The solution of the bOU?Jdary value problem for harmonic functions leads readily to the conformal mapping of simply connected domains on the interior or a circle.. the corresponding pola[r]

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contains only those numbers whose base a expansion consists of zeros and ones, FS(Γ(a)) is of density zero (so Γ(a) is incomplete), while if α ∈ T is an irrational whose base a expansion is missing some digit, then the set Γ(a)α is nowhere dense in T (so Γ(a) is not dispersing). So it makes sense to augment the semigroup property with some information on the size of the set in question: the **sets** Γ(a, b) are larger than the **sets** Γ(a), and in general it is easier for larger **sets** to be **complete** and dispersing. In the case of the **sets** Γ(a, b), information on the size is provided by the following lemma due to Furstenberg:

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By contrast, the notion of a “dispersing” set has not appeared explicitly in the literature before. It bears some resemblance to the notion of a “Glasner set” (cf. [12, 3], and see [2] for a generalization). 1 However, the differences between these definitions are significant, and we will not discuss Glasner **sets** in this paper. Although their definitions are very different, the notions of completeness and dispersion do share some relation. Both describe some notion of “largeness” of a set of integers which measures not just the growth rate but also in some sense the arithmetical properties of the set in question. This is manifested in the following classical results about **complete** and dispersing **sets**, which are due to Birch and Furstenberg, respectively:

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As far as effective usability of the computational tool, the **complete** **sets** of data collected and the feedback provided by clinicians indicated that AVF.SIM system is easy to use and well accepted in clinical routine, with a limited additional workload for pre-operative examin- ation by the clinicians. Actually the computed prediction of patient-specific BFVs after VA maturation are ob- tained using the AVF.SIM system on the basis of demo- graphic information, clinical data and pre-operative DUS measurements, that are today all standard of care for patients in need of VA for HD treatment. Time required to fill up the pre-operative form and send data to the Simulation Center was estimated to extend the routine vascular mapping of approximately 10–15 min only. Pre-operative physical examination and DUS evaluation currently recommended by international guidelines [16] have several potential benefits, but only a system that takes into account the complex interplay of demographic and clinical factors, as well as vessel dimensions and local BFV, could really help the surgeon in identifying the best site for AVF placement, as well as in preventing very low or very high BFV, likely associated with VA complications. The establishment of effective usability of AVF.SIM in the clinical environment is a significant step forward allowing computer assisted clinical decision making on type and location of AVF, as well as for the

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of genes [34]. The distributions typically are broad but fol- low a general bell-shaped form (some are bimodal), in support of the notion of an intrinsic rate of domain evo- lution for which the mean (or median) of the respective distribution can be taken as a reasonable proxy. Domains present within multidomain proteins do not show sys- tematic differences in evolutionary rates compared to solo domains as illustrated by two examples in Figure 3. We then addressed the issue of homogenization of the rates of evolution of domains that is predicted by the MIM hypothesis to result from the fusion of domains within a multidomain protein. Figure 4 shows anecdotal evidence for two proteins, each consisting of 3 distinct domains. For one of these proteins, homogenization is obvious (Figure 4A) whereas the other one shows no obvious sign of homogenization (Figure 4B). These examples are char- acteristic of the diversity of the evolutionary regimes of domains, so that homogenization is seen in many but by no means all multidomain proteins, and some actually display the opposite trend (Additional Files 1 and 2, and see below). This striking variability notwithstanding, the results of the analysis of the **complete** **sets** of domains unequivocally reveal substantial homogenization as illus- trated by the comparison of the probability density func- tions for the difference (ratio) of the evolutionary rates for all domain combinations and for domain pairs fused within multidomain proteins. The difference in evolu- tionary rates between a pair of domains within a multid- omain protein tends to be substantially less than the difference between rates for the same pair of domains found in different proteins in both human (Figure 5A) and Arabidopsis (Figure 5B).

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coactivators and corepressors to activate and suppress target gene expression [4]. The transcriptional activities of about half of the NRs studied in humans are regulated by small lipophilic ligands whereas the other, so-called orphan, receptors await ligand identification. Because of their poten- tial for modulation by exogenous compounds and their central roles in metabolism, these receptors are extremely important targets in human disease. Additionally, NRs rep- resent possible new targets for the control of invertebrate pests in agriculture. The latter approach would be especially promising if certain classes of NRs are shown to be specific to selected invertebrate subgroups. Comparison of **complete** **sets** of NRs from evolutionarily divergent organisms should tell us more about the feasibility of these approaches as well as shed light on general phylogenetic relationships among NRs and among species.

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Ten assessment portfolios were selected for analysis on the basis that these provided the most **complete** **sets** of data identified for close analysis: rationales for their topic selection, on-going reflective commentaries in the form of learning logs and a final evaluation of the topic in terms of both their own and the pupils’ learning. Ethical approval procedures were completed and all teachers gave permission for their work to be analysed and findings published and

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A group G of order p m is said to be of maximal class if m > 2 and the nilpotence class of G is m − 1. It is well known (for example see Theorem 1.2 and Corollary 1.7 of [3]) that the 2-groups of maximal class are dihedral, semidihedral and generalised quaternion. Moreover, by [3, Theorem 1.2], if G is a 2-group of maximal class of order at least 16, then G/Z(G) is dihedral of order 1 2 | G | . A detailed examination of locally maximal product-free **sets** in 2-groups of maximal class, which among other things results in an alternative proof of Lemma 3.3, appears in [1]. However, since we need the result here we thought it would be useful to include a short proof for ease of reference.

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Denote by B (X ), C (X ) and C B (X), the totality of fuzzy **sets** which satisfy that for each α ∈ I, the α-cut of µ is nonempty bounded, nonempty closed and nonempty bounded closed in X , respectively. Next, we give the definition of the function M ∞ on F (X ) × F (X), which is induced by H M

We have that an intuitionistic fuzzy topological space can be associated with two fuzzy topological spaces and vice versa [1]. . If (X, 𝜏) is an IFTS and 𝜏₁= { μₐ / ∃ γₐ ∊ Iˣ such that (μₐ , γₐ) ∊ 𝜏 }, 𝜏₂ = { 1- γₐ / ∃ μₐ ∊ Iˣ such that (μₐ, γₐ) ∊ 𝜏},then (X, 𝜏₁) and (X, 𝜏₂) are fuzzy topological spaces. Similarly if (X, 𝜏₁) and (X, 𝜏₂) are two fuzzy topological space, 𝜏 = {(u,1- v)/ u∊ 𝜏₁, v∊ 𝜏₂ and u ⊆ v} is an intuitionistic fuzzy topology and (X, 𝜏) is an intuitionistic fuzzy topological spaces. We study some relationships connecting the closures and interiors of an intuitionistic fuzzy set in an intuitionistic fuzzy topological space and the closures and interiors of its co- ordinate fuzzy **sets** in its corresponding fuzzy topological spaces. .

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This inequality is sharp when q = 2 or when q = 3 and each part has at least two vertices. Proof. The inequality comes from substituting k = 1 into the above corollary. To see that equality holds when q = 2, the **complete** bipartite case, we first note that MOF(G) ≤ n − 1 for any non-empty graph G. This is because there is always a vertex v with in-degree at least one in such cases, and the set of all vertices other than v is a forcing set. To see that MOF(G) ≥ n − 1, let A and B be the two parts and direct all edges from A to B. Now each vertex of A is necessarily in any forcing set and nothing can be forced unless at least |B| − 1 vertices from B are included. Therefore, MOF(G) = n − 1 = n − q + 1, as claimed.

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they are isomorphic to the fuzzification of the complete lattice of crisp objects subsets, closed sets, subalgebras, etc... FUZZY SUBSETS Theorem 5.1.[r]

In traditional game theory, ﬁxed point theory in topological spaces or metric spaces has been an essential tool for the proof of the existence of Nash equilibria of noncooperative games, in which the payoﬀ functions of the players take real values (see [–]). Recently, the concept of nonmonetized noncooperative games has been introduced where the pay- oﬀ functions of the players take values in ordered **sets**, on which the topological structure may not be equipped. The existence of generalized and extended Nash equilibria of non- monetized noncooperative games has been studied by applying ﬁxed point theorems to ordered **sets**. These games are also named generalized games by some authors (see [– ]). Naturally, ﬁxed point theory on ordered **sets** has revealed its crucial importance in this new subject in game theory.

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• The black horizontal lines above and below the box (which are in most cases connected to the box via dashed vertical lines) represent the maximimum and minimum value, respectively, excluding outliers. These figures provide a wealth of information concerning the variability and identifiability of parameter estimates. Individually, the box plots within each figure can be used to determine a median parameter estimate and to visualize the variation (spread) in parameter estimates for a given donor and cell type when multiple five-day data **sets** are considered. Each box plot can also be used to conclude whether or not a particular parameter is likely to be identifiable for any particular donor and cell type. For example, Figures 16 and 17 reveal extremely large spreads in many of the box plots, indicating that the parameters E

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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.

In this paper, we study the undirected graph Γ 𝐼𝐼 (𝑀𝑀) of Gamma near rings for any completely reflexive ideal I of M. Throughout this paper M stands for a non zero Gamma near -ring with zero element and I is a completely reflexive ideal of M. For distinct vertices x and y of a Graph G, let d(x, y) be the length of the shortest path from x to y. The diameter of a connected graph is the supremum of the distances between vertices. For any graph G, the girth of G is the length of a shortest cycle in G and is denoted by gr(G). If G has no cycle, we define the girth of G to be infinite. A clique of a graph is a maximal **complete** subgraph and the number of graph vertices in the largest clique or graph G, denoted by ω(G) is called the clique number of G. A graph G is bipartite with vertex classes 𝑉𝑉 1 , 𝑉𝑉 2 if the set of all vertices of G is 𝑉𝑉 1 ∪ 𝑉𝑉 2 , 𝑉𝑉 1 ∩ 𝑉𝑉 2 = ∅ , and edge of G joins a vertex from 𝑉𝑉 1 to a vertex of 𝑉𝑉 2 .

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Generating Sets of the Complete Semigroups of Binary Relations Defined by Semilattices of the Class Σ 2 X ,4 Bariş Albayrak1, Omar Givradze2, Guladi Partenadze2 Department of Banking a[r]

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As we have seen, the difficulty in finding Brouwerian counterexamples to The- orems 1 and 2 arises in finding abelian groups which are ‘nice’ enough to be **complete**, but which are not periodic or noncompact, respectively. In fact, our one successful Brouwerian counterexample concerning the generalisation of COP required us to assume a non-constructive principle (LLPO) in order to establish the completeness of the abelian group under consideration. Given that being **complete** appears to impose so much structure on our **complete** abelian groups, it seems likely that Theorems 1 and 2 admit constructive proofs. On the other hand, it is interesting to note that there is some similarity between the extra hypothesis assumed in Theorems 18 and 36. In Theorem 18 we are concerned with showing that G is ‘small’, in that it is periodic and therefore the image, under a uniformly continuous mapping, of a compact space. Here we assume that θ(0, ∞) is open, in which case θ(−∞, 0) is also open, so S 1 is

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The aim of this paper is to show the existence of Nash equilibria in mixed strategies for a class of two-player discontinuous games with **complete** in- formation in which the strategy **sets** are non-compact. The problem of the existence of equilibria in discontinuous games has been already addressed by Dasgupta and Maskin (1986a, 1986b), Maskin (1986), Simon (1987), Simon and Zame (1990), and, more recently by Reny (1999). Unlike the existing pa- pers, we construct Nash equilibria in mixed strategies when players’ strategy **sets** coincides with the set of real numbers.

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A nonempty subset 𝑆 of 𝑉(𝐺), where 𝐺 is any graph, is a clique in 𝐺 if the graph 𝐺 [𝑆] = 〈 𝑆 〉 induced by 𝑆 is **complete**. A clique 𝑆 in 𝐺 is a \textit{clique dominating} set if it is a dominating set. It is a secure clique dominating set in 𝐺 if for every 𝑢 ∈ 𝑉(𝐺) ∖ 𝑆 , there exists 𝑣 ∈ 𝑆 ∩ 𝑁 𝐺 (𝑢) such that (𝑆 ∖ { 𝑣 }) ∪ { 𝑢 } is a clique dominating set in 𝐺 . The secure clique domination number of 𝐺 , denoted by 𝛾 𝑠𝑐𝑙 (𝐺), is the minimum cardinality of a secure clique dominating set of 𝐺.

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