Convexity and level sets for interval-valued fuzzy sets

Convexity is a deeply studied concept since it is very useful in many fields of mathematics, like optimization. When we deal with imprecision, the convexity is required as well and some important applications can be found fuzzy optimization, in particular convexity of fuzzy sets. In this paper we have extended the notion of convexity for interval-valued fuzzy sets in order to be able to cover some wider area of imprecision. We show some of its interesting properties, and study the preservation under the intersection and the cutworthy property. Finally, we applied convexity to decision-making problems.Slovak grant agency VEGAVedecka grantova agentura MSVVaS SR a SAV (VEGA) [1/0150/21]; Spanish Ministry of Science and TechnologySpanish Government [TIN-201787600-P, PGC2018-098623-B-I00]; FICYT [IDI/2018/000176]Ministerio de Ciencia y Tecnología, MICYT: PGC2018-098623-B-I00, TIN-2017-87600-P; Fundación para el Fomento en Asturias de la Investigación Científica Aplicada y la Tecnología, FICYT: IDI/2018/00017

Poset-valued preference relations

summary:In decision processes some objects may not be comparable with respect to a preference relation, especially if several criteria are considered. To provide a model for such cases a poset valued preference relation is introduced as a fuzzy relation on a set of alternatives with membership values in a partially ordered set. We analyze its properties and prove the representation theorem in terms of particular order reversing involution on the co-domain poset. We prove that for every set of alternatives there is a poset valued preference whose cut relations are all relations on this domain. We also deal with particular transitivity of such preferences

The effect of selected hydrocolloids on the rheological properties of processed cheese analogues made with vegetable fats during the cooling phase

The aim of this work was to observe the influence of different hydrocolloids on changes in the rheological properties of processed cheese and their analogues within the process of cooling depending on the type of the fat used (butter, coconut fat and palm oil). κ-carrageenan, ι-carrageenan, λ-carrageenan, arabic gum and locust bean gum were used at a concentration of 1.0% w/w. With the decreasing temperature during the cooling period, an increase in the complex modulus (G*) was observed in all samples tested. Within the cooling period, the sample with the addition of κ-carrageenan showed both the most significant increase in G* in comparison with the control and the highest values of hardness after a 7-day storage period regardless of the type of the fat used. The samples with coconut fat were assessed as the hardest. On the other hand, the samples with palm oil showed the lowest hardness (with the same hydrocolloid used). In the gelling hydrocolloids (κ- and ι-carrageenan), only small changes in temperature of coil-to-helix transition were observed (in range of 2-7 °C) as a result of the addition of different types of fat. © 2013 Society of Dairy Technology

Objective measurements of skin surface roughness after microdermabrasion treatment

Background: The aim of this article is to present a new methodology for assessment of skin topology using a three-dimensional image (3D). Methods: The measurement of the skin surface roughness is based on 3D scanning of silicone replicas by chromatic aberration length technique in a contactless manner, i.e. by a polychromatic light beam. Analysis of the skin surface reprints was performed using Talymap, Gold version. Results were analysed by fractal geometry, which allows to evaluate changes of the skin surface before and after application of cosmetics and instrumental cosmetological techniques. The methodology was applied for objective assessment of the effects of diamond microdermabrasion on the skin surface roughness. Measurements were performed on 23 volunteers in the age group of 31–67 years. Results: Based on the results of skin surface scanning after the treatment with diamond microdermabrasion it may be concluded that inequalities of the skin surface are reduced immediately after exfoliation. However, this effect mostly diminishes within 14 days after treatment. The entire study ultimately suggests that the instrumental method used only leads to improvement of the skin surface immediately after its application. Thermo vision images of the skin surface temperature were obtained during the application of the abrasive method. The experimental results showed that the skin is rather cooled than heated by the treatment. Conclusion: This study is focused on the development of a methodology for objective measurement of changes in treated skin relief using 3D scanning. The results are evaluated using fractal dimension. The output may also include also an enlarged model of the skin surface made by 3D printer, which can serve for illustrative communication with the client. © 2017 John Wiley & Sons A/S. Published by John Wiley & Sons Ltd[IGA/FT/2014/009

Scanning and evaluation of biological surfaces using the technique of rapid prototyping

One of the most important characteristics of a cosmetic product is maintain or improving the condition and characteristics of the skin surface. In dermatological, respectively cosmetic practice, there is very often a problem of an objective assessment of the obtained data, which corresponds to visual changes in skin relief. With technical advancements has it been proposed many approaches of solution: mechanical needle devices, optical interference devices, and microscopic, holographic, laser and photometers ones. Each of them has its advantages and negative qualities. The newest devices allow not only two-dimensional assessment (2D), but also spatial one (3D). Scientific literature such as Measuring the Skin recommends for these cases scanning the surface in 3D, using the contact scanners for made imprints. Here, however, errors that may be caused by mechanical method of scanning skin relief are not taken into consideration and there is also not clearly described the material suitable for the production of custom impression. The following article focuses precisely on this issue, specifically on the implementation of the biological surface relief imprint using a silicone impression material, its 3D scanning without contact and forming an enlarged replica by rapid prototyping technology. © (2014) Trans Tech Publications, Switzerland

Generalized convexities related to aggregation operators of fuzzy sets

summary:We analyze the existence of fuzzy sets of a universe that are convex with respect to certain particular classes of fusion operators that merge two fuzzy sets. In addition, we study aggregation operators that preserve various classes of generalized convexity on fuzzy sets. We focus our study on fuzzy subsets of the real line, so that given a mapping $F: [0,1] \times [0,1] \rightarrow [0,1]$, a fuzzy subset, say $X$, of the real line is said to be $F$-convex if for any $x, y, z \in \mathbb{R}$ such that $x \le y \le z$, it holds that $\mu_X(y) \ge F(\mu_X(x),\mu_X(z))$, where $\mu_X: \mathbb{R} \rightarrow [0,1]$ stands here for the membership function that defines the fuzzy set $X$. We study the existence of such sets paying attention to different classes of aggregation operators (that is, the corresponding functions $F$, as above), and preserving $F$-convexity under aggregation of fuzzy sets. Among those typical classes, triangular norms $T$ will be analyzed, giving rise to the concept of norm convexity or $T$-convexity, as a particular case of $F$-convexity. Other different kinds of generalized convexities will also be discussed as a by-product

Generalized convexities related to aggregation operators of fuzzy sets

summary:We analyze the existence of fuzzy sets of a universe that are convex with respect to certain particular classes of fusion operators that merge two fuzzy sets. In addition, we study aggregation operators that preserve various classes of generalized convexity on fuzzy sets. We focus our study on fuzzy subsets of the real line, so that given a mapping $F: [0,1] \times [0,1] \rightarrow [0,1]$, a fuzzy subset, say $X$, of the real line is said to be $F$-convex if for any $x, y, z \in \mathbb{R}$ such that $x \le y \le z$, it holds that $\mu_X(y) \ge F(\mu_X(x),\mu_X(z))$, where $\mu_X: \mathbb{R} \rightarrow [0,1]$ stands here for the membership function that defines the fuzzy set $X$. We study the existence of such sets paying attention to different classes of aggregation operators (that is, the corresponding functions $F$, as above), and preserving $F$-convexity under aggregation of fuzzy sets. Among those typical classes, triangular norms $T$ will be analyzed, giving rise to the concept of norm convexity or $T$-convexity, as a particular case of $F$-convexity. Other different kinds of generalized convexities will also be discussed as a by-product

Comparison of metrological techniques for evaluation of the impact of a cosmetic product containing hyaluronic acid on the properties of skin surface

The aim of this research was to evaluate mutual interchangeability of four principally different biometric instrumental techniques designed for objective measurement of changes in the physical, mechanical, and topographical properties of the skin surface treated with commercial antiaging cosmetic products with hyaluronic acid. The following instrumental devices were used: Visioscope PC 35, Corneometer Multiprobe Adapter MPA 6, Reviscometer RVM 600, and 3D scanner Talysurf CLI 500. The comparison of the individual methods was performed using cluster analysis. The study involved 25 female volunteers aged 40-65. Measurements were taken before and after 30 daily in vivo applications of an antiaging preparation to the skin surface in the periorbital area. A slight reduction in skin surface roughness was recorded in 55% of the volunteers. On the contrary, a worsening from their initial states was detected in 25% of the subjects, while for 20%, no significant change was reported. Cluster analysis confirmed that the mentioned methodologies can be divided into two basic clusters, namely, a cluster of methods recording the changes in skin relief by means of optical techniques, and a cluster of methods investigating changes in hydration and anisotropy. In practice, the techniques in different clusters are not interchangeable and should be assessed separately. © 2017 American Vacuum Society.TBU in Zlin [IGA/FT/2015/009

Effect of 1-monoglycerides on viscoelastic properties of processed cheese

Konzistence je velmi důležitým senzorickým znakem tavených sýrů. Konzistence tavených sýrů může být ovlivněna přídavky substancí, které jsou schopny ovlivnit proces emulgace tuku, jako například různé emulgátory. V práci byly po 14 dnech skladování při 6 ? 2 oC pomocí dynamické oscilační reometrie analyzovány a srovnávány vzorky tavených sýrů s přídavky 1-monoacylglycerolů kyselin kaprylové, laurové a stearové. S rostoucím počtem uhlíků v molekule mastné kyseliny, která je esterově vázaná v 1 monoacylglycerolu, se zvyšoval zejména elastický modul pružnosti G´ (storage modulus). Rovněž průsečík křivek (crossover point) elastického modulu pružnosti G´ a ztrátového modulu pružnosti G´´ (loss modulus) se s rostoucím počtem uhlíku posouval do nižších frekvencí. Po 120 dnech skladování při teplotě 6 ? 2 oC došlo ke zvýšení jak elastického G´ tak i ztrátového G´´ modulu pružnosti, z čehož lze usuzovat na zvýšení tuhosti v průběhu skladování po dobu 4 měsíců. Jako nejstabilnější byl hodnocen tavený sýr s přídavkem 1 monoacylglycerolu kyseliny stearové - došlo u něj k nejmenším změnám sledovaných parametrů.This paper deals with the influence of selected 1-monoglycerides (1 monocaprylin, 1-monocaprin, 1-monolaurin, 1 monopalmitin and 1-monostearin) in concentration of 0.25 % w/w on viscoelastic and sensory properties of the processed cheese with 45 and 50 % fat in dry matter after 14 days of storage at temperature of 6 ? 2 °C. With increasing number of carbons in the molecule of fatty acid, which is ester-bound in 1 monoglycerides, both the storage G´ and the loss G´´ moduli increased in whole frequency range. This enhancement is relatively smaller in the group of processed cheeses with 45 % fat in dry matter compared to the group of processed cheeses with 50 % fat in dry matter. The additions of 1 monocaprylin, 1 monocaprin and 1 monolaurin were not sensory acceptable due to off-flavour. The concentration experiments with 1-monopalmitin and 1-monostearin (from 0.1 to 0.5 % w/w) were performed and discussed