Functional analysis was an  important development in the viewpoint of mathematicians in examining the behavior of functions as a set (space of functions). The important concept of single space and the distance of its points, invented by ‎Fréchet‎ in the early 20th century, and the fusion of algebraic and topological concepts eventually led to the establishment of this branch of mathematics around 1933. The first Iranian mathematician to make a mark in this field was Heydar Radjavi, who contributed to the field in Iran from 1963 to 1968. Another branch of this field is called nonlinear functional analysis, and Behzad Djafari-Rouhani was the first specialist in this branch who worked in the country from 19‎87‎ onwards. Another key figure in the development and expansion of this branch in the country is Jafar Zafarani. This article, after outlining the history of functional analysis, will provide a brief overview of his contributions to this field. Through a conversation with him, we will delve into his experiences and insights from various aspects of his professional life.

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A pro le is a relation between one response variable and one or more explanatory variables that represent the quality of a product or the performance of a process. Process Capability Indices (PCI) is measured to evaluate processes in producing conforming products. All existing methods that measure process capability indices in a simple linear pro le consider response variables at some levels of explanatory variable and ignore all ranges of x-values. In this paper, a functional approach is proposed to measure the process capability index of simple linear pro les in all ranges of explanatory variable. This new approach follows the traditional de nition of process capability indices and leads to their accurate values for a simple linear pro le. The functional approach uses a reference pro le, functional speci cation limits and functional natural tolerance limits to present a functional form of process capability indices. This functional form results in measuring the process capability at each level of the explanatory variable in a simple linear pro le, as well as the unique value of a process capability index for a simple linear pro le. A comparison study using a non-conforming proportion method shows the better performance of functional process capability indices in measuring the process capability in a simple linear pro le.

The quality of some processes is defined by a simple linear function between a response and an explanatory variable, which is named a simple linear profile. Various capability indices have been introduced in the literature for these processes. Some of them are functional, that use all domain of the explanatory variable instead of its levels values. Therefore, these methods may fail to account capability of the process and overstate it. This paper applies fuzzy logic to deal with the explanatory variable domain and introduces two capability indices. Then, investigates and compares the performance of the proposed indices and some existing ones, using a simulation study. Results show that one of the new proposed indices has the best performance based on the mean square error and mean absolute error criteria.

In the present work we continue studying the solvability of the linear functional equations åNj=1Cj F o aj=H and also the strong and weak stability of the corresponding operator P (see the definition below). By analogy with the Cauchy and Jensen operators once more model operator pˆis considered, and the stability problems as well as some solvability problems for pˆ are studied. Several unsolved problem of a general character are formulated.

This paper introduces a functional mixed effect random model to model spa-tial data. In this model, the spatial locations form the index set, while the contributing effects to the response variable are set as a linear mixture of fixed and random effects. These fixed and random effects are linear combinations of L2 functions and random elements, respectively. However, the corresponding linear factors depend on the spa-tial location variable. Therefore, we develop estimation procedures to estimate the fixed and random coeficients, using spatial functional principal component analysis. Then, we perform prediction by adapting the functional universal kriging method to our model.

Purpose: Brain connectivity studies unveil the intricate interactions within neural networks. Various approaches exist to explore brain connectivity, yet the debate between the efficacy of linear versus non-linear methods remains unresolved due to the advantages and limitations of each. This study aims to provide a comprehensive evaluation of neuroimaging data analysis to gain insights into the functional aspects of the brain, particularly in the context of Alzheimer's Disease (AD). The objective is to identify potential pathways for early intervention and prevention, despite the controversies arising from diverse neuroimaging modalities and analytical techniques. Materials and Methods: Using fMRI data, both linear and non-linear approaches are investigated. The linear approach employs the Pearson Correlation Coefficient (PCC) to create whole-brain graphs. For non-linear approaches, Distance Correlation (DC) and the kernel trick are utilized. Functional brain networks are constructed and sparsified for each AD stage, followed by calculating global graph measures. Results: The findings indicate that non-linear approaches are more effective in distinguishing between different stages of AD. Among these, the kernel trick method performs better than the DC technique. Polynomial kernel (degree 3) showed better group separability, with significantly different graph measures such as clustering, transitivity, modularity, and small-worldness. Kernel analysis revealed that within-region connectivity was more disrupted in AD. Notably, the functional graphs of the brain are more significantly degraded in the early stages of AD. Conclusion: In the initial phases of AD, both functional integration and segregation of the brain are compromised, with a more pronounced decline in functional segregation as the disease progresses. The clustering coefficient, indicative of brain functional segregation, emerges as the most distinguishing feature across all stages of AD, highlighting its potential as a biomarker for early diagnosis.