a square-reduced residue system (mod r) as the set of integers a (mod r) such that the greatest common divisor of a and r, denoted by (a, r), is a perfect square ≥ 1 and contained in a residue system (mod r). This leads to a Class-division of integers (mod r) based on the 'square-free' divisors of r. The number of elements in a square-reduced residue system (mod r) is denoted by b(r). It is shown that(1)(2)In view of (2), b(r) is said to be 'specially multiplicative'. The exponential sum associated with a square-reduced residue system (mod r) is defined bywhere the summation is over a square-reduced residue system (mod r).B(n, r) belongs to a new class of multiplicative functions known as 'Quasi-symmetric functions' and(3)As an application, the sum is considered in terms of the Cauchy-composition of even functions (mod r). It is found to be multiplicative in r. The evaluation of the above sum gives an identity involving Pillai's arithmetic function

Detection of pitted morningglory (Ipomoea lacunosa) by hyperspectral remote sensing. I. Effects of tillage and cover crop residue

Weed Science ◽  
2004 ◽  
Vol 52 (2) ◽  
pp. 222-229 ◽  
Author(s):  
Clifford H. Koger ◽  
David R. Shaw ◽  
Krishna N. Reddy ◽  
Lori M. Bruce
Keyword(s):  
Cover Crop ◽  
Crop Residue ◽  
Field Experiments ◽  
Leaf Growth ◽  
Plant Size ◽  
Hyperspectral Reflectance ◽  
Discrimination Accuracy ◽  
Reflectance Measurement ◽  
Pitted Morningglory ◽  
Residue Systems

Field experiments were conducted to evaluate the potential of hyperspectral reflectance data collected with a hand-held spectroradiometer to discriminate soybean intermixed with pitted morningglory and weed-free soybean in conventional till and no-till plots containing rye, hairy vetch, or no cover crop residue. Pitted morningglory was in the cotyledon to six-leaf growth stage. Seven 50-nm spectral bands (one ultraviolet, two visible, four near-infrared) derived from each hyperspectral reflectance measurement were used as discrimination variables. Pitted morningglory plant size had more influence on discriminant capabilities than tillage or cover crop residue systems. Across all tillage and residue systems, discrimination accuracy was 71 to 95%, depending on the size of pitted morningglory plants at the time of data acquisition. The versatility of the seven 50-nm bands was tested by using a discriminant model developed for one experiment location to test discriminant capabilities for the other experiment, with discrimination accuracy across all tillage and residue systems of 55 to 73%, depending on pitted morningglory plant size.

Greatest common divisor of matrices one of which is a disappear matrix

2014 ◽  
Vol 66 (3) ◽  
pp. 479-485 ◽  
Author(s):  
A. M. Romaniv ◽  
V. P. Shchedryk
Keyword(s):  
Greatest Common Divisor
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