The Triple Product Rule is Incorrect

The triple product rule, also known as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, relates the partial derivatives of three interdependent variables, and often finds application in thermodynamics. It is shown here that its derivation is wrong, and that this rule is not correct; hence, the Mayer's relation and the heat capacity ratio, which describe the difference between isobaric and isochoric heat capacities, are also untrue. Also, the relationship linking thermal expansion and isothermal compressibility is wrong. These results are confirmed by many experiments and by the previous theoretical findings of the author.

The Triple Product Rule is Incorrect

The triple product rule, also known as the cyclic chain rule, cyclic relation, cyclical rule or Euler’s chain rule, relates the partial derivatives of three interdependent variables, and often finds application in thermodynamics. It is shown here that its derivation is wrong, and that this rule is not correct; hence, the Mayer’s relation and the heat capacity ratio, which describe the difference between isobaric and isochoric heat capacities, are also untrue. Also, the relationship linking thermal expansion and isothermal compressibility is wrong. These results are confirmed by many experiments and by the previous theoretical findings of the author.

Mathematics Refresher

The chapter “Mathematics Refresher” provides a brief reminder of operations with logarithms, matrices, and calculus, for student reference. It starts off by reviewing the differences between regular logarithms and natural logarithms and provides some examples of common operations with logarithms. It then introduces derivatives and integrals (although it is never necessary to compute an integral in this book, it is still useful to know what an integral is) and explains the sum rule, the product rule, the quotient rule, and the chain rule. Next, it provides a brief overview of matrices and matrix operations, including matrix dimensions, and addition and multiplication of matrices. It concludes with a discussion of the identity matrix.

Study of Fractional Analytic Functions and Local Fractional Calculus

In this present paper, the role of fractional analytic function in local fractional calculus is studied. Some important properties and theorems in local fractional calculus are discussed, such as product rule, quotient rule, chain rule, fundamental theorem of local fractional calculus, change of variable, integration by parts and so on. In addition, we propose several examples and formulas of local fractional calculus.

Calculating Probable Theoretical Offspring Genotype in Fruit Flies

Being able to calculate an offspring's theoretical genotype is critical in genetic sciences. We calculate the theoretical genotype and phenotype of fruit fly offspring. Using the product rule, we determine the probability for each trait and then for each genotype. In conclusion, we calculate 64 different genotypes that are supposed to be possible, but only 8 phenotypes are possible.

Classification of electrocardiogram signal using an ensemble of deep learning models

PurposeAccording to the World Health Organization, arrhythmia is one of the primary causes of deaths across the globe. In order to reduce mortality rate, cardiovascular disease should be properly identified and the proper treatment for the same should be immediately provided to the patients. The objective of this paper was to implement a better heartbeat classification model which will work better than the other implemented heartbeat classification methods.Design/methodology/approachIn this paper, the ensemble of two deep learning models is proposed to classify the MIT-BIH arrhythmia database into four different classes according to ANSI-AAMI standards. First, a convolutional neural network (CNN) model is used to classify heartbeats on a raw data set. Secondly, four features (wavelets, R-R intervals, morphological and higher-order statistics) are extracted from the data set and then applied to a long short-term memory (LSTM) model to classify the heartbeats. Finally, the ensemble of CNN and LSTM model with sum rule, product rule and majority voting has been used to identify the heartbeat classes.FindingsAmong these, the highest accuracy obtained is 98.58% using ensemble method with product rule. The results show that the ensemble of CNN and BLSTM has offered satisfactory performance compared to other techniques discussed in this study.Originality/valueIn this study, we have developed a new combination of two deep learning models to enhance the performance of arrhythmia classification using segmentation of input ECG signals. The contributions of this study are as follows: First, a deep CNN model is built to classify ECG heartbeat using a raw data set. Second, four types of features (R-R interval, HOS, morphological and wavelet) were extracted from the raw data set and then applied to the bidirectional LSTM model to classify the ECG heartbeat. Third, combination rules (sum rules, product rules and majority voting rules) were tested to ensure the accumulated probabilities of the CNN and LSTM models.

Calculating the Probability of Constitutional Isomers of Pentane

Depending on the reagent, and orientation of collisions within a chemical reaction, organic molecules can be present as different constitutional isomers of the same molecule. We can analyze the likelihood of getting a mixture of pentane with certain conformers. Based on this, we find that there are 16 potential conformers, but 13 are identical structures, meaning only three are distinct from each other. Using the product rule, we then demonstrate how to go about calculating the probability of specific conformers, including specific identical structures, being present in a mixture, and then we demonstrate that process is strictly within the three different conformers.