In [14], Qiu et al. intuitively showed a method of finding the inverse operation in the **quotient** **space** of fuzzy numbers based on the Mareˇs equivalence relation [9], [10], which have the desired group properties for the addition operation [7], [13], [27]. As an application of the main results, it is shown that if we identify every fuzzy number with the

In this paper, we have researched the convergence of successive approximations for fuzzy differential equations in the **quotient** **space** of fuzzy numbers. We have solved the convergence of successive approximations of the initial value problem for the fuzzy differential equations, provided that f is a continuous with respect to d sup , of uniformly

In this paper, we study the fuzzy diﬀerential equations in the **quotient** **space** of fuzzy numbers. We solve the initial value problem for fuzzy diﬀerential equations provided that the involved mappings are continuous, of uniformly bounded variation, and are bounded functions. Then we establish a variety of comparison results for the solutions of fuzzy diﬀerential equations.

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We introduce an order in a **quotient** **space** of strictly monotone continuous functions on a real interval and show that a new average function on this **quotient** **space** is order-preserving. We also apply this new order-preserving function to derive a ﬁnite form of Jensen type inequality with negative weights.

**Quotient** **space** theory and rough set theory are granular computing models, and are discussed in the framework of set theory. They regard the granularity as a subset of the problem domain. They are not mutually exclusive, but different in main focus. From the model point of view, both are described in the model of the ability of human to deal with things in different granularity. They believe that the concepts can be expressed in subsets. Different concepts of granularity can be used different sizes of subsets. All of these representations can be described by equivalence relations. Form the research object point of view, the **quotient** **space** theory and rough set theory all take the collection of objects discussed as the problem domain. When discussing the relationship between the objects, they are different. The prototype of rough set theory is probably derived through relational database abstraction, that is, use different attribute values of the elements to describe the relationship between the elements and represent different concept granularity according to the classification of different attributes. The prototype of the **quotient** **space** theory is Hierarchical method that in addition to the properties of the elements, it also introduced the relationship between the elements. In our approach, we use rough set theory to predict the traffic flow and use **quotient** **space** theory to predict the traffic congestion. The results show the effectiveness and efficiency of our methods.

In this paper, we firstly review data **space**, data science, and researches on BDP, and talk about the source, form, sig- nificance, and research works of brain big data. We propose the three mechanisms of MGrC and discuss their relation- ship with five major models of MGrC, i.e., fuzzy set, rough set, **quotient** **space**, cloud model, and deep learning. We also discussed the key issues of current BDP and the reasons why MGrC can tackle them. Then we propose the potential of exploring brain big data with MGrC. Future research may include representing the brain big data from real world with MGrR and conducting intelligent computation based on it to offer effective solution to the problems to do successful research in brain BDP.

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to identifiability. Thus numerical approximations are postponed to the estimation phase of an analysis. For example rather than checking numerically if the rank of the design/model matrix for a candidate model is maximal, one computes a basis of the **quotient** **space**. This might be advantageous or disadvantageous according to the practical situations. We find that the information embedded in the ideal of a design or of its cone are useful in visualising the constraints imposed on the power terms by the design.

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Let ˜ P be the **space** of n × n SPDMs, and let P be its subset of matrices with determinant one. The idea is to first identify the **space** P with the **quotient** **space** SL(n)/SO(n) and borrow the Riemannian struc- ture from the latter directly. Then, one can straightfor- wardly extend the Riemannian structure on P to ˜ P . The process starts by choosing a Riemannian metric on G as follows: for any point G ∈ SL(n) the metric is defined by pulling back the tangent vectors under G −1 to I, and then using the trace metric ( see more details in the Section 1 of Supplementary Material I ). This definition leads to expressions for the exponential map, its inverse, parallel transport of tangent vectors, and the Riemannian curvature tensor on SL(n). It also induces a Riemannian structure on the **quotient** **space** SL(n)/SO(n) in a natural way because the chosen met- ric is invariant to the action of SO(n) on SL(n). Finally, these results are transferred to P using the mapping

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Now, using what we know about the **quotient** topology and group acitons, an informal definition of a **quotient** **space** will be given. Let Γ be the topological group acting on a **space** X. If we say points on X are equivalent if and only if they lie in the same orbit, then X/Γ is the **quotient** **space** of X under the action of Γ. Fundamental domains are useful because they allow us to intuitively visualize X/Γ. The following examples are of well known fundamental domains: the torus (using E 2 ) and the punctured torus (using U 2 ).

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We have attempted to improve upon previous defi- nitions of distance between languages in a language **space**. After considering previous work by Vianu (1977) which defined a language distance using the density of their symmetric set difference, we pro- gressed to a new adaptation of a pseudometric in- spired by Besicovitch (1932). In a language **space**, the Besicovitch pseudometric was developed which is essentially the upper density of the set-difference between languages. By lifting to the **quotient** **space** Q ζ using Besicovitch equivalence, a natural metric topology was developed and shown to be perfect but not compact. Another step of lifting brought us a compact “upper” **quotient** **space** N ζ homeomorphic to the unit interval. The ideals of this upper **space** were studied, also invoking the notion of word ideal defined herein. In the last section it was shown that neither the finite nor locally testable languages are dense in N ζ . Finally, the regular languages were

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This map tries very hard to be a homeomorphism. The map is continuous, onto, and it is almost one-to-one with a continuous inverse. It fails in this endeavor only where we join the left and right edges: the points (0, y) and (1, y) in I 2 both get sent by p to the point (1, 0, y) . But p is nice enough to induce a homeomorphism between the cylinder and a modied version of the domain I 2 , obtained by dividing out of I 2 the mapping redundancies so that the result is one-to-one. The new version of I 2 is called a **quotient** **space**. We develop **quotient** spaces in this section because all surfaces and candidate three-dimensional universes can be viewed as **quotient** spaces. We need the notion of an equivalence relation on a set. To get this, we need the notion of a relation.

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Sternberg (2008: 67) explains intellectual intelligence is the ability to learn from experience, think using metacognitive processes, and the ability to adapt to the surrounding environment. Intellectual intelligence is the ability to analyze, logic and a person's ratio. Thus, this is related to speaking skills, intelligence of **space**, awareness of something visible, and mastery of mathematics. IQ measures our speed in learning new things, focusing on various tasks and exercises, storing and recalling objective information, engaging in thought processes, working with numbers, abstract and analytical thinking, and solving problems and applying pre-existing knowledge (Anastasia, 2007: 58).

In §4 we independently verify that the metric on the **space** of numerical stability conditions on a smooth complex projective curve of genus ≥ 1 is complete. We compute this metric as follows. There is a natural action of the universal cover G of GL + 2 R on any **space** of stability conditions. When the phases of semistable objects are dense for a stability condition σ, the orbit through σ is free and the restriction d G of the metric to it is independent of σ and can be

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Role of Parents in development of Emotional **Quotient** 1) Personal attention towards their children’s should be provided. 2) Try to develop secure emotional relationship with children. 3) Understand their emotions and give respect to their emotions. 4) Motivate to develop good hobbies in children’s.

A third kind of neural organization makes it possible for us to do creative, insightful, rule- making/-breaking thinking [13]. It is heart-to-heart thinking, it is the demystified spiritual—not necessarily religious--thinking with which we reframe and transform our previous thinking---our Spiritual Intelligence, our Spiritual **Quotient** (SQ), our quantum quest for meaning, our hyperlink to everything. It is in its transformative power that SP differs from EQ, as emotional intelligence allows you to judge what situation you are in and to behave appropriately within the boundaries of the situation, allowing the situation to guide you. Your spiritual intelligence allows you to ask if you want to be in this particular situation in the first place! In our mathematical shorthand, beyond IQ and EQ is SQ, which can develop our capacity for vision, meaning, and value, our dreams, our happiness, our intuition, our beliefs and our subsequent actions. Ideally, based upon our brain design, our three basic intelligences work together and support one another, but each of them has its own area of power and can function in a stand-alone mode.

The results of the descriptive analysis of job characteristics, emotional **quotient**, spiritual **quotient**, job satisfaction, and nurses performance at Islamic Hospitals in Gresik are all perceived to be high, this is indicated by the average score of the answers in the range of 3.4 - 4.2 (high), this shows that the average nurse at Islamic Hospital in Gresik has a good job characteristics, good emotional **quotient**, good spiritual **quotient**, good job satisfaction, and good performance.

(vii) Contra sbĝ-continuous map[4] if (V) is sbĝ-closed in (X,τ) for every open set V in (Y,σ). (viii) sbĝ – homeomorphism[3] if f is both sbĝ – continuous map and sbĝ – open map. (ix) Strongly sbĝ-continuous[3] if (V) is closed in (X,τ) for every sbĝ-closed set V in (Y,σ). (x) **Quotient**[12] if f is continuous and (V) is open in X implies V is open in Y.

Single inductive (and **quotient** inductive) sets are simply elements of hSet . Induc- tive families [17] indexed over some ﬁxed type A are families A → hSet . For the inductive-inductive deﬁnitions we are considering, the situation is more compli- cated, since we allow very general dependency structures. Our only requirement is that there is no looping dependency, since this is easily seen to lead to contra- dictions, e.g. we do not allow the deﬁnition of a family A : B → hSet mutually with a family B : A → hSet (whatever this would mean). Concretely, we will ensure that the collection of type formation rules (the type signatures) is given in a valid order, and we refer to the types used as family indices as the sorts of the deﬁnition. Hence our ﬁrst step towards a speciﬁcation of general QIITs is to explain what a valid speciﬁcation of the sorts is.

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Abstract. We modify and apply new property obtained recently in (Udo-utun, Fixed Point Theory and Applications 2014, 2014:65) and results in (Berinde, Carpath. J. Math. 19(1):7-22, 2003; Nonlinear Anal. Forum 9(1):43- 53, 2004) on (δ , k)−weak contractions to obtain asymptotic fixed point theorems for bi-Lipschitz mappings and Lipschitz **quotient** mappings in Banach spaces. Our results complement and improve several fixed point theorems for Lipschitzian mappings.

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terms, coming from the critical points of the superpotential. This vacuum moduli **space** M is typically a high dimensional object of subtle structure and consists of many branches, such as mesonic versus baryonic, and Higgs versus Coulomb, etc. Conceptually, M is a **quotient** of F ♭ by the gauge symmetries prescribed by D ♭ . In this short summary of a companion

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