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关键词： 18M15 (primary)   05E10   57K14 (secondary)  

摘要： We explicitly compute a monoidal subcategory of the monoidal center of Deligne's interpolation category Re_p(St), for t not necessarily a natural number, and we show that this subcategory is a ribbon category. For t=n, a natural number, there exists a functor onto the braided monoidal category of modules over the Drinfeld double of Sn which is essentially surjective and full. Hence the new ribbon categories interpolate the categories of crossed modules over the symmetric groups. As an application, we obtain invariants of framed ribbon links which are polynomials in the interpolating variable t. These polynomials interpolate untwisted Dijkgraaf-Witten invariants of the symmetric groups.

关键词： smooth manifold   principal bundle   covering   fundamental group  

摘要： We investigate the category of regular coverings of a given smooth principal bundle. A covering map of one principal bundle to another principal bundle is understood as a morphism in the category of principal bundles consisting of covering maps of their structural groups, total spaces and bases. The main results are: the construction of an invariant of a regular covering, which is a subsequence of the homotopy sequence of the base bundle;the existence theorem of a covering with a given invariant;criteria for morphisms and isomorphisms;a theorem on the automorphism group of a given covering.

关键词： Monoidal categories   Brauer category   Schauenburg tensor product   Topological quantum field theories   Semirings   Fold maps   Kervaire spheres  

摘要： The Brauer category is a symmetric strict monoidal category that arises as a (horizontal) categorification of the Brauer algebras in the context of Banagl's framework of positive topological field theories (TFTs). We introduce the chromatic Brauer category as an enrichment of the Brauer category in which the morphisms are component-wise labeled. Linear representations of the (chromatic) Brauer category are symmetric strict monoidal functors into the category of real vector spaces and linear maps equipped with the Schauenburg tensor product. We study representation theory of the (chromatic) Brauer category, and classify all its faithful linear representations. As an application, we use indices of fold lines to construct a refinement of Banagl's concrete positive TFT based on fold maps into the plane.

摘要： An important goal in studying the relations between unitary VOAs and conformal nets is to prove the equivalence of their ribbon categories. In this article, we prove this conjecture for many familiar examples. Our main idea is to construct new structures associated to conformal nets: the categorical extensions. Let V be a strongly-local unitary regular VOA of CFT type, and assume that all V-modules are unitarizable. Then V is associated with a conformal net A(V) by Carpi et al. (From vertex operator algebras to conformal nets and back, Vol. 254, No. 1213, Memoirs of the American Mathematical Society, 2018). Let Rep(u)(V) and Rep(ss)(A(V)) be the braided tensor categories of unitary V-modules and semisimple A(V)-modules respectively. We show that if one can find enough intertwining operators of V satisfying the strong intertwining property and the strong braiding property, then any unitary V-module W-i can be integrated to an A(V)-module H-i, and the functor F : Rep(u)(V) -> Rep(ss)(A(V)), W-i bar right arrow H-i induces an equivalence of the ribbon categories Rep(u)(V) ->similar or equal to F(Rep(u)(V)). This, in particular, shows that F(Rep(u)(V)) is a modular tensor category. We apply the above result to all unitary c < 1 Virasoro VOAs (minimal models), many unitary affine VOAs (WZW models), and all even lattice VOAs. In the case of Virasoro VOAs and affine VOAs, one further knows that F(Rep(u)(V)) = Rep(ss)(A(V)). So we've proved the equivalence of the unitary modular tensor categories Rep(u)(V) approximate to Rep(ss) (A(V)). In the case of lattice VOAs, besides the equivalence of Rep(u)(V) and F(Rep(u)(V)), we also prove the strong locality of V and the strong integrability of all (unitary) V-modules. This solves a conjecture in Carpi et al. (From vertex operator algebras to conformal nets and back, Vol. 254, No. 1213, Memoirs of the American Mathematical Society, 2018).

关键词： Price drop   Product quality   Brand reputation   EFA   MANOVA  

摘要： A price drop is a very common promotional tool used by brands and marketers to stimulate impulse buying as well as to improve stagnant sales. This study is primarily directed to study the effect of the price drop on the customer's perceived evaluation across selected product categories. For this study, the top three product categories based on the category wise sales data revealed by various online as well as offline shopping players during price drop sales have been selected, namely apparel, footwear, and electronics, as these product categories were most preferably shopped by customers and witnessed frequent price drop during festival seasons. For collecting data in National Capital Region (NCR), using purposive sampling 250 shoppers were surveyed who shopped for these product categories at Big Bazaar, Pantaloons, Spencer, Reliance trends, Amazon, Flipkart during discount days. Exploratory factor analysis (EFA) has been run to identify the factors of customers' perceived evaluation in case of a price drop. Further Multivariate Analysis Of Variance (MANOVA) has been used to check the effect of a price drop on identified perceived evaluation across selected product categories. This study explored that in case of price drop customers become concerned about the product quality, value, brand reputation, and innovativeness. Based on the mean value, the study found that customers were more concerned about perceived quality in case of price drop followed by perceived innovativeness, perceived value, and perceived brand reputation. The research further demonstrated that product categories were significantly related to perceived evaluators in case of a price drop.

关键词： amputation   claudication   critical limb ischemia   chronic limb-threatening ischemia   endovascular therapy   mortality   peripheral artery disease   revascularization   Rutherford category  

摘要： Purpose: To report the 3-year results of the LIBERTY 360 study, which investigated outcomes of endovascular treatment of symptomatic peripheral artery disease (PAD). Materials and Methods: The LIBERTY trial (*** NCT01855412) was a prospective, observational, core laboratory-assessed, multicenter study of endovascular interventions enrolling >1200 participants treated at 51 sites. Data from 1189 patients were stratified according to Rutherford category (RC) and analyzed [RC 2-3 (n=500), RC 4-5 (n=589), and RC 6 (n=100)]. The primary outcomes were major amputation of the target limb and all-cause death;secondary outcomes were target vessel revascularization (TVR) or target lesion revascularization (TLR);major adverse events (MAEs;death within 30 days, TVR or TLR, and major amputation);death or major amputation combined;and change in self-reported quality of life (QoL) measures (VascuQol-25). The Kaplan-Meier (KM) method was employed to estimate the outcomes;estimates are presented with the 95% confidence intervals (CI). Predictors of 3-year MAE, death, TVR, and major amputation were analyzed using Cox proportional hazard regression modeling. Results: The 36-month KM survival rates were 86.0% in RC 2-3, 79.8% in RC 4-5, and 62.0% in RC 6 groups. The KM estimates of freedom from major amputation at 36 months were 98.5% in RC 2-3, 94.0% in RC 4-5, and 79.9% in RC 6. The 36-month KM estimates for freedom from TVR/TLR were 71.1% in RC 2-3, 64.2% in RC 4-5 and 61.9% in RC 6 groups. Patients with claudication at baseline were at lower risk for MAEs compared with RC 4-5 and RC 6 patients during the 36-month follow-up. Vascular QoL improved from baseline and persisted up to 36 months in all patients. Conclusion: Endovascular therapy is a viable treatment option for patients with symptomatic PAD, with sustained improved quality of life in both claudicants and patients with chronic limb-threatening ischemia. These results provide important point estimate

关键词： Vertex tensor categories   Virasoro algebra   Non C-2 vertex tensor category   Universal Virasoro vertex algebras  

摘要： We show that there is a braided tensor category structure on the category of C-1-cofinite modules for the (universal or simple) Virasoro vertex operator algebras of arbitrary central charge. In the generic case of central charge c = 13 - 6(t + t(-1)), with t is not an element of Q, we prove semisimplicity, rigidity and non-degeneracy and also compute the fusion rules of this tensor category. (C) 2021 Elsevier Inc. All rights reserved.

关键词： Relative category   Morse decomposition   Semiflows   Wazewski pair  

摘要： We employ the relative category to develop relations between the Wazewski pair (N, E) and the Morse decomposition of the maximal invariant set in (N\E) over bar for infinite-dimensional dynamical systems. Via these relations, we can detect connecting trajectories between Morse sets and obtain a dynamical-system version of critical point theorem with relative category. (c) 2021 Elsevier B.V. All rights reserved.

关键词： Simplicial complex   Δ   -complex   Poset   Small category   Simple homotopy   Nerve theorem  

摘要： We study a combinatorial homotopy theory for small categories without a loop (loopfree categories) which is closely related to Whitehead's simple homotopy theory for regular CW-complexes with triangular cells. Quillen's theorem A and the nerve theorem for loopfree categories are considered from the viewpoint of the simple homotopy theory. Moreover, we extend the classical nerve theorem and discuss an application to the topological data analysis for data-sets lying in a non-convex field. (c) 2021 Elsevier B.V. All rights reserved.