The following section gathers several known results from the theory of functions of bounded directional variation. In Section “Directional variation, perimeter and covariogram of **measurable** **sets**”, the main results relating both the derivative at the origin and the Lipschitz continuity of the covariogram of a set to its directional variations and its perimeter are stated. Finally, in the last section, applications of these results to the theory of random closed **sets** are discussed and illustrated.

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These considerations lead us to a general analysis of finitely subadditive outer measures on PX and finitely superadditive inner measures on 7X [For a finite outer measure countably suba[r]

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menting paths of length at most 2i + 1 in an arbitrary **measurable** way until none remains. This works by the observation of Lyons and Nazarov [25, Re- mark 2.6] that (13) and (14) are satisfied automatically provided G has the expansion property (i.e. there is ε > 0 such that the measure of the neighbour- hood of X is at least (1 +ε)λ(X) for every set X occupying at most half of one part in measure). As shown in [10], the expansion property applies to a wide range of pairs A, B. For example, one of the results in [10] is that two bounded Lebesgue **measurable** **sets** A, B ⊆ R k for k > 3 are measurably equidecompos-

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menting paths of length at most 2i + 1 in an arbitrary **measurable** way until none remains. This works by the observation of Lyons and Nazarov [25, Re- mark 2.6] that (13) and (14) are satisfied automatically provided G has the expansion property (i.e. there is ε > 0 such that the measure of the neighbour- hood of X is at least (1 +ε)λ(X) for every set X occupying at most half of one part in measure). As shown in [10], the expansion property applies to a wide range of pairs A, B. For example, one of the results in [10] is that two bounded Lebesgue **measurable** **sets** A, B ⊆ R k for k > 3 are measurably equidecompos-

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In section 2, based on Tanaka (2009) [7], the fundamentals of lattice sigma algebra, lattice measure on a lattice sigma algebra were described. Further based on Anil kumar etrl (2011) [2] the concepts of lattice **measurable** set, lattice measure space and lattice σ – finite measure were defined. Here some elementary properties of lattice **measurable** **sets** were derived.

the interplay between the measurable sets associated with these outer measures, regularity properties of the measures, smoothness properties of the measures, and lattice topological prop[r]

At present there is no conclusive explanation of 3D perception in the absence of **measurable** stereo-acuity. However, we observe three key differences between 3D entertainment media and clinical stereotests. First, the artistic/monocular/pictorial cues in the image can cause monocular stereopsis, the illusion of depth from a flat image. Second, moving images provide additional monocular and stereoscopic motion cues. Third, the 3D display technology itself, the methods of 3D presenta- tion, generate both supraliminal and subliminal artefacts. Depth can be identified through a number of pictorial cues such as linear perspective, relative size, texture gradient, height in the visual field, aerial perspective (blur), occlusion, lighting and shadow. 16 While pictorial

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Attanassov, K.T., Gargov, G. 1989. Interval valued intuitionstics fuzzy **sets**; Fuzzy **sets** and system. 343-349. Bustince, H., Buriuo, P. 1996. Vague **sets** are intutionistics fuzzy **sets**. Fuzzy **sets** and system 79 403-405. Dubois, D. and Prade , H. 1980. Fuzzy **sets** and systems : Theory and Applications, Academic press, New York. Gau, W. L., Daniel, J. B. 1993. Vague **sets**, IEEE transactions on systems, man Cybernetics 23 610-614. Zadeh, L.A. 1965. fuzzy **sets**. Information and control 8 338-353.

The goals and objectives component of the IPP provides a functional, working document that a teacher can use in planning, developing, implementing and assessing programming for individual students. Goals and objectives that are meaningful, **measurable** and manageable allow teachers to plan, organize and deliver instruction to meet those goals and objectives, and greatly increase students’ chances of success. Meaningful, **measurable** and manageable goals and objectives also ensure that everyone on a student’s learning team has the same expectations for what the student will be doing over the school year, and make it easier to see and

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In this section we use the theoretical results obtained previously, to conduct a study of convex integral functlonals which are defined on Lebesgue-Bochner spaces.. dimensions and Bismut[r]

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For the rest of the paper, (Ω, Σ, µ) is assumed to be a finite measure space, E is a order continuous K¨ othe function space over (Ω, Σ, µ). At first we introduce the kind of **measurable** bundles of Banach spaces, which we need. Liftable mea- surable bundle of Banach spaces X over (Ω, Σ, µ) is said to have a Generalized M-basis or GM-basis for short if there is a family (ϕ i , ϕ ⋆ i ) i∈I **measurable** section,

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Through evaluation procedures and with 85 accuracy, the student will ask discerning and searching questions when gathering information. Through evaluation procedures and with 85 ac[r]

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(b) Wave motion: Wave motion is a continuous reoccurrence of a disturbance as it travels with or without a physical medium through space e.g. the continuous disturbance we generate wh[r]

The trust concept itself is a complicated notion with different meanings depending on both participants and situations and influenced by both measurable and non-measurable factors. Ther[r]

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5. Personal management, including time management, interpersonal skills, integrity, ethics. 6. Knowledge of workplace rules and requirements (e.g., worker safety, sexual harassment). Supervisory Skill Set (Supervisors and Managers of the Income and Rent Function) Note: Front-line supervisors require the skills **sets** identified above for rent specialists as well as the following skill **sets**.

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Given the universe of set theory, V , and a set I we can define as V I the class of functions with domain I. Even though the equivalence classes modulo U are not **sets** but proper classes we can turn them into **sets** by cutting them at the least level of the cumulative hierarchy in which they are non-empty (this trick was first applied by Dana Scott and it is known as Scott’s Trick). Thus we can define V I /U as above (which is extensional because if some [f ] is cut at height

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The novelty of the proofs of Corollary 3.5 and Theorem 3.11 is applying the perspective of a one variable convex function of **measurable** operators. See [18] for more information and details on noncommutative perspectives. Moreover, one more perspective yields the following result, which is also a special case of Theorem 3.11.

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In this article we consider τ -**measurable** operators aﬃliated with a ﬁnite von Neumann algebra equipped with a normal faithful ﬁnite trace τ . By virtue of the method of Lin and Zhan [8, 10], based on the notion of generalized singular value studied by Fack and Kosaki [5], we obtain generalizations of results in [8] and [10] with regard to Anderson–Taylor type inequalities for τ -**measurable** operators case.

We look at the process from its finalized form, that is we see a set of ingredients with a partial order (or a more general relation if wanted) covered by finite **sets** of aspects describing the ingredients completely (or as good as possible) with aspect relations defining in some sense (to be clarified later) the underlying relation between ingredients which are shadow objects, i.e., satisfactory approximations of some reality. We also see how this blue-print evolved in time with temporary shadow objects, temporary **sets** of aspects and relations; this time evolution can be encoded by giving to each component in the final picture a function of the time interval for the process to the interval of numbers between 0 and 1, expressing for example the percentage of realization of the component at that moment in time. The learning process based on the shadow objects as ingredients is thus evolving in time growing in a discrete way within the final form framework step by step realizing each final component at certain (different) moments in time.

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Once into the the computer tions between variables will the and data a second factorial full experiment composite design is operating points will The time, variables development this pla[r]

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