In this paper, we construct invariants of 3-manifolds 'à la Reshetikhin-Turaev' in the setting of non-semi-simple ribbon tensor categories. We give concrete examples of such categories that lead to a family of 3-manifold invariants indexed by the integers. We prove that this family of invariants has several notable features, including: they can be computed via a set of axioms, they distinguish homotopically equivalent manifolds that the standard Witten-Reshetikhin-Turaev invariants do not and they allow the statement of a version of the Volume Conjecture and a proof of this conjecture for an infinite class of links. n i=1 V i ⊕ W where V i is a simpleŪ -module with non-zero quantum dimension and W is aŪ -module with zero quantum dimension. By quotienting the category of finite-dimensionalŪ -modules byŪ -modules with zero quantum dimension, one obtains a modular category D. Loosely speaking, a modular category is a semi-simple ribbon category with a finite number of isomorphism classes of simple objects satisfying some axioms. Let M be a manifold obtained by surgery on L. If the ith component of L is labeled by a

It has been known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for instance, every 3-manifold has a surgery diagram. There are several proofs of this fact, but little attention has been paid to the complexity of the 4-manifold produced. Given a 3-manifold M 3 of complexity n, we construct a 4-manifold bounded by M of complexity O(n 2 ), where the 'complexity' of a piecewise-linear manifold is the minimum number of n-simplices in a triangulation.The proof goes through the notion of 'shadow complexity' of a 3-manifold M . A shadow of M is a well-behaved 2-dimensional spine of a 4-manifold bounded by M . We further prove that, for a manifold M satisfying the geometrization conjecture with Gromov norm G and shadow complexity S, we have c1 G S c2 G 2 , for suitable constants c1 , c2 . In particular, the manifolds with shadow complexity 0 are the graph manifolds.In addition, we give an O(n 4 ) bound for the complexity of a spin 4-manifold bounding a given spin 3-manifold. We also show that every stable map from a 3-manifold M with Gromov norm G to R 2 has at least G/10 crossing singularities, and if M is hyperbolic there is a map with at most c3 G 2 crossing singularities.To make this question more precise, let us make some definitions. Definition 1.2. A ∆-complex is the quotient of a disjoint union of simplices by identifications of their faces. (See [13, Section 2.1] for a complete definition. These are also semi-simplicial complexes in the sense of Eilenberg and Zilber [6].) A ∆-triangulation is a ∆-complex whose underlying topological space is a manifold. Definition 1.3. The complexity of a piecewise-linear oriented n-manifold M n is the minimal number of n-simplices in a ∆-triangulation of M , C(M n ) = min Triang. T of M no. of n-simplices in T . ( 1 .1) Remark 1.4. Since the second barycentric subdivision of a ∆-triangulation is an ordinary simplicial triangulation, C(M ) would change only by at most a constant factor if we insisted on simplicial triangulations in Definition 1.3.Definition 1.5. The 3-dimensional boundary complexity function G 3 (k) is the minimal complexity such that every 3-manifold of complexity at most k is bounded by a 4-manifold of complexity at most G 3 (k).We can think of G 3 (k) as a kind of topological isoperimetric inequality. We can now give a concrete version of our original Question 1.1.Question 1.6. What is the asymptotic growth rate of G 3 ?The first main result of this paper is that G 3 (k) = O(k 2 ). More precisely, we have the following theorem, which appears in Section 5.Theorem 5.2. If a 3-manifold M has a ∆-triangulation with t tetrahedra, then there exists a 4-manifold W such that ∂W = M and W has a ∆-triangulation with O(t 2 ) simplices. Moreover, W has 'bounded geometry'. That is, there exists an integer c (not depending on M and W ) such that each vertex of the triangulation of W is contained in fewer than c simplices.

Abstract. We construct and study a new family of TQFTs based on nilpotent highest weight representations of quantum sl(2) at a root of unity indexed by generic complex numbers. This extends to cobordisms the non-semi-simple invariants defined in [12] including the Kashaev invariant of links. Here the modular category framework does not apply and we use the "universal construction". Our TQFT provides a monoidal functor from a category of surfaces and their cobordisms into the category of graded finite dimensional vector spaces and their degree 0-morphisms and depends on the choice of a root of unity of order 2r. The functor is always symmetric monoidal but for even values of r the braiding on GrVect has to be the super-symmetric one, thus our TQFT may be considered as a super-TQFT. In the special case r = 2 our construction yields a TQFT for a canonical normalization of Reidemeister torsion and we re-prove the classification of Lens spaces via the non-semi-simple quantum invariants defined in [12]. We prove that the representations of mapping class groups and Torelli groups resulting from our constructions are potentially more sensitive than those obtained from the standard Reshetikhin-Turaev functors; in particular we prove that the action of the bounding pairs generators of the Torelli group has always infinite order.

No abstract

In the literature, many applications of Digital Twin methodologies in the manufacturing, construction and oil and gas sectors have been proposed, but there is still no reference model specifically developed for risk control and prevention. In this context, this work develops a Digital Twin reference model in order to define conceptual guidelines to support the implementation of Digital Twin for risk prediction and prevention. The reference model proposed in this paper is made up of four main layers (Process industry physical space, Communication system, Digital Twin and User space), while the implementation steps of the reference model have been divided into five phases (Development of the risk assessment plan, Development of the communication and control system, Development of Digital Twin tools, Tools integration in a Digital Twin perspective and models and Platform validation). During the design and implementation phases of a Digital Twin, different criticalities must be taken into consideration concerning the need for deterministic transactions, a large number of pervasive devices, and standardization issues. Practical implications of the proposed reference model regard the possibility to detect, identify and develop corrective actions that can affect the safety of operators, the reduction of maintenance and operating costs, and more general improvements of the company business by intervening both in strictly technological and organizational terms.

Background: Ghrelin, a peptide mainly derived from the stomach, plays a pivotal role in the regulation of food intake, energy metabolism, and storage, as well as in insulin sensitivity. Ghrelin circulates in acylated (A-Ghr) and nonacylated (NA-Ghr) forms, and their potential differential associations with insulin resistance (IR) in childhood obesity remain undefined. Objective: We investigated the associations of ghrelin forms with IR in normal weight and obese children and the impact of metabolic syndrome (MS) on their plasma values. Design: A total of 210 children in four subgroups of normal weight/obese children with and without components of MS were studied. Fasting blood glucose, insulin, lipid profile, and acylated and total ghrelin were examined. IR was determined by a homeostasis model assessment (HOMA) of IR. Results: In the entire population, plasma insulin and HOMA-IR were associated negatively with T-Ghr and NA-Ghr, but positively with the ratio of A/NA-Ghr after adjustment for age, gender, and Tanner stage. Obese metabolically abnormal children had lower T-Ghr and NA-Ghr, but comparable A-Ghr and a higher A/NA-Ghr ratio than obese metabolically normal subjects. Compared with lean healthy children, lean metabolically abnormal subjects had higher A-Ghr and the A/NA-Ghr ratio, but comparable T-Ghr and NA-Ghr. A multiple regression analysis showed that A-Ghr and the A/NA-Ghr ratios were positively associated with HOMA-IR, independent of age, gender, Tanner stage, and body mass index (or waist circumference) and other components of MS. Conclusions: A-Ghr excess may negatively modulate insulin action in obese and nonobese children, and may contribute to the association of IR and MS.

In this paper we consider the representation theory of a non-standard quantization of sl(2). This paper contains several results which have applications in quantum topology, including the classification of projective indecomposable modules and a description of morphisms between them. In the process of proving these results the paper acts as a survey of the known representation theory associated to this non-standard quantization of sl(2). The results of this paper are used extensively in [4] to study Topological Quantum Field Theory (TQFT) and have connections with Conformal Field Theory (CFT).

18316j -symbols, hyperbolic structures and the volume conjecture FRANCESCO COSTANTINOWe compute the asymptotical growth rate of a large family of U q .sl 2 / 6j -symbols and we interpret our results in geometric terms by relating them to volumes of hyperbolic truncated tetrahedra. We address a question which is strictly related with S Gukov's generalized volume conjecture and deals with the case of hyperbolic links in connected sums of S 2 S 1 . We answer this question for the infinite family of fundamental shadow links.