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Fraction frenzy : fractions and decimals
\"Introduces the reader to fractions and decimals.\"-- Provided by publisher.
Book
Sums of Reciprocals of Fractional Parts and Multiplicative Diophantine Approximation
There are two main interrelated goals of this paper. Firstly we investigate the sums
eBook
Meanings of savarṇana in Indian Arithmetic
In Indian mathematical texts the term savarnana \"reduction to the same color\" is usually found in the context of calculation for fractions. A number of explanations for the term have been offered in previous studies, but they slightly differ from each other. The Trišatībhāsva is an anonymous commentary on Srīdhara's Trišatī. In the present paper, I survey the meanings of savarnana in each text and the usage of it in the Trišatībhāsva.
Journal Article
Piece=part=portion : fractions=decimals=percents
Explains how in the language of mathematics, fractions, decimals and percents are three different ways of describing the same parts of things.
Book
The correlation of left atrial diameter with preserved ejection fraction, reduced ejection fraction, and mid‐range ejection fraction
Background
In patients with heart failure, left atrial remodeling often occurs to varying degrees. Left atrial enlargement has been proved to be an important predictor of cardiovascular‐related adverse events. However, the relationship between left atrial diameter (LAD) with heart failure (HF) with preserved ejection fraction (HFpEF), reduced ejection fraction (HFrEF) and mid‐range ejection fraction (HFmrEF) remains unclear.
Hypothesis
We want to explore the correlation between left atrial diameter and HFpEF, HFmrEF, and HFrEF.
Methods
A total of 210 patients with heart failure who underwent hospitalization in Ningbo Medical Center Lihuili Hospital, Zhejiang, China, from January 1, 2020, to June 31, 2021, were reviewed. The basic demographic characteristics, blood test, and the related indexes of echocardiography of the subjects were collected and analyzed.
Results
There is a significant difference between HFpEF and HFrEF group in LAD (p = .007), and LAD is negatively correlated with left ventricular ejection fraction (LVEF) (p = .002, r = −.209).
Conclusion
LAD is negatively correlated with LVEF, which may predict the prevalence of HFrEF.
We studied the data of 210 individuals with heart failure. There is a significant difference between HFpEF and HFrEF groups in LAD (p = .007). And LAD is negatively correlated with LVEF (p = .002, r = −.209), which may predict the prevalence of HFrEF.
Journal Article
Ready for fractions and decimals
\"Learn the basic fundamentals of adding, subtracting, multiplying and dividing using fractions and decimals\"--Provided by publisher.
Book
6-010 AI-driven global longitudinal strain and long-axis ejection fraction by cardiac MRI provide a better assessment to serially monitor heart function compared to manually derived short axis EF in breast cancer patients on trastuzumab
IntroductionTrastuzumab is associated with cardiotoxicity, necessitating interval cardiac monitoring. Our centre utilises cardiac magnetic resonance (CMR) measured ejection fraction (EF) for serial heart function assessment. This pilot study evaluated the accuracy and reliability of an AI-powered method using Global Longitudinal Strain (GLS) and long-axis EF (LA EF) compared to conventional volume-derived EF (SA EF) from CMR in patients undergoing Trastuzumab treatment for breast cancer.MethodsWe retrospectively analysed CMR imaging of five patients who had serial scans as per protocol between 2021–2024. AI-assisted software (Q Mass) generated GLS and LA EF from three long-axis views (4-chamber, 3-chamber, and 2-chamber), which were compared to SA EF derived manually from CMR, the gold standard for clinical decisions.ResultsFive patients underwent 34 scans during the monitoring period. Bland-Altman analysis showed a linear regression with no significant proportional bias between SA EF and LA EF (p = 0.377). GLS and SA EF demonstrated a strong inverse correlation (Spearman coefficient = 0.77, p < 0.0001) with a worsening GLS (i.e. becoming less negative) corresponding to a lower EF. ROC analysis indicated that GLS had a moderate ability to detect SA EF < 50% (AUC = 0.667).ConclusionEF derived by LA EF and SA EF methods showed consistent differences, supporting the reliability of AI-generated LA EF for serial LV function assessment. AI-based LA EF provides a rapid and reproducible monitoring method, reducing interobserver variability. While GLS and EF are strongly correlated, GLS shows moderate predictive ability for EF decline below 50%. Further studies are warranted to enhance its clinical utility.Conflict of InterestNone
Journal Article
Ergodicity of Iwasawa continued fractions via markable hyperbolic geodesics
We prove the convergence and ergodicity of a wide class of real and higher-dimensional continued fraction algorithms, including folded and
$\\alpha $
-type variants of complex, quaternionic, octonionic, and Heisenberg continued fractions, which we combine under the framework of Iwasawa continued fractions. The proof is based on the interplay of continued fractions and hyperbolic geometry, the ergodicity of geodesic flow in associated modular manifolds, and a variation on the notion of geodesic coding that we refer to as geodesic marking. As a corollary of our study of markable geodesics, we obtain a generalization of Serret’s tail-equivalence theorem for almost all points. The results are new even in the case of some real and complex continued fractions.
Journal Article