The Tchebysheffian approximation of one rational function by another

1964 ◽  
Vol 60 (4) ◽  
pp. 877-890 ◽  
Author(s):  
A. Talbot
Keyword(s):  
Rational Function ◽  
Finite Interval ◽  
Algebraic Method ◽  
Method Of Solution

In a previous paper we discussed a uniform algebraic method of solution of problems in which a prescribed real rational function (or polynomial) g(x) was to be approximated in a given finite interval by a real rational function (or polynomial)f(x) with prescribed numerator and denominator degrees, the approximation being Tchebysheffian, i.e. such as to make the ‘deviation’ of f, max |f − g| in the interval, as small as possible.

scholarly journals Algebraic Method of Solution of Schrödinger’s Equation ofa Quantum Model

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Wave Function ◽  
Wave Equation ◽  
Quantum Dot ◽  
Direct Method ◽  
Algebraic Method ◽  
Time Dependent ◽  
Quantum Model ◽  
Decomposition Technique ◽  
Method Of Solution ◽  
Schrödinger's Equation

This work is aiming to show the advantage of using the Lie algebraic decomposition technique to solvefor Schrödinger’s wave equation for a quantum model, compared with the direct method of solution. The advantageis a two-fold: one is to derive general form of solution, and, two is relatively manageable to deal with the case oftime-dependent system Hamiltonian. Specifically, we consider the model of 2-level optical atom and solve for thecorresponding Schrödinger’s wave equation using the Lie algebraic decomposition technique. The obtained formof solution for the wave function is used to examine computationally the atomic localization in the coordinate space.For comparison, the direct method of solution of the wave function is analysed in order to show its complicationwhen dealing with time-dependent Hamiltonian.The possibility of using the Lie algebraic method for a qubit model(a driven quantum dot model) is briery discussed, if Schrödinger’s wave function is to be examined for the qubitlocalization.

Coin Tossing

1996 ◽  
Vol 89 (8) ◽  
pp. 642-645
Author(s):  
Mako E. Haruta ◽  
Mark Flaherty ◽  
Jean McGivney ◽  
Raymond J. McGivney
Keyword(s):  
North Carolina ◽  
Rational Function ◽  
Related Problem ◽  
Analytic Solution ◽  
Geometric Probability ◽  
Coin Tossing ◽  
North Carolina School ◽  
Method Of Solution ◽  
Widespread Availability ◽  
And Mathematics

The idea for this article came from a problem that was published before the widespread availability of graphing calculators (North Carolina School for Science and Mathematics 1988). In that publication, Geometric Probability, an interesting analytic solution to the problem about comparing the areas of squares was given and is described later in Algebraic-Geometric Solution. We have adapted the problem and have used successfully another, equally interesting method of solution in numerous classes from seventh-grade prealgebra through precalculus, as well as with several groups of teachers. This article presents our solution and, in addition, an example of how a rational function, a type not commonly found in applications of mathematics at this level, can model a solution of a related problem. The problem that we use with our students follows:

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

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

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