scholarly journals Unimodular transformations for DAE initial trajectory problems

PAMM ◽  
2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Stephan Trenn ◽  
Benjamin Unger
Keyword(s):  
Unimodular Transformations

Unimodular Transformations

2011 ◽  
pp. 2103-2112
Author(s):  
Allan Gottlieb ◽  
Utpal Banerjee ◽  
Gianfranco Bilardi ◽  
Geppino Pucci ◽  
William Carlson ◽  
...  
Keyword(s):  
Unimodular Transformations

GENERALISING THE UNIMODULAR APPROACH TO RESTRUCTURE IMPERFECTLY NESTED LOOPS

1996 ◽  
Vol 06 (03) ◽  
pp. 401-414
Author(s):  
Jingling Xue
Keyword(s):  
Code Generation ◽  
Lexicographic Order ◽  
Generation Algorithm ◽  
Loop Nest ◽  
Loop Nests ◽  
Iteration Space ◽  
New Concepts ◽  
Affine Constraints ◽  
Nested Loops ◽  
Unimodular Transformations

Although overcoming some limitations of the generate-and-test approach, unimodular transformations are limited to perfect loop nests only. Extending the unimodular approach, this paper describes a framework that enables the use of unimodular transformations to restructure imperfect loop nests. The concepts used previously for perfect loop nests, such as iteration vector, iteration space and lexicographic order, are generalised and some new concepts like preorder tree are introduced. Multiple unimodular transformations are allowed, one each statement in the loop nest. A code generation algorithm is provided that produces a possibly imperfect loop nest to scan an iteration space that is given as a union of sets of affine constraints.

scholarly journals On the Compatibility of Bézout Coefficients between Pythagorean Pairs under Unimodular Transformations

2019 ◽  
pp. 1-13
Author(s):  
Cherng-tiao Perng ◽  
Maila Brucal-Hallare
Keyword(s):  
Simple Proof ◽  
Unimodular Transformations

In a recent preprint, Gullerud and Walker proved a theorem and made a conjecture about thecorrectness of efficiently generating Bézout trees for Pythagorean pairs. In this note, we give a simple proof of their theorem, conrm that their conjecture is true, and furthermore we give a generalization.

On the Convergence of Mean Values Over Lattices

1958 ◽  
Vol 10 ◽  
pp. 103-110 ◽  
Author(s):  
Wolfgang Schmidt
Keyword(s):  
Linear Transformation ◽  
Measurable Function ◽  
Dimensional Space ◽  
Fundamental Region ◽  
Geometry Of Numbers ◽  
Linear Transformations ◽  
Mean Values ◽  
Unimodular Transformations

Recently C. A.Rogers (2, Theorem 4) proved the following theorem which applies to many problems in geometry of numbers: Let f(X1,X2, … , Xk) be a non-negative B or el-measurable function in the nk-dimensional space of points (X1,X2. Further, let Λo be the fundamental lattice, Ω a linear transformation of determinant 1, F a fundamental region in the space of linear transformations of determinant 1, defined with respect to the subgroup of unimodular transformations and μ(Ω) the invariant measure1 on the space of linear transformations of determinant 1 in Rn.

Index Reduction via Unimodular Transformations

2018 ◽  
Vol 39 (3) ◽  
pp. 1135-1151 ◽  
Author(s):  
Satoru Iwata ◽  
Mizuyo Takamatsu
Keyword(s):  
Index Reduction ◽  
Unimodular Transformations

From integer Lorentz transformations to Pythagoras

2004 ◽  
Vol 88 (511) ◽  
pp. 16-21 ◽  
Author(s):  
Christopher J. Bradley
Keyword(s):  
Diophantine Equations ◽  
Indefinite Metric ◽  
Lorentz Transformations ◽  
Repeated Application ◽  
Pythagorean Triples ◽  
Unimodular Transformations

In this article we show how to obtain by recurrence all primitive Pythagorean triples from the single basic (4, 3, 5) triple. It is all done by repeated application of two integer unimodular transformations that leave the indefinite metric x2 + y2 - z2 invariant, together with the additional transformations (i) that change the sign of x and (ii) that change the sign of y. These alone would restrict the triples to those in which x is even, y is odd and z is positive, so we then include two further transformations (iii) that exchange x and y and (iv) that change the sign of z, thereby accounting for all primitive triples. There is a bonus in extending the method beyond the first objective in that the theory, which we outline, of the Lorentz transformations involved not only provides necessary background explanation of the first part, but it also enables us to show how to solve by recurrence Diophantine equations of the form x2 + y2 = z2 + n, where n is any integer.

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