Lattice-valued bornological systems

Paseka, Jan; Solovyov, Sergey A.; Stehlík, Milan

doi:10.1016/j.fss.2014.09.006

Cited by 15 publications

(33 citation statements)

References 16 publications

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“…At present there are only few works dealing with this problem, see [1], [42], [30]. It is quite different from the situation in "Fuzzy Topology", which is a very well developed area of theoretical mathematics with many fundamental concepts, constructions and results, and which has important applications to other fields of mathematics, see e.g.…”

Section: Bornologies and Bornological Spacesmentioning

confidence: 99%

Bornological structures on many-valued sets

Šostak¹,

Uljane²

2017

Rad Hrvatske akademije znanosti i umjetnosti

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Abstract. We introduce an approach to the concept of bornology in the framework of many-valued mathematical structures and develop the basics of the theory of many-valued bornological spaces and initiate the study of the category of many-valued bornological spaces and appropriately defined bounded "mappings" of such spaces. A scheme for constructing many-valued bornologies with prescribed properties is worked out. In particular, this scheme allows to extend an ordinary bornology of a metric space to a many-valued bornology on it.In the academic year 1978/79 I was lucky to get a post-doc research position at the University of Zagreb and to spend this time working in collaboration with professor Sibe Mardešić in shape theory. It was a real pleasure to work together and to communicate generally with this outstanding mathematician, talented teacher and a very kind and nice man. Unfortunately, at present I am quite far from shape theory. Therefore the paper I would like to dedicate to Sibe Mardešić, written together with my former PhD student Ingrīda Uljane, is in the field of so called many-valued mathematical structures -the area of my main mathematical interests at present.

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Section: Bornologies and Bornological Spacesmentioning

confidence: 99%

Bornological structures on many-valued sets

Šostak¹,

Uljane²

2017

Rad Hrvatske akademije znanosti i umjetnosti

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“…In particular, they showed that L-Born is a topological construct [1,Theorem 5.3]. In [21], we provided the necessary and sufficient condition on the complete lattice L for the category L-Born to be topological, thereby correcting a flaw in [1,Theorem 5.3]. We also introduced the category L-Born of variable-basis lattice-valued bornological spaces (in the sense of S.E.…”

Section: Introductionmentioning

confidence: 99%

“…Motivated by the theory of bornological vector spaces of [13], in this paper we introduce the category L-VBorn of L-bornological vector spaces over a complete lattice L, and extend our result from [21], to show that for certain complete lattices L, L-VBorn is topological over the category Vec of vector spaces. With the help of this achievement, we arrive at the conclusion that L-VBorn is a topologically algebraic construct [2].…”

Section: Introductionmentioning

confidence: 99%

“…With the help of this achievement, we arrive at the conclusion that L-VBorn is a topologically algebraic construct [2]. Moreover, stimulated by the concept of L-bornological system of [21] (which is a bornological analogue of topological systems of S. Vickers [36], which provide a common setting for both point-set and point-free topologies, and whose lattice-valued versions were considered in, e.g., [6,7,[28][29][30]32,33]), we introduce the category L-VBornSys of L-bornological vector systems, and show that the category L-VBorn is isomorphic to a full reflective subcategory of L-VBornSys. We also introduce the notions of convex L-bornological vector space and system (cf., e.g., [13]).…”

Section: Introductionmentioning

confidence: 99%

“…The main purpose of our investigation (started in [21,22]) is the development of a lattice-valued analogue of the Hausdorff dimension in convex bornological spaces of [3], to provide a new technique for the study of cancer-related diseases in humans and animals. This new machinery should additionally take into consideration the presence of fuzziness in the real world phenomena (in the sense of L.A. Zadeh [37]).…”

Section: Introductionmentioning

confidence: 99%

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On the category of lattice-valued bornological vector spaces

Paseka

Solovyov

Stehlík

2014

Journal of Mathematical Analysis and Applications

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Fractal and stochastic geometry inference for breast cancer: a case study with random fractal models and Quermass-interaction process

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Fractals are models of natural processes with many applications in medicine. The recent studies in medicine show that fractals can be applied for cancer detection and the description of pathological architecture of tumors. This fact is not surprising, as due to the irregular structure, cancerous cells can be interpreted as fractals. Inspired by Sierpinski carpet, we introduce a flexible parametric model of random carpets. Randomization is introduced by usage of binomial random variables. We provide an algorithm for estimation of parameters of the model and illustrate theoretical and practical issues in generation of Sierpinski gaskets and Hausdorff measure calculations. Stochastic geometry models can also serve as models for binary cancer images. Recently, a Boolean model was applied on the 200 images of mammary cancer tissue and 200 images of mastopathic tissue. Here, we describe the Quermass-interaction process, which can handle much more variations in the cancer data, and we apply it to the images. It was found out that mastopathic tissue deviates significantly stronger from Quermass-interaction process, which describes interactions among particles, than mammary cancer tissue does. The Quermass-interaction process serves as a model describing the tissue, which structure is broken to a certain level. However, random fractal model fits well for mastopathic tissue. We provide a novel discrimination method between mastopathic and mammary cancer tissue on the basis of complex wavelet-based self-similarity measure with classification rates more than 80%. Such similarity measure relates to Hurst exponent and fractional Brownian motions. The R package FractalParameterEstimation is developed and introduced in the paper.

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Lattice-valued bornological systems

Cited by 15 publications

References 16 publications

Bornological structures on many-valued sets

Bornological structures on many-valued sets

On the category of lattice-valued bornological vector spaces

Fractal and stochastic geometry inference for breast cancer: a case study with random fractal models and Quermass-interaction process

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