Ebubekir İnan

Fuzzy soft rings and fuzzy soft ideals

Our aim in this paper is to introduce and study the fuzzy soft ring and

Bold-Independent Computational Entropy Assesses Functional Donut-Like Structures in Brain fMRI Images.

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PRIMARY EVIDENCE OF A DONUT-LIKE, FOURTH SPATIAL DIMENSION IN THE BRAIN

-fuzzy soft subring that is generalization of fuzzy soft ring. We also study some of their basic properties and give several examples.

ENTROPY IN PRIMARY SENSORY AREAS LOWER THAN IN ASSOCIATIVE ONES: THE BRAIN LIES IN HIGHER DIMENSIONS THAN THE ENVIRONMENT

Contrary to common belief, the brain appears to increase the complexity from the perceived object to the idea of it. Topological models predict indeed that: (a) increases in anatomical/functional dimensions and symmetries occur in the transition from the environment to the higher activities of the brain, and (b) informational entropy in the primary sensory areas is lower than in the higher associative ones. To demonstrate this novel hypothesis, we introduce a straightforward approach to measuring island information levels in fMRI neuroimages, via Rényi entropy derived from tessellated fMRI images. This approach facilitates objective detection of entropy and corresponding information levels in zones of fMRI images generally not taken into account. We found that the Rényi entropy is higher in associative cortices than in the visual primary ones. This suggests that the brain lies in dimensions higher than the environment and that it does not concentrate, but rather dilutes messages coming from external inputs.

Near Groups of Weak Cosets on Nearness Approximation Spaces

We introduce a novel method for the measurement of information level in fMRI (functional Magnetic Resonance Imaging) neural data sets, based on image subdivision in small polygons equipped with different entropic content. We show how this method, called maximal nucleus clustering (MNC), is a novel, fast and inexpensive image-analysis technique, independent from the standard blood-oxygen-level dependent signals. MNC facilitates the objective detection of hidden temporal patterns of entropy/information in zones of fMRI images generally not taken into account by the subjective standpoint of the observer. This approach befits the geometric character of fMRIs. The main purpose of this study is to provide a computable framework for fMRI that not only facilitates analyses, but also provides an easily decipherable visualization of structures. This framework commands attention because it is easily implemented using conventional software systems. In order to evaluate the potential applications of MNC, we looked for the presence of a fourth dimensions distinctive hallmarks in a temporal sequence of 2D images taken during spontaneous brain activity. Indeed, recent findings suggest that several brain activities, such as mind-wandering and memory retrieval, might take place in the functional space of a four dimensional hypersphere, which is a double donut-like structure undetectable in the usual three dimensions. We found that the Rényi entropy is higher in MNC areas than in the surrounding ones, and that these temporal patterns closely resemble the trajectories predicted by the possible presence of a hypersphere in the brain.

Erratum to: Fuzzy soft rings and fuzzy soft ideals

In 1999, Molodtsov introduced the theory of soft sets, which can be seen as a new mathematical approach to vagueness. In 2002, near set theory was initiated by J. F. Peters as a generalization of Pawlaks rough set theory. In the near set approach, every perceptual granule is a set of objects that have their origin in the physical world. Objects that have, in some degree, affinities are considered perceptually near each other, i.e., objects with similar descriptions. Also, the concept of near groups has been investigated by Inan and Ozturk [30]. The present paper aims to combine the soft sets approach with near set theory, which gives rise to the new concepts of soft nearness approximation spaces SNAS, soft lower and upper approximations. Moreover, we give some examples and properties of these soft nearness approximations.

Fuzzy proximal relator spaces

We introduce a novel method for the measurement of information in fMRI neuroimages, i.e., nucleus clustering’s Rényi entropy derived from strong proximities in feature-based Voronoï tessellations, e.g., maximal nucleus clustering (MNC). We show how MNC is a novel, fast and inexpensive image-analysis technique, independent from the standard blood-oxygen-level dependent signals, which facilitates the objective detection of hidden temporal patterns of entropy/information in zones of fMRI images generally not taken into account by the subjective standpoint of the observer. In order to evaluate the potential applications of MNC, we looked for the presence of a fourth dimension’s distinctive hallmarks in a temporal sequence of 2D images taken during spontaneous brain activity. Indeed, recent findings suggest that several brain activities, such as mind-wandering and memory retrieval, might take place in the functional space of a four dimensional hypersphere, which is a double donut-like structure undetectable in the usual three dimensions. We found that the Rényi entropy is higher in MNC areas than in the surrounding ones, and that these temporal patterns closely resemble the trajectories predicted by the possible presence of a hypersphere in the brain.

Approximately Semigroups and Ideals: An Algebraic View of Digital Images

The brain, rather than integrate sensory inputs and concentrate them into concepts as currently believed, appears to increase the complexity from the perceived object to the idea of it. Topological models predict indeed an increase in dimensions and symmetries from the environment to the higher activities of the brain. Models predict that informational entropy in the primary sensory areas must be lower than in the higher associative ones. In order to demonstrate the novel hypothesis, we introduce a method for the measurement of information in fMRI neuroimages, i.e., nucleus clustering’s Rényi entropy derived from strong proximities in feature-based Voronoï tessellations, e.g., maximal nucleus clustering. The technique facilitates the objective detection of entropy/information in zones of fMRI images generally not taken into account. We found that the Rényi entropy is higher in associative cortices than in the visual primary ones. It suggests that the brain lies in higher dimensions than the environment and that it does not concentrate, but rather dilutes the message coming from external inputs.