generalizes the concept of fuzzy subalgebra to a much larger class of functions in a natural way. In addition, they discussed a method of fuzzification of a wide class of algebraic systems onto 0, 1 along with some consequences. Jun et al. 13 discussed implicative ideals of BCK- algebras based on the fuzzy sets and the **theory** of falling **shadows**. Also, Jun et al. 14 used the **theory** of a falling shadow for considering falling d-subalgebras, falling d-ideals, falling d # -ideals, and falling BCK-ideals in d-algebras.

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Based on the **theory** of falling **shadows** and fuzzy sets, the notion of a falling fuzzy implicative ideal of a BCK-algebra is introduced. Relations among falling fuzzy ideals, falling fuzzy implicative ideals, falling fuzzy positive implicative ideals, and falling fuzzy commutative ideals are given. Relations between fuzzy implicative ideals and falling fuzzy implicative ideals are provided.

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In the western sense community is about sharing similar beliefs while residing in the same place and respecting differences in values and ways of life. (Carlson, Keith Thor. The Power of Place, the Problem of Time: Aboriginal Identity and Historical Consciousness in the Cauldron of Colonialism (2010) University of Toronto Press). The second part of the term is curriculum. Curriculum also has its origins from the Latin and is described as the action of running, course of action, race (competition). (http://dictionary.reference.com/browse/curriculum). Typically the term is associated with schools and their programs of study. Kerr defines curriculum as, âall the learning that is planned and guided by the school, whether it is carried on in groups or individually inside or outside the school.â (Kelly, A.V. (1983; 1999.) The curriculum. **Theory** and Practise, 4e, London; Paul Chapman) (http://www.infed.org\biblio\b.curric.htm). As Aoki states, it is truly a tool for managing schools and ensures that what the government decides is going to be taught is taught.

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In an insightful link in volume I, Hurford puts forward the connection bet- ween the ventral and dorsal neural pathways, and predicateâargument structure respectively (a connection already suggested by Jackendoff & Landau 1992: 121â 123). In a nutshell, the dorsal pathway (âwhere-streamâ) is said to identify the location of an object; the ventral pathway (âwhat-streamâ) gives all the properties necessary to identify it. External objects delivered by the dorsal stream are given individual variables (x, y, z); categorical judgments about objectsâ properties are delivered by the ventral stream and considered predicates (red, cat, Mary). Additionally, two types of psychological attention take part in this process: global attention delivers one-place judgments about the whole scene, while local attention delivers one-place judgments about the objects within each scene; the two processes, global and local, operate in parallel. The notation Hurford pro- poses follows that of the Discourse Representation **Theory** (Kamp & Reyle 1993) regarding the use of boxes; he depicts each one containing the conflated information of both the dorsal and the ventral stream through the two attentional processes; e.g., FLY - SMALL (local attention), or FLY FLY - SMALL (global and local).

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Monika Kostera is Professor in Management and Organization **Theory** at Warsaw University. She has been a visiting professor and visiting researcher at a number of various institutions such as Utrecht University of Humanist Studies, Erasmus University in Rotterdam, Wissenschaftszentrum Berlin, Lund University. She has published several books and chapters, edited books, and has been the guest editor of one special issue of Studies in Cultures, Organizations and Societies. She has published over 20 articles in refereed journals such as: Organization; Management Learning; Studies in Cultures, Organizations, and Societies; Qualitative Sociology; Organization Studies; Human Resource Development International; Journal of Organizational Behavior; Scandinavian Journal of Management, and others. She is a member of editorial boards of several journals, among them: Human Relations; Management Learning; Tamara. Her current research interests are: ethnography of NGOs, aesthetics and spirituality of organizing, critical perspectives management in East European countries.

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In the western sense community is about sharing similar beliefs while residing in the same place and respecting differences in values and ways of life. (Carlson, Keith Thor. The Power of Place, the Problem of Time: Aboriginal Identity and Historical Consciousness in the Cauldron of Colonialism (2010) University of Toronto Press). The second part of the term is curriculum. Curriculum also has its origins from the Latin and is described as the action of running, course of action, race (competition). (http://dictionary.reference.com/browse/curriculum). Typically the term is associated with schools and their programs of study. Kerr defines curriculum as, âall the learning that is planned and guided by the school, whether it is carried on in groups or individually inside or outside the school.â (Kelly, A.V. (1983; 1999.) The curriculum. **Theory** and Practise, 4e, London; Paul Chapman) (http://www.infed.org\biblio\b.curric.htm). As Aoki states, it is truly a tool for managing schools and ensures that what the government decides is going to be taught is taught.

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Development in humans is characterised by such domains as physical development, cognitive development, emotional development, social development and personality development. This paper focuses on the domain of personality development. There are at least two common theories that have been used to explain personality development. These are the psychosexual **theory** and the psychosocial **theory**. The psychosexual **theory** was advocated by Sigmund Freud, whereas the psychosocial **theory** was advanced by Erik Erikson. The paper therefore focuses on Erik Eriksonâs eight stages of the psychosocial **theory** which are then used to explain personality development at Early Childhood Development (ECD) level. The paper discusses the relevance of the **theory** to education and how an appreciation of the **theory** can contribute to a better personality development. This was done through examining the role of the school in the development of the child. The paper concludes that the relevance of Eriksonâs psychosocial **theory** is found in its emphasis on the development of the ego and the relevance of life experiences in developing personality.

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In Forlan, the usual objects of formal language **theory**âautomata, reg- ular expressions, grammars, labeled paths, parse trees, etc.âare defined as abstract types, and have concrete syntax. The standard algorithms of formal language **theory** are implemented in Forlan, including conversions between different kinds of automata and grammars, the usual operations on automata and grammars, equivalence testing and minimization of deter- ministic finite automata, etc. Support for the variant of the programming language Lisp that we use (instead of Turing machines) as a universal pro- gramming language is planned.

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According to (7), Onsagerâs model diffusion in the concentrated liquid mixtures presents the motion of the components, which are mutually rubbed with. the force[r]

Selection by the instructor of topics of interest will be very important, because normally Iâve found that I canât cover anywhere near all of this material in a semester. A reasonable choice for a first try might be to begin with Chapter 2 (recursive algorithms) which contains lots of motivation. Then, as new ideas are needed in Chapter 2, one might delve into the appropriate sections of Chapter 1 to get the concepts and techniques well in hand. After Chapter 2, Chapter 4, on number **theory**, discusses material that is extremely attractive, and surprisingly pure and applicable at the same time. Chapter 5 would be next, since the foundations would then all be in place. Finally, material from Chapter 3, which is rather independent of the rest of the book, but is strongly connected to combinatorial algorithms in general, might be studied as time permits.

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We restrict the computational tasks to yes-or-no problems; this is not too much of a restric- tion, and pays oïŹ in what we gain simplicity of presentation. Note that the task of computing any output can be broken down to computing its bits in any reasonable binary representation. Most of this chapter is spent on illustrating how certain computational tasks can be solved within given resource contraints. We start with the most important case, and show that most of the basic everyday computational tasks can be solved in polynomial time. These basic tasks include tasks in number **theory** (arithmetic operations, greatest common divisor, modular arithmetic) linear algebra (Gaussian elimination) and graph **theory**. (We cannot in any sense survey all the basic algorithms, especially in graph **theory**; weâll restrict ourselves to a few that will be needed later.

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One of the nondeterministic instructions is an assignment instruction of the form "y := ?"; the other is a looping instruction of the form "do or until " An input of a given program is a[r]

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Another major choice is the use of asymptotic analysis. Specically, we con- sider the complexity of an algorithm as a function of its input length, and study the asymptotic behavior of this function. It turns out that structure that is hidden by concrete quantities appears at the limit. Furthermore, depending on the case, we classify functions according to dierent criteria. For example, in case of time complexity we consider classes of functions that are closed under multiplication, whereas in case of space complexity we consider closure under addition. In each case, the choice is governed by the nature of the complexity measure being consid- ered. Indeed, one could have developed a **theory** without using these conventions, but this would have resulted in a far more cumbersome **theory**. For example, rather than saying that nding a satisfying assignment for a given formula is polynomial- time reducible to deciding the satisability of some other formulae, one could have stated the exact functional dependence of the complexity of the search problem on the complexity of the decision problem.

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SUMMARY OF MATHEMATICAL TERMS A finite set of objects called symbols An input to a function A relation whose domain is a set of pairs An operation on Boolean values The values TRUE or FA[r]

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The second **theory** (cf., [16, 17]), due to Solomonov [22], Kolmogorov [15] and Chaitin [3], is rooted in computability **theory** and specically in the notion of a universal language (equiv., universal machine or computing device). It measures the complexity of objects in terms of the shortest program (for a xed universal machine) which generates the object. Like Shannon's the- ory, Kolmogorov Complexity is quantitative and perfect random objects appear as an extreme case. Interestingly, in this approach one may say that a single object, rather than a distribution over ob- jects, is perfectly random. Still, Kolmogorov's approach is inherently intractable (i.e., Kolmogorov Complexity is uncomputable), and { by denition { one cannot generate strings of high Kolmogorov Complexity from short random seeds.

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Proof. This is very easy indeed. With a universal Turing machine we can simulate any finite automaton. To decide whether it accepts an input we need just watch it for a number of steps equal to the length of that input. So far, so good. In order to decide things a bit more intricate than membership though, we need a very useful technical lemma which seems very strange indeed until we begin to use it. It is one of the most important results in finite automata **theory** and it provides us with a handle on the finiteness of finite automata. It tells us that only the information stored in the states of a finite automaton is available during computation. (This seems obvious, but the proof of the following lemma points this out in a very powerful manner.)

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decoded, since every number has a unique prime factorization, and intuitively the decoding function is computable. If we had exponentiation as a primitive of RE, it would be quite easy to see that the decoding function is recursive; but we do not have it as a primitive. Although GĂ¶del did not take exponentiation as primitive, he found a trick, using the Chinese Remainder Theorem, for carrying out the above coding with only addition, multiplication and successor as primitive. We could easily have taken exponentiation as a primitive â it is not essential to recursion **theory** that the language of RE have only successor, addition and multiplication as primitive and other operations as defined. If we had taken it as primitive, our proof of the easy half of Church's thesis, i.e. that all r.e. relations are semi-computable, would still have gone through, since exponentiation is clearly a computable function. Similarly, we could have added to RE new variables to range over finite sets of numbers, or over finite sequences. In fact, doing so might have saved us some time at the beginning of the course. However, it is traditional since GĂ¶delâs work to take quantification over

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At this point, the reader is probably thinking of familiar games such as Chess, Go, Checkers etc. and wondering whether complexity **theory** may help differentiate between themâfor example, to justify the common intu- ition that Go is more difficult than Chess. Unfortunately, formalizing these issues in terms of asymptotic complexity is tricky because these are finite games, and as far as the existence of a winning strategy is concerned, there are at most three choices: Player 1 has has a winning strategy, Player 2 does, or neither does (they can play to a draw). However, one can study general- izations of these games to an n Ă n board where n is arbitrarily large âthis may involve stretching the rules of the game since the definition of chess seems tailored to an 8 Ă 8 boardâ and then complexity **theory** can indeed by applied. For most common games, including chess, determining which player has a winning strategy in the n Ă n version is PSPACE-complete (see [?]or [?]). Note that if NP 6= PSPACE then in general there is no short certificate for exhibiting that either player in the TQBF game has a winning strategy, which is alluded to in Evens and Tarjanâs quote at the start of the chapter.

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David Bolter is called Turing's Man University of North Caroline Press, 1984, and Brian Rotman's critique of traditional mathematical concepts of infinity Ad lrifinitum Stanford Universi[r]

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