Direct products of bounded fuzzy lattices realized by triangular norm operators without zero divisors

In this note we continue the work of Chon, as well as Mezzomo, Bedregal, and Santiago, by studying direct products of bounded fuzzy lattices arising from fuzzy partially ordered sets. Chon proved that fuzzy lattices are closed under taking direct products defined using the minimum triangular norm operator. Mezzomo, Bedregal, and Santiago extended Chon's result to the case of bounded fuzzy lattices under the same minimum triangular norm product construction. The primary contribution of this study is to strengthen their result by showing that bounded fuzzy lattices are closed under a much more general construction of direct products; namely direct products that are defined using triangular norm operators without zero divisors. Immediate consequences of this result are then investigated within distributive and modular fuzzy lattices

Fuzzy Convergence Sequence and Fuzzy Compact Operators on Standard Fuzzy Normed Spaces

الغرض الرئيسي من هذا العمل هو تقديم بعض أنواع متتابعات التقارب الضبابية للمؤثرات المعرفة  على الفضاءات المعيارية الضبابية والبحث في  بعض الخصائص والعلاقات بين هذه المفاهيم  في البداية , يتم تقديم تعريف متتابعة التقارب  (SFN-spaces)القياسية الحد نفسه  تتقارب تقارباً ضبابيا ضعيفا مع  الضبابي الضعيف بدلالة الدوال الخطية الضبابية المقيدة. بعد ذلك تم برهان أن المتتابعة  ( fuzzy Baire's and uniform fuzzyboundedness   تم ذكروبرهان نظريتين مهمتين(  متقاربة ضبابياً. في حالة أن المتتابعة  للمؤثرات وهذه النظريات ضرورية  في الاتصال مع التقارب الضبابي الضعيف. يتم تقديم مفاهيم متتابعات التقارب الضبابي القوي والضعيف   حيث  متتابعة متقاربة ضبابياً بقوة مع   . على وجه الخصوص, إذا كان واثبات النظريات الأساسية المتعلقة بهذه المفاهيم  تنتمي    فأن  الى الفضاء المعياري  الضبابي القياسي  المؤثر الخطي من الفضاء المعياري الضبابي القياسي التام الى مجموعة كل المؤثرات الخطية الضبابية المقيدة. . اضافة الى ذلك , تم تقديم مفهوم المؤثر الخطي المتراص الضبابي في الفضاء المعياري الضبابي القياسي. أيضا ، تتم دراسة العديد من النظريات الأساسية للمؤثرات الخطية المتراصة الضبابية على نفس الفضاء. بتعبير أدق ، ثبت   فضاء معياري ضبابي قياسي. و أن كل مؤثر خطي متراص ضبابي يكون مقيد ضبابي بحيث أن كل من على الفضاء المعياري الضبابي القياسي  والمؤثر الخطي الضبابي المقيد    في نهاية البحث ، أثبتنا أن المؤثر الخطي الضبابي المتراص يجب أن يكون متراصاً ضبابياً.  The main purpose of this work is to introduce some types of fuzzy convergence sequences of operators defined on a standard fuzzy normed space (SFN-spaces) and investigate some properties and relationships between these concepts. Firstly, the definition of weak fuzzy convergence sequence in terms of fuzzy bounded linear functional is given. Then the notions of weakly and strongly fuzzy convergence sequences of operators  are introduced and essential theorems related to these concepts are proved. In particular, if ( ) is a strongly fuzzy convergent sequence with a limit  where linear operator from complete standard fuzzy normed space  into a standard fuzzy normed space  then  belongs to the set of all fuzzy bounded linear operators . Furthermore, the concept of a fuzzy compact linear operator in a standard fuzzy normed space is introduced. Also, several fundamental theorems of fuzzy compact linear operators are studied in the same space. More accurately, every fuzzy compact linear operator  is proved to be fuzzy bounded  where  and  are two standard fuzzy normed space

Properties of Fuzzy Compact Linear Operators on Fuzzy Normed Spaces

In this paper the definition of fuzzy normed space is recalled and its basic properties. Then the definition of fuzzy compact operator from fuzzy normed space into another fuzzy normed space is introduced after that the proof of an operator is fuzzy compact if and only if the image of any fuzzy bounded sequence contains a convergent subsequence is given. At this point the basic properties of the vector space FC(V,U)of all fuzzy compact linear operators are investigated such as when U is complete and the sequence ( ) of fuzzy compact operators converges to an operator T then T must be fuzzy compact. Furthermore we see that when T is a fuzzy compact operator and S is a fuzzy bounded operator then the composition TS and ST are fuzzy compact operators. Finally, if T belongs to FC(V,U) and dimension of V is finite then T is fuzzy compact is proved

A survey on fuzzy fractional differential and optimal control nonlocal evolution equations

We survey some representative results on fuzzy fractional differential equations, controllability, approximate controllability, optimal control, and optimal feedback control for several different kinds of fractional evolution equations. Optimality and relaxation of multiple control problems, described by nonlinear fractional differential equations with nonlocal control conditions in Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN: 0377-0427. Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication 20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515

Unsharp Degrees of Freedom and the Generating of Symmetries

In quantum theory, real degrees of freedom are usually described by operators which are self-adjoint. There are, however, exceptions to the rule. This is because, in infinite dimensional Hilbert spaces, an operator is not necessarily self-adjoint even if its expectation values are real. Instead, the operator may be merely symmetric. Such operators are not diagonalizable - and as a consequence they describe real degrees of freedom which display a form of "unsharpness" or "fuzzyness". For example, there are indications that this type of operators could arise with the description of space-time at the string or at the Planck scale, where some form of unsharpness or fuzzyness has long been conjectured. A priori, however, a potential problem with merely symmetric operators is the fact that, unlike self-adjoint operators, they do not generate unitaries - at least not straightforwardly. Here, we show for a large class of these operators that they do generate unitaries in a well defined way, and that these operators even generate the entire unitary group of the Hilbert space. This shows that merely symmetric operators, in addition to describing unsharp physical entities, may indeed also play a r{\^o}le in the generation of symmetries, e.g. within a fundamental theory of quantum gravity.Comment: 23 pages, LaTe

MODELLING EXPECTATIONS WITH GENEFER- AN ARTIFICIAL INTELLIGENCE APPROACH

Economic modelling of financial markets means to model highly complex systems in which expectations can be the dominant driving forces. Therefore it is necessary to focus on how agents form their expectations. We believe that they look for patterns, hypothesize, try, make mistakes, learn and adapt. AgentsÆ bounded rationality leads us to a rule-based approach which we model using Fuzzy Rule-Bases. E. g. if a single agent believes the exchange rate is determined by a set of possible inputs and is asked to put their relationship in words his answer will probably reveal a fuzzy nature like: "IF the inflation rate in the EURO-Zone is low and the GDP growth rate is larger than in the US THEN the EURO will rise against the USD". æLowÆ and ælargerÆ are fuzzy terms which give a gradual linguistic meaning to crisp intervalls in the respective universes of discourse. In order to learn a Fuzzy Fuzzy Rule base from examples we introduce Genetic Algorithms and Artificial Neural Networks as learning operators. These examples can either be empirical data or originate from an economic simulation model. The software GENEFER (GEnetic NEural Fuzzy ExplorER) has been developed for designing such a Fuzzy Rule Base. The design process is modular and comprises Input Identification, Fuzzification, Rule-Base Generating and Rule-Base Tuning. The two latter steps make use of genetic and neural learning algorithms for optimizing the Fuzzy Rule-Base.