All the above examples can be framed in turn into the general setting proposed here, and seen as particular instances of the general definition of **L**-**fuzzy** notion. But with this new general formulation, we do not restrict ourselves to the con- texts of IVF and AIF **sets**, and therefore we can cite additional examples from the literature that could not be included in our initial formulation proposed in [1], like the notions of inclusion and similarity between T2F **sets** introduced by Yang and Lin in [27], the notions of similarity and distance between T2F **sets** considered by Hung and Yang [28], **different** axiomatisations of similarity, distance and entropy between hesitant **fuzzy** **sets** (see [14], [29], for instance) inclusion **measures** between hesitant **fuzzy** **sets** [29], divergences between hesitant **fuzzy** **sets** [10]. Further examples in **different** generalised contexts can be found. Notwithstanding, an exhaustive list of the ax- iomatic definitions that can be regarded as particular instances of Definition 2 would fall outside the scope of this manuscript. One of the main features of this new formulation is that we do not need separate equations for every particular framework, but the same formula applies in **different** contexts, each of them referring to a particular instance of the lattice **L**. Thus, a single general **L**-**fuzzy** notion encompasses the original **fuzzy** notion, together with its possible extensions to more complex frameworks.

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In this section we will deal with set-valued extensions of concepts originally **defined** within the context of **fuzzy** **sets**. In [11], we have reviewed three **different** but related construction methods of set-valued generalizations, the so- called set-valued, max-min and max-min-varied extensions. Each of those construction methods includes, as particular cases, several particular definitions from the literature. Al- though less common, we can also find some works in the literature proposing lists of axioms that certain set-valued **measures** should satisfy. This is the case of the notion of set-valued inclusion **measures** of Cornelis-Kerre [10], and the three variants of the notion of interval-valued similarity **measures** respectively introduced by Bustince [6] Galar et al. [18] and Stachowiak-Dyczkowski [37]. Let us recall Galar et al. definition [18] as an illustration of the general definition to be provided in this section (Definition 6):

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µ and ν A = 1 . A denotes the IFS complement of A . Based on the definition of **fuzzy** entropy and similarity measure, numerous entropy and similarity measure have been introduced [6-8]. The usefulness also verified through previous results. Comparison between IFSs similarity measure was done by Y. Li, D.L. Olson, and Z. Qin [158]. They have compared conventional similarity **measures**. Furthermore, other **fuzzy** entropy on IFSs has been also **defined** by Hung and Yang [16]. In their definition, IP2 and IP4 are **different** from that of their properties. Difference of two definitions has their own characteristics. Next, entropy and similarity measure between IFSs has been derived based on Definition 2.4. Results are somewhat similar with that of **fuzzy** set.

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A simulation study is conducted to compare the es- timation performances of the considered distance mea- sures. Considering the overall statement of the simu- lation results, we reached minimum M AE values with taking into account the distance measure described by Kaufmann and Gupta [13] and Heilpern-2 [12] for Case- II. Besides, the distance measure described by Chen and Hsieh [4] gives minimum M AE values for Case- III. It is demonstrated that the distance measure used by Abdalla and Buckley [1][2] is not convenient to esti- mate **fuzzy** linear regression model parameters with MC methods. Since all maximum values of M AE are calcu- lated with the distance measure that is considered by Abdalla and Buckley [1][2] for both Case-II and Case- III.

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Theories of level II **fuzzy** **sets** and type II **fuzzy** **sets** are generalizations of **fuzzy** set theory for modeling higher order of uncertainty. Real world problem involve many kinds of imperfections in the data. This imperfect information can be classified in following five categories- Uncertain information, imprecise information, vague information, inconsistent information and incomplete information [4]. **Fuzzy** **sets** linguistically deal with above types of information but value of each element being crisp cannot handle higher level of uncertainty [2]. Level II **fuzzy** **sets** are mainly used to precise clearly the difference between fuzziness and uncertainty [5]. Details will be discussed in the section 2, while type II **fuzzy** **sets** linguistically deal capturing uncertainties with words with

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several applications in the areas such as signal processing, decision making, control theory, pattern recognition, computer version and so on.The complexities of modeling uncertain data are the main problem in engineering, environmental science, social sciences, health and medical sciences etc. The **fuzzy** set theory [2], probability theory, rough set theory [10] etc. are well known and useful mathematical tools which describes uncertainty but each of them has its own limitation pointed out by Molodstov. Therefore, Molodstov introduced the theory of soft **sets** [1] to model vague and uncertain information.A soft set is a parameterized collection of subsets of a universe of discourse.A huge amount of literature can be seen in [1,2,3,5,9,10,11].

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Abstract: In this note, we will use a compactness type condition in connection with the weak topology to prove the existence of weakly continuous solutions for a functional integral equa[r]

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Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In the objects studied in discrete mathematics – such as integers, graphs, and statements in logic [1] – do not vary smoothly in this way, but have distinct, separated values[2]. Discrete mathematics therefore excludes topics in ―continuous mathematics" such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable **sets** [3]. Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are useful in studying and describing objects and problems in branches of computer science, such as computer algorithms, programming, cryptography, automated theorem proving, and software development. In mathematics, **fuzzy** **sets** whose elements have degrees of membership. **Fuzzy** mathematics is the study of **fuzzy** structures, such mathematical structures that at some points replace the two classical truth values 0 and 1 with a larger structure of degrees. Most of our traditional tools for formal modelling, reasoning, and computing are crisp, deterministic, and precise in character. Crisp means dichotomous, that is, yes-or-no type

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[5] Kandil A., Biproximities and **fuzzy** bitopological spaces, Simon Steven 63(1989) ,45-66. [6]Kovar, M.,(2000).On 3-Topological version of Thet-Reularity. Internat.J.Matj,Sci. ,23(6),393-398. [7] Levine N.(1963). semi open **sets** and semi –continuity in topological spaces, Amer. Math., 70,36- 41.

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5IFTBNFBSHVNFOUTDBOCFBQQMJFEUPNBONBEFTZTUFNT"QMBUGPSN BSDIJUFDUVSFQBSUJUJPOTBTZTUFNJOUPTUBCMFDPSFDPNQPOFOUTBOEWBSJBCMF QFSJQIFSBMDPNQPOFOUT#ZQSPNPUJOHUIFSFVTFPGDPSFDPNQPOFOUTTVDI QBS[r]

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The technology in the storage of data is developing every day. Though the process of storing the data is the challenging process and many new techniques are in the developing stage. The paper focuses about the **Fuzzy** C- Means Clustering and the **different** techniques that are incorporated with it. The discussion part gives the comparative result of FCM and K-Means with respect to time. Even if the number of iterations in FCM is more when compared to K-Means the working time is small in FCM.

In this paper, mutant **fuzzy** **sets** in semigroups with respect to product of two **fuzzy** **sets** given any semigroup are established and its basic properties are studied. In Section 3, t-norm based mutation for **fuzzy** **sets** on any crisp set are discussed. Prominent t-norm based operations among **fuzzy** **sets** such as algebraic product, bounded product, drastic product are used, and some results are given.

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τ τ **fuzzy** pre-open set and characterized a **fuzzy** pair wise pre continuous mappings on a **fuzzy** bitopological space. We have [12] introducedfuzzy tri semi-open **sets** and **fuzzy** tri pre-open **sets**, **fuzzy** tri continuous function, **fuzzy** tri semi-continuous function and **fuzzy** tri pre-continuous functions and their basic properties.

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In this paper, we define a new kind of **fuzzy** set which is intuition **fuzzy** set, by combining the **fuzzy** starshaped set and intuition **fuzzy** set. By the basic definition of in- tuitionistic **fuzzy** starshaped set, we introduce some new **different** types of starshapedness. We discuss the rela- tionships among these **different** types of starshapedness,

Other extensions are as follows. First, while our baseline model assumes that the interest rate in the capital market is given, we analyze how our results are a¤ected if we close the model and endogenize the interest rate. Our results for global optimality do not change. In the case of national optimality, however, countries may have an incentive to change the interest rate, depending on whether they are capital exporters or capital importers, so that the nationally optimal tax policy may be distorted. Second, we explore the impact of taxing foreign source income only when it is repatriated to the home country. In line with the "new **view**" of dividends, we show that the investment from earnings retained abroad does not depend on the home country taxation of foreign source income. Third, we examine the case of competition amongst acquirers, which allows the owners of the target company to capture part of the surplus generated by an acquisition. Again we show that our results are una¤ected. Fourth, we allow for positive spillovers of activity by the multinational company in the foreign country. This does not a¤ect national optimality for the home country, since the spillovers accrue to foreign residents. We show that global optimality requires a modi…cation of the cross-border cash ‡ow tax, with higher allowances being required to promote additional outbound investment.

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In general, monotone **measures** lose additivity, they appear to be much looser than the classical **measures**. Thus, it is quite difficult to develop a general theory of monotone **measures** and nonlinear integral without any additional condition. The concepts mentioned above replace additivity play important roles in genera- lized measure theory (see [6] [18] [19] [26] [27] [28]). So, the results we ob- tained here are significant when applying monotone measure theory to expert systems, decision making, pattern recognition, image processing, risk analysis, and other areas.

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Other extensions are as follows. First, while our baseline model assumes that the interest rate in the capital market is given, we analyze how our results are a¤ected if we close the model and endogenize the interest rate. Our results for global optimality do not change. In the case of national optimality, however, countries may have an incentive to change the interest rate, depending on whether they are capital exporters or capital importers, so that the nationally optimal tax policy may be distorted. Second, we explore the impact of taxing foreign source income only when it is repatriated to the home country. In line with the "new **view**" of dividends, we show that the investment from earnings retained abroad does not depend on the home country taxation of foreign source income. Third, we examine the case of competition amongst acquirers, which allows the owners of the target company to capture part of the surplus generated by an acquisition. Again we show that our results are una¤ected. Fourth, we allow for positive spillovers of activity by the multinational company in the foreign country. This does not a¤ect national optimality for the home country, since the spillovers accrue to foreign residents. We show that global optimality requires a modi…cation of the cross-border cash ‡ow tax, with higher allowances being required to promote additional outbound investment.

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Based on level **sets** of **fuzzy** **sets**, we propose deﬁnitions of limits of sequences of **fuzzy** **sets**, and limits and derivatives of **fuzzy** set-valued mappings. Then, their properties are derived. Limits of sequences of **fuzzy** **sets**, and limits and derivatives of **fuzzy** set-valued mappings are fuzziﬁed ones of them for crisp **sets**, where ‘crisp’ means ‘non-**fuzzy**’.

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The aim of this paper is to review **different** hardness **measures** **defined** in the literature, and to provide **unified** characterisations for these **measures** in terms of Prover-Delayer games and **sets** of partial assignments satisfying some consistency conditions. These **unified** characterisations allow elegant proofs of basic relations between the **different** hardness **measures**. Unlike in the works [8,11,48], our emphasis here is not on trade-off results, but on exact relations between the **different** **measures**. For a clause-set F we will consider the following **measures**: (i) size **measures**: the depth dep(F ) and the hardness hd(F) (of best resolution refutations of F ); (ii) width **measures**: the symmetric and asymmetric width wid(F) and awid(F); (iii) clause-space **measures**: semantic space css(F ), resolution space crs(F ) and tree-resolution space cts(F ).

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This paper also serves to characterize the asymptotic behavior of a memoryless source. New results on estimating the source distribution and entropy which are independent of the alphabet size are obtained. These results can improve some existing results for those source distribution that have a finite but long tail. They are also instrumental in proving the main results in this paper, and they may have further applications in **different** problems.

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