Algebraic Method for Solving System of Linear Congruences

2017 ◽  
Vol 3 (1) ◽  
pp. 93-99
Author(s):  
Polemer Cuarto ◽  
Keyword(s):  
Algebraic Method ◽  
Linear Congruences

An algebraic method for solving Hartree-Fock-Roothaan equations

1990 ◽  
Vol 87 ◽  
pp. 2017-2025 ◽  
Author(s):  
Lac Malbouisson ◽  
JDM Vianna
Keyword(s):  
Algebraic Method ◽  
Hartree Fock

The elliptic products of Jacobi and the theory of linear congruences

1926 ◽  
Vol 23 (4) ◽  
pp. 337-355
Author(s):  
P. A. MacMahon
Keyword(s):  
Elliptic Functions ◽  
Linear Congruences ◽  
Image Position ◽  
Theory Of Numbers

In the application of Elliptic Functions to the Theory of Numbers the two formulae of Jacobiare of great importance.

scholarly journals Categorification of the Müller-Wichards System Performance Estimation Model: Model Symmetries, Invariants, and Closed Forms

Systems ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 6
Author(s):  
Allen D. Parks ◽  
David J. Marchette
Keyword(s):  
Closed Form ◽  
System Performance ◽  
Algebraic Method ◽  
Expressive Power ◽  
Computer Applications ◽  
Performance Estimation ◽  
Parallel Computer ◽  
Estimation Model ◽  
Data Movement ◽  
Fundamental Symmetry

The Müller-Wichards model (MW) is an algebraic method that quantitatively estimates the performance of sequential and/or parallel computer applications. Because of category theory’s expressive power and mathematical precision, a category theoretic reformulation of MW, i.e., CMW, is presented in this paper. The CMW is effectively numerically equivalent to MW and can be used to estimate the performance of any system that can be represented as numerical sequences of arithmetic, data movement, and delay processes. The CMW fundamental symmetry group is introduced and CMW’s category theoretic formalism is used to facilitate the identification of associated model invariants. The formalism also yields a natural approach to dividing systems into subsystems in a manner that preserves performance. Closed form models are developed and studied statistically, and special case closed form models are used to abstractly quantify the effect of parallelization upon processing time vs. loading, as well as to establish a system performance stationary action principle.

scholarly journals Some remarks on the Zarankiewicz problem

2021 ◽  
pp. 1-7
Author(s):  
DAVID CONLON
Keyword(s):  
Upper Bound ◽  
Algebraic Method ◽  
Zarankiewicz Problem

Abstract The Zarankiewicz problem asks for an estimate on z(m, n; s, t), the largest number of 1’s in an m × n matrix with all entries 0 or 1 containing no s × t submatrix consisting entirely of 1’s. We show that a classical upper bound for z(m, n; s, t) due to Kővári, Sós and Turán is tight up to the constant for a broad range of parameters. The proof relies on a new quantitative variant of the random algebraic method.

On the Theory of Thermally Weighted Electron-Phonon Transition Probabilities

1980 ◽  
Vol 35 (9) ◽  
pp. 902-914
Author(s):  
J. Schupfner
Keyword(s):  
Transition Probabilities ◽  
Algebraic Method ◽  
Calculation Technique ◽  
Radiative Transitions ◽  
Special Cases ◽  
Explicit Formulae ◽  
The Common ◽  
Common Transition ◽  
Franck Condon ◽  
Refined Calculation

Abstract We present a refined calculation method for the phonon part (Franck-Condon Overlaps) of the transition probabilities of electron-phonon radiative and non-radiative transitions in crystals. The evaluation of the thermal averaged Franck-Condon integrals is a purely algebraic method and the transition probabilities we use are derived from first principles and completely atomistic. For the electronic transitions we take into account the frequency shift of the lattice and the change of the phonon normal coordinates. Explicit formulae of the phonon parts are derived and it is shown that the common transition probabilities used in literature are special cases of our functional calculation technique.

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