Journal of AppliedMath

The relation between continuous functions and random vectors is revealed in the paper that the main meaning is described as, for any given continuous function, there must be a sequence of probability spaces and a sequence of random vectors where every random vector is defined on one of these probability spaces, such that the sequence of conditional mathematical expectations formed by the random vectors uniformly converges to the continuous function. This is a random vector representation of continuous functions, which is regarded as a bridge to be set up between real function theory and probability theory. By means of this conclusion, an interesting result about function approximation theory can be obtained. The random vector representation of continuous functions is of important applications in physics. Based on the conclusion, if a large proportion of certainty phenomena can be described by continuous functions and random phenomena can also be described by random variables or vectors, then any certainty phenomenon must be the limit state of a sequence of random phenomena. Then, in the approximation from a sequence of random vectors to a continuous function, the base functions are appropriately selected by us, and an important conclusion for quantum mechanics is deduced: classical mechanics and quantum mechanics are unified. Particularly, an interesting and very important conclusion is introduced as the fact that the mass point motion of a macroscopical object possesses a kind of wave characteristic curve, which is called wave-mass-point duality.

Rendering musical notes randomly does not create music. To generate music, there has to be a pattern which makes the musical notes dependent. It is therefore of interest to know whether the probability of the next note depends on the current note only or whether it depends on the note(s) prior to the current note. In other words, it is important to explore the orderof Markov chain in the musical piece. In the context of Hindustani classical music, does this order depend on the raga or the composition? The present work addresses this fascinating question and attempts to answer it through Akaike’s information criterion (AIC). It appears, interestingly, that the order of Markov chain is dependent on the raga which has a well defined melodic structure with fixed notes and a set of rules characterizing a particular mood that is conveyed by performance. As long as these rules are maintained, as in a raga bandish, the order of the Markov chain is invariant over the raga compositions.

Handwritten digit recognition systems are integral to diverse applications such as postal services, banking, and document processing in our digitally-driven society. This research addresses the challenges posed by evolving datasets and dynamic scenarios in handwritten digit recognition by proposing an approach based on incremental support vector machines (ISVM). ISVM is an extension of traditional support vector machines (SVM) designed to handle scenarios where new data points become available over time. The dataset includes handwritten images (numbers “0” to “6”) and trials introducing new classes (“7”, “8”, and “9”). Evaluation utilizes k-fold cross-validation for robustness. Digital image processing involves converting images into numeric data using the histogram method. The result showed the positive outcomes of using ISVM in handwritten digit recognition, emphasizing its adaptability to incremental learning and its ability to maintain robust performance in the face of evolving datasets, which is crucial for real-world applications.

The paper proposes a topological approach in order to explore several data tables simultaneously. These data tables of quantitative and/or qualitative variables measured on different homogeneous themes, collected from the same individuals. This approach, called topological analysis of multiple tables (TAMT), is based on the notion of neighborhood graphs in the context of a joint analysis of several data tables. It allows the simultaneous study of possible links between several thematic tables. The structure of the correlations or associations of the variables in each thematic table is analyzed according to quantitative, qualitative or mixed variables considered. Like the multiple factorial analysis (MFA), the TAMT allows several tables of variables to be analyzed simultaneously, and to obtain results, in particular graphical representations, which make it possible to study the relationship between individuals, variables and tables of data. These can also be tables of temporal data, collected at different times on the same individuals. The proposed TAMT approach is illustrated using real data associated with several and different homogeneous themes. Its results are compared to those from the MFA method.