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The influence of roughness on the hydro-mechanical behavior of rock discontinuities has long been recognized. As a result, several definitions and measures of roughness have been developed. According to the ISRM (Int J Rock Mech M...
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The influence of roughness on the hydro-mechanical behavior of rock discontinuities has long been recognized. As a result, several definitions and measures of roughness have been developed. According to the ISRM (Int J Rock Mech Min Sci Geomech Abstr 15(6):319-368, 1978), discontinuity roughness comprises large-scale (wav-iness) and small-scale (unevenness) components. However, the division between these scales is not clear and most investigations of surface roughness have been restricted to small fracture surfaces (<1 m~2). Hence, the large-scale components of roughness are often neglected. Furthermore, these investigations typically define roughness using two-dimensional profiles rather than three-dimensional surfaces, which can lead to biased estimates of roughness.
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Automata theory plays a key role in computational theory as many computational problems can be solved with its help. Formal grammar is a special type of automata designed for linguistic purposes. Formal grammar generates formal la...
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Automata theory plays a key role in computational theory as many computational problems can be solved with its help. Formal grammar is a special type of automata designed for linguistic purposes. Formal grammar generates formal languages. Rough grammar and rough languages were introduced to incorporate the imprecision of real languages in formal languages. These languages have limitations on uncertainty. The authors have considered both uncertainty and approximations to define rough fuzzy grammar and rough fuzzy languages. Under certain restrictions, their grammar reduces to formal grammar. Furthermore, the authors have proposed definition of rough fuzzy automata that accepts rough fuzzy regular language.
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In this paper, the concept of rough representation of a rough topological group on a Banach space is explored. Mainly, the continuity and the irreducibility of rough representations are studied.
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Rough Set Theory (RST) is a mathematical tool to deal with uncertain data. Combining RST to Algebra is a way to apply uncertainty in Algebra. Group, Ring & modules have been presented by some authors based on rough set theory. In this paper, we have introduced the notion of Rough Exact Sequences over an R-module, and some properties....
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Rough Set Theory (RST) is a mathematical tool to deal with uncertain data. Combining RST to Algebra is a way to apply uncertainty in Algebra. Group, Ring & modules have been presented by some authors based on rough set theory. In this paper, we have introduced the notion of Rough Exact Sequences over an R-module, and some properties.
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In this paper we explore the interrelations between rough set theory and group theory. To this end, we first define a topological rough group homomorphism and its kernel. Moreover, we introduce rough action and topological rough g...
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In this paper we explore the interrelations between rough set theory and group theory. To this end, we first define a topological rough group homomorphism and its kernel. Moreover, we introduce rough action and topological rough group homeomorphisms, providing several examples. Next, we combine these two notions in order to define topological rough homogeneous spaces, discussing results concerning open subsets in topological rough groups.
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Rough set is a new approach to uncertainties in spatial analysis. In this paper, rough set symbols are simplified and standardized in terms of rough interpretation and specialized indication. Rough spatial entities and their topol...
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Rough set is a new approach to uncertainties in spatial analysis. In this paper, rough set symbols are simplified and standardized in terms of rough interpretation and specialized indication. Rough spatial entities and their topological relationships are also proposed in rough space, thus a universal intersected equation is developed, and rough membership function is further extended with the gray scale in our case study. We complete three works. First, a set of simplified rough symbols is advanced on the basis of existing rough symbols. Second, rough spatial entity is put forward to study the real world as it is, without forcing uncertainties into crisp set. Third, rough spatial topological relationships are studied by using rough matrix and their figures. The relationships are divided into three types, crisp entity and crisp entity (CC), rough entity and crisp entity (RC), and rough entity and rough entity (RR). A universal intersected equation is further proposed. Finally, the maximum and minimum maps of river thematic classification are generated via rough membership function and rough relationships in our case study.
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We present simulations of X-ray resonant magnetic re-flectivity (XRMR) spectra of the surface magnetic dead layer in La_(1-x)Sr_xMn0_3 (LSMO) films that take in account the effect of different forms of roughness that can be encoun...
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We present simulations of X-ray resonant magnetic re-flectivity (XRMR) spectra of the surface magnetic dead layer in La_(1-x)Sr_xMn0_3 (LSMO) films that take in account the effect of different forms of roughness that can be encountered experimentally. The results demonstrate a method to distinguish between surface (morphological) roughness, and two generic kinds of magnetic roughness at the buried interface between the surface dead layer and the fully magnetic bulk part of the film. We show that the XRMR technique can distinguish between different types of magnetic roughness at the dead layer/bulk interface only if the sample surface is nearly atomically flat (the morphological roughness is one unit cell or less). Furthermore, to distinguish between the two types of magnetic roughness, the simula-tions show that fitting of XRMR spectra out to very high incidence angles must be performed. In the specific case of LSMO films with a dead layer with average thickness of 4 unit cells, this corresponds to an incidence angle >50°.
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This paper investigates the general roughness bounds for rough set operations. Compared with set-oriented rough sets, the results prove that the same upper bound of the roughness for the union, difference and complement operation ...
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This paper investigates the general roughness bounds for rough set operations. Compared with set-oriented rough sets, the results prove that the same upper bound of the roughness for the union, difference and complement operation could be determined by the roughness of the two operand sets. However, the lower roughness bounds of set-oriented rough sets operations do not hold for other rough sets. We provide an example to show the derived bounds from the operand's roughness.
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Rough set theory, initiated by Pawlak, is a mathematical tool in dealing with inexact and incomplete information. Numerical characterizations of rough sets such as accuracy measure, roughness measure, etc, which aim to quantify th...
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Rough set theory, initiated by Pawlak, is a mathematical tool in dealing with inexact and incomplete information. Numerical characterizations of rough sets such as accuracy measure, roughness measure, etc, which aim to quantify the imprecision of a rough set caused by its boundary region, have been extensively studied in the existing literatures. However, very few of them are explored from the viewpoint of rough logic, which, however, helps to establish a kind of approximate reasoning mechanism. For this purpose, we introduce a kind of numerical approach to the study of rough logic in this paper. More precisely, we propose the notions of accuracy degree and roughness degree for each formula in rough logic with the intension of measuring the extent to which any formula is accurate and rough, respectively. Then, to measure the degree to which any two formulae are roughly included in each other and roughly similar, respectively, the concepts of rough inclusion degree and rough similarity degree are also proposed and their properties are investigated in detail. Lastly, by employing the proposed notions, we develop two types of approximate reasoning patterns in the framework of rough logic.
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