r such factors, allowing us to switch the order of a(z), b(w) by their locality. The terms with 0 ≤ s ≤ r have (z − u) appearing to a power of at least r, which allows us to move c(u) through a(z) while also still having (w − u) to the rth power, so that we can move c(u) through b(w). Similarly, on the right hand side, the terms with r < s ≤ 2r will vanish, **and** the other terms give us the same expression as on the left hand side. This establishes (3.4) **and** the lemma.

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bracket (2.22) **and** the quadratic part of the action (3.1) expanded around a constant back- ground suggests that two-point correlation functions of suitable boundary **vertex** operators yield the extension of Kontsevich’s star product formula to higher **forms** on a manifestly covariant as well as explicit form, at least in the case of R n target space topology. Moreover, assuming the supercurrent to be anomaly free we expect its charge to be a deformation of the de Rham differential into a nilpotent **operator** that is compatible with the star product. Clearly, the first steps in this direction are to reproduce the higher-form generalized Poisson bracket (2.22) at order ~ **and** then verify the bi-differential **operator** found in [11, 12] at order ~ 2 , which we leave for separate considerations. 5 More precisely, we propose that the BRST cohomology of the model contains a ring generated by the constant modes of (φ α , θ α ) that realizes the star product deformation of the space Ω(N ) of differential **forms**

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of the space of conformal blocks with chiral insertion of type V i at the ith marked point of Σ g , n .
In the particular case that the category C is ﬁnitely semisimple, the structure of a **modular** functor is reasonably well understood. Speciﬁcally, precise conditions are known under which the representa- tion category of a **vertex** algebra V is a **modular** tensor category. In this case the Reshetikhin–Turaev construction allows one to obtain a **modular** functor just on the basis of C as an abstract category. In a remarkable development, Lyubashenko **and** others (see [KL] **and** references cited there) have ex- tended many aspects of this story to a larger class of monoidal categories that are not necessarily semisimple any longer. In particular, given an abstract monoidal category with adequate additional properties, one can still construct representations of mapping class groups.

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In section 4, following [2], we introduce functors relating the module categories of A **and** A/G when G acts freely on A. The main results in these notes (section 5) relate the representation types of A **and** A/G (which was the original purpose of the introduction of Galois coverings). Indeed, given a Galois covering F : A → A/G = B with B a finite dimensional k-algebra, then B is representation-finite if **and** only if A is locally representation-finite (that is, for each i ∈ A 0 , there are only finitely

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The breakthrough in the p-adic representation theory of p-adic groups was made in papers by P. Schneider **and** J. Teitelbaum. Motivated by examples of p-adic representations, which they found in their study of cohomologies of Drinfeld upper-half planes, they studied locally analytic p-adic groups **and** their p-adic representations in general. They studied categories of continuous representations of p-adic groups in Banach spaces over K (the orbit maps are continuous), locally-analytic representations (orbit maps are locally-analytic) of p-adic groups in topological Frechet K-vector spaces of compact type (i.e. inductive limits of p-adic Banach spaces, endowed with inductive limit topology) **and** relations between these two types of representations. They singled out “good” categories of representations of both types, which they called admissible representations.

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We comply with the provisions in the directives of the European Parlament **and** from the Council 2006/42/CE, on 17th May 2006, regarding the approach of several legislations of the member states on machinery **and** 2004/108/CE, on 15th December 2004, regarding the approach of several legislations of the member states about electromagnetic compatibility, **and** it has been manufactured in accordance with the following norms: - UNE-EN ISO 12100 - 1:2004. Security on machines. Basic concepts, main principles for its design. Part 1: Basic terminology, methodology.

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1 . So far we constructed a bialgebra structure on the group algebra kG. Truth is, the construction applies to weaker structures such as monoids. Consider A = k N as a bialgebra **and** suppose that there exists an antipode over A. We have, by definition of the antipode, that S ( n ) + N n = 0 for each n, this means that S ( n ) = − n which would lead to a contradiction since − n is not in A (We emphasize the fact that “ + N ” is the addition operation of the natural numbers, different from “ + ” the sum opperation in A). Lemma 1.1.16. Let A be a graded connected k-bialgebra with decomposition A = M

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Second, the introduction of a quorum might either increase or decrease the number of attendees in equilibrium, depending on whether the second meeting is ever reached. More speci cally, if the quorum requirement is higher than the number of members who would attend if there was no quorum, the decision is taken in the rst meeting when individuals are harmed \enough" by delaying the decision. In that case, the quorum is binding, **and** its introduction yields to an increase in the equilibrium number of attendees with respect to the no-quorum case. Conversely, if the second meeting is ever reached, **and** provided that individ- uals value strictly less a decision that has been postponed, the introduction of a quorum has the opposite e ect of lowering the number of attendees in equilibrium. Third, given that we focus on symmetric individuals' favorite policies, intro- ducing a quorum has in most cases no e ect on the chosen policy, no matter whether the second meeting is reached or not. In terms of welfare, it turns out that introducing a quorum, while having no direct policy e ects, always yields to a loss of aggregate welfare, even in the cases for which less individuals attend the meeting as compared to the no-quorum equilibrium.

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(5) Behrend’s interpretation of Donaldson-Thomas invariants as weighted Euler characteristics [2].
The credit for the development of motivic Hall **algebras** as a tool for studying moduli spaces of sheaves on Calabi-Yau threefolds is due jointly to Joyce **and** to Kontsevich **and** Soibelman. Joyce introduced motivic Hall **algebras** in a long series of papers [17, 18, 19, 20, 21, 22]. He used this framework to define generalizations of the naive Donaldson-Thomas invariants considered above, which apply to moduli stacks containing strictly semistable sheaves. He also worked out the wall-crossing formula for these invariants **and** proved a very deep no-poles theorem. Kontsevich **and** Soibelman [27] constructed an alternative theory which incorporates motivic vanishing cycles, **and** therefore applies to genuine DT invaraints **and** motivic versions thereof. They also produced a more conceptual statement of the wall-crossing formula. Some of their work was conjectural **and** is still being developed today. Joyce **and** Song [24] later showed how to directly incorporate the Behrend function into Joyce’s framework, **and** so obtain rigorous results on DT invariants.

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23 Formalmente, la abolición del instituto de la esclavitud se produce por parte de las grandes potencias coloniales a lo largo del siglo XIX: Inglaterra con una ley de 1833 que entró en vigor en 1838, Francia en 1848 con un Decreto del gobierno revolucionario, los Estados Unidos en 1865, mientras España y Portugal (países donde non se había desarrollado un movimiento abolicionista consistente) fueron los últimos Estados europeos en declarar ile- gal la esclavitud, en 1876 y en 1878, respectivamente. Cfr., por último, J. ALLAIN, “125 años de abolicíon: el derecho de la esclavitud y la explotación humana”, en e. PÉREZ ALONSO (dir.), El derecho ante las formas contemporáneas de esclavitud, cit., pp. 147-182. Véase también S. DRESCHER, Abolition: A History of Slavery **and** Antislavery, Cambridge University Press, Cambridge, 2009.

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We have generalised the notions of interacting Frobenius **algebras** [12, 16] **and** interacting Hopf **algebras** [7] to the non-commutative case, **and** in the process shown that they are rather common structures. This work could be viewed as an extension of classical results showing that concrete Hopf **algebras** over finite dimensional vector spaces are also Frobenius **algebras** [26]. Another perspective is that we make precise how much ambient symmetry is required to obtain a Hopf- Frobenius algebra. The original setting of interacting Frobenius **algebras** [12] was a †-compact category, which provides a lot of duality on top of the commutative **algebras** themselves. We show that none of this structure is necessary: all that is required is one-sided half-dual for the carrier object. The major question that remains is to pin down exactly when the Frobenius condition holds; as Lemma 3.12 shows, this is tightly related to the existence of integrals. Compact closure

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Proof. Suppose f : X → X 0 is an isomorphism. Then X e **and** X e 0 are isomorphic. Conversely, suppose |f| : X e → X e 0 is an isomorphism. |f | : X e → X e 0 lifts to the universal covers, giving an isomorphism f e : C → C which fixes the origin. By a standard result from one-variable complex analysis, f e is multiplication by a nonzero scalar λ. If N = 1, define the super Lie algebra morphism ϕ : g → g by C 7→ λC 0 , Z 7→ √ λZ 0 . Clearly ϕ| g 0 = d|f |, so F := (|f |, ϕ) is an isomorphism of

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same referent that has been tracked **and** maintained throughout all of Manuel’s turns since its initial introduction through the ethnonym peechulla. In this way Humberto confirms his uptake of the pronominal frame **and** by extension the social validity of offering an ethnonym to stand for a reference to a single individual by way of his social category. At a very high level of granularity, it confirms the status of the category as a social phenomenon. The conversation between Manuel **and** Humberto continued for several more minutes, all concerning the same referent but never using any other recognitional terms for him, only third person pronouns **and**, more often, predicates unmarked for person marking of any kind. In Cha’palaa discourse more generally reference is highly underspecified at the clause or sentence level **and** relies heavily on discourse structure for disambiguation, notably more so than languages with obligatory person marking. Once a referential framework has been established between speakers, that framework is present to be exploited by speakers for disambiguating their underspecified utterances (until they are modified by the introduction of new referents or by the switch reference system of the grammar). Understanding Manuel **and** Humberto’s conversation in those terms helps to show how Manuel successfully drew on the intersubjective social reality of race to acheive person reference in interaction. Picking up at a later part of the conversation we can obvserve that the same framework has been maintained throughout the duration of the conversation, in which all clauses reference the same Black man under discussion. Throughout much of the conversation not even third person pronouns (ya) are required for tracking co-reference, **and** person is grammatically unmarked in all but one clause below:

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c) Lipoatrofia. La diabetes lipoatrófica frecuentemente se asocia a hipertricosis. La forma congénita autosó- mica recesiva se denomina síndrome de Berardine- lli-Seip. Las manifestaci[r]

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For countries that have accomplished a socialist revolution, a positive response to this question is clearly of decisive importance. In particular, it implies that the importing or "reproducing" (or even "perfecting") of the proven techniques of the most industrialized capitalist countries can create an obstacle to the development of the productive forces, since these forces have characteristics that call for new relations of production **and** enable these relations to really open up a new stage of history by revolutionizing the productive forces. It does seem that, at the level of political technique, what distinguishes Soviet from Chinese industrialization (which is achieved under the slogan: autonomy, independence, **and** development through one's own resources) is a recognition of the necessity for not taking capitalist technique as a model, while at the same time deriving the maximum of what is utilizable from it in the construction of socialism. This is expressed in an immense economy in the requirements for accumulation, thanks to which the industrialization of China can be achieved without pressure being exerted on the standard of living of the peasant masses; on the contrary, this standard has been greatly improved, **and** is progressing steadily. In contrast, in the Soviet Union the technical line followed (which was determined by a set of ideological **and** political conditions) has led to the realization of an onerous "primitive socialist accumulation," whose economic **and** political consequences have been so considerable that, in the final

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The purpose ofthis section is to prove that ifthe lines €0 , ..• , f 3 are quadratically dependent, then: a the quadratic form Q uhao; zero asan cigenvalue if and only ifthe four points [r]

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From the perspective of economic sociology, look- ing back to 40,000 years B.C., Wood (2000) has shown that the developed nations are in the sixth wave (from 1975 to 2010), correspond[r]

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A ring has IBN if any two bases of a finitely generated free module have the same number of elements. Fields, division rings, commutative rings, Noetherian rings all have IBN. A classical example of a ring without IBN is the algebra of endomorphisms of a countably infinite dimensional vector space. The free module of rank 1 over this ring has bases of n elements for any positive integer n. In the early 60s William Leavitt asked **and** then answered this question: Given any positive integers m < n is there a ring R having a free module with a basis of m elements **and** another basis with n elements but no bases with k elements if k < n **and** not equal to m? Such a ring is said to be non-IBN of type (m, n).

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mined by just two values: its lower **and** upper bound. The aim of this work is to develop a logic that has intervals as truth values. The intended semantics are residuated lattices on the set of closed subintervals of the unit interval. We call this set the triangularization of the unit interval. A particular subset of this triangularization is its so-called diagonal, consisting of those intervals for which the lower **and** upper bound coincide. These intervals are called exact intervals **and** rep- resent truth values of propositions about which the knowledge is complete. Intuitively, the truth values of formulas constructed with these propositions should be exact intervals as well (because in these cases, the situation is similar to working with formulas in fuzzy logics). The semantics of so-called interval-valued fuzzy logics have already been examined by different authors. Especially interval-valued triangular norms, triangular conorms **and** implicators have received ample atten- tion. Most of these authors [1, 7, 26, 37, 44] only consider interval-valued operations that map the diagonal on the diagonal, although the most general definitions of triangular norms, triangu- lar conorms **and** implicators allow other operations as well [15, 21, 19, 48]. Generally speaking, interval-valued operations satisfy not as many properties as operations on the unit interval. For example, standard interval-valued residuated lattices can never satisfy prelinearity [16]. A lot of other properties can hold though. There are even interval-valued implicators that satisfy all the Smets-Magrez axioms [17, 70].

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