Spherical DG-functors

For two DG-categories A and B we define the notion of a spherical Morita quasi-functor A -> B. We construct its associated autoequivalences: the twist T of D(B) and the co-twist F of D(A). We give powerful sufficiency criteria for a quasi-functor to be spherical and for the twists associated to a collection of spherical quasi-functors to braid. Using the framework of DG-enhanced triangulated categories, we translate all of the above to Fourier-Mukai transforms between the derived categories of algebraic varieties. This is a broad generalisation of the results on spherical objects in [ST01] and on spherical functors in [Ann07]. In fact, this paper replaces [Ann07], which has a fatal gap in the proof of its main theorem. Though conceptually correct, the proof was impossible to fix within the framework of triangulated categories.Comment: 53 pages; v2: An inaccuracy in the definition of homotopy action maps fixed by tensoring everything in sight with bar-complexes; several twisted complex computations correcte

Orthogonally spherical objects and spherical fibrations

We introduce a relative version of the spherical objects of Seidel and Thomas. Define an object E in the derived category D(Z x X) to be spherical over Z if the corresponding functor from D(Z) to D(X) gives rise to autoequivalences of D(Z) and D(X) in a certain natural way. Most known examples come from subschemes of X fibred over Z. This categorifies to the notion of an object of D(Z x X) orthogonal over Z. We prove that such an object is spherical over Z if and only if it has certain cohomological properties similar to those in the original definition of a spherical object. We then interpret this geometrically in the case when our objects are actual flat fibrations in X over Z.Comment: 29 pages; v2: A missing assumption reinstated in Prop. 3.7, some notation cleaned up. The final version to appear in Adv. in Mat

On adjunctions for Fourier-Mukai transforms

We show that the adjunction counits of a Fourier–Mukai transform Φ:D(X1)→D(X2) arise from maps of the kernels of the corresponding Fourier–Mukai transforms. In a very general setting of proper separable schemes of finite type over a field we write down these maps of kernels explicitly –facilitating the computation of the twist (the cone of an adjunction counit) of Φ. We also give another description of these maps, better suited to computing cones if the kernel of Φ is a pushforward from a closed subscheme Z⊂X1×X2. Moreover, we show that we can replace the condition of properness of the ambient spaces X1 and X2 by that of Z being proper over them and still have this description apply as is. This can be used, for instance, to compute spherical twists on non-proper varieties directly and in full generality

What Strategies do Learners Use to Remember the Spelling of Newly Learned Words?

Many studies suggest that a native language (L1) may influence the second language acquisition (L2). This study is interested in the possible impact of L1 orthography on the choice of spelling strategies of Chinese, Arabic, and French speaking learners. Data was collected through a short test in which participants were asked to memorize new English words. Afterwards, they reported strategies which were used in order to learn the spelling. After calculating individual and group average of employed strategies, the most commonly used among them were determined for the members of the same language group, and for all participants as a group. I also wanted to identify which language group would be the most successful in spelling orthographically challenging English words. This empirical study provides evidence that regardless of a native language, the same strategies were used most of the time by all participants. The findings related to accuracy of each language group support the influence of L1 on the spelling process in L2. Implications for ESL teachers are discussed

Artemisia argyi Leveil. et Vaniot (Asteraceae) в Києві та Криму: біологічні особливості, інтродукція, хімічний склад, можливості використання

The results of using alien plant  – Artemisia argyi Leveil. et Vaniot in Kyiv and its introduction in Crimea are represented. Artemisia argyi botanical description is given. Chemical composition of essential oil on basis of chromatographic analysis is established. The potentialies of plant using are determine.Представлено результати вивчення адвентивної рослини – Artemisia argyi Leveil. et Vaniot, знайденої в Києві, та її інтродукції в Криму. Наведено ботанічний опис Artemisia argyi. За результатами хроматографічного аналізу визначено компонентний склад ефірної олії. Встановлено можливості використання рослини