scholarly journals Algorithmic Determination of a Large Integer in the Two-Term Machin-like Formula for π

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2162
Author(s):  
Sanjar M. Abrarov ◽  
Rajinder K. Jagpal ◽  
Rehan Siddiqui ◽  
Brendan M. Quine
Keyword(s):  
Rational Number ◽  
Large Integer ◽  
Simple Iteration ◽  
Alternative Method

In our earlier publication we have shown how to compute by iteration a rational number u2,k in the two-term Machin-like formula for π of the kind π4=2k−1arctan1u1,k+arctan1u2,k,k∈Z,k≥1, where u1,k can be chosen as an integer u1,k=ak/2−ak−1 with nested radicals defined as ak=2+ak−1 and a0=0. In this work, we report an alternative method for determination of the integer u1,k. This approach is based on a simple iteration and does not require any irrational (surd) numbers from the set ak in computation of the integer u1,k. Mathematica programs validating these results are presented.

Determination of Myocardial FFA Elimination Rates by Functional Images of Uncorrected Half-Time Values

1984 ◽  
Vol 23 (06) ◽  
pp. 277-282 ◽  
Author(s):  
A. Van Lingen ◽  
G. Westera ◽  
M. van ◽  
W. Den Hollander ◽  
E. E. Van der Wall ◽  
...  
Keyword(s):  
Coronary Artery ◽  
Regions Of Interest ◽  
Heptadecanoic Acid ◽  
Half Time ◽  
Alternative Method ◽  
Functional Images ◽  
Functional Image ◽  
Back Ground ◽  
Objective Determination

SummaryThis paper presents an alternative method of demarcating regions of in terest over the myocardium after ad ministration of 123I-heptadecanoic acid to patients with coronary artery disea se. In a matrix of 32 × 32 pixels the elimination rates of the radioactivity, which are not corrected for back ground activity, are visualized per pixel in a functional image. The func tional image showed areas in the myocardium with high values of uncorrected elimination rates. These areas corresponded with the tracer defects on the scintigram. Corrected elimination rates obtained from re gions of interest of functional images were comparable with those of scinti grams. Thus based on functional im ages of uncorrected elimination rates a reliable, objective determination of regions of interest over normal and abnormal myocardium can be made.

Determination of Ammonia (Alternative Method)

1984 ◽  
pp. 14-17 ◽  
Author(s):  
TIMOTHY R. PARSONS ◽  
YOSHIAKI MAITA ◽  
CAROL M. LALLI
Keyword(s):  
Alternative Method

scholarly journals An estimate for trigonometrical sums over square-free integers with a constant number of prime factors

1967 ◽  
Vol 15 (4) ◽  
pp. 249-255
Author(s):  
Sean Mc Donagh
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Prime Number ◽  
Rational Number ◽  
Large Integer ◽  
Constant Number ◽  
The Real ◽  
Prime Factors ◽  
Squarefree Number ◽  
Image Position

1. In deriving an expression for the number of representations of a sufficiently large integer N in the formwhere k: is a positive integer, s(k) a suitably large function of k and pi is a prime number, i = 1, 2, …, s(k), by Vinogradov's method it is necessary to obtain estimates for trigonometrical sums of the typewhere ω = l/k and the real number a satisfies 0 ≦ α ≦ 1 and is “near” a rational number a/q, (a, q) = 1, with “large” denominator q. See Estermann (1), Chapter 3, for the case k = 1 or Hua (2), for the general case. The meaning of “near” and “arge” is made clear below—Lemma 4—as it is necessary for us to quote Hua's estimate. In this paper, in Theorem 1, an estimate is obtained for the trigonometrical sumwhere α satisfies the same conditions as above and where π denotes a squarefree number with r prime factors. This estimate enables one to derive expressions for the number of representations of a sufficiently large integer N in the formwhere s(k) has the same meaning as above and where πri, i = 1, 2, …, s(k), denotes a square-free integer with ri prime factors.

Exploring MBTH as a spectrophotometric probe for the determination of total phenolic compounds in beverage samples

2018 ◽  
Vol 10 (19) ◽  
pp. 2197-2204 ◽  
Author(s):  
Bruna da Silva Granja ◽  
José Ricardo Honório de Mendonça Filho ◽  
Woodland de Souza Oliveira ◽  
Josué Carinhanha Caldas Santos
Keyword(s):  
Phenolic Compounds ◽  
Total Phenolic ◽  
Total Phenolic Compounds ◽  
Beverage Samples ◽  
Alternative Method

An alternative method using MBTH as a spectrophotometric probe for the determination of total phenolic compounds in samples of wines (red and white), coffees (instant and brewed), teas and infusions.

Alternative Method for the Determination of a Full S-N Fatigue Profile

2016 ◽  
Vol 250 ◽  
pp. 209-216 ◽  
Author(s):  
Przemysław Strzelecki ◽  
Janusz Sempruch ◽  
Tomasz Tomaszewski
Keyword(s):  
Fatigue Life ◽  
Linear Regression ◽  
Fatigue Limit ◽  
Fatigue Curve ◽  
The Other ◽  
Staircase Method ◽  
Alternative Method ◽  
Random Fatigue

This paper presents two methods for estimating the S-N fatigue curve. The first is the traditional linear regression and staircase method. The other, alternative, method is based on random fatigue life, fatigue limit and probability. The both methods provide similar results but the latter one requires fewer test samples

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