Maruyoshi-Song flows and defect groups of $$ {\mathrm{D}}_{\mathrm{p}}^{\mathrm{b}} $$(G) theories

We study the defect groups of $$ {D}_p^b $$ D p b (G) theories using geometric engineering and BPS quivers. In the simple case when b = h∨(G), we use the BPS quivers of the theory to see that the defect group is compatible with a known Maruyoshi-Song flow. To extend to the case where b ≠ h∨(G), we use a similar Maruyoshi-Song flow to conjecture that the defect groups of $$ {D}_p^b $$ D p b (G) theories are given by those of G(b)[k] theories. In the cases of G = An, E6, E8 we cross check our result by calculating the BPS quivers of the G(b)[k] theories and looking at the cokernel of their intersection matrix.

Half-hypermultiplets and incomplete/complete resolutions in F-theory

We consider resolutions of codimension-two enhanced singularities from SO(12) to E7 and from E7 to E8 in six-dimensional F-theory, where a half-hypermultiplet arises for generic complex structures achieving them. The exceptional fibers at the enhanced point exhibit different structures depending on how the colliding 7-brane approaches the stack of gauge 7-branes, as previously observed by Morrison and Taylor in the case of the enhancement from SU(6) to E6. When the colliding brane approaches them as O(s), where s is the coordinate of the base space along the gauge 7-branes, the resolution process ends up with fewer exceptional fibers than naively expected from the Kodaira classification, with a non-Dynkin intersection matrix including half-integral intersection numbers. We confirm that the exceptional fibers at the enhanced point form extremal rays of the cone of the positive weights of the relevant pseudo-real representation, explaining why a half-hypermultiplet arises there. By altering the ordering of the singularities blown up in the process, we obtain, for both SO(12) → E7 and E7 → E8, the intersection diagram on every other row of the corresponding box graphs. We present detailed derivations of the intersection diagrams of the exceptional fibers at the singularity enhanced points by examining how an exceptional curve is lifted up on the chart arising due to the subsequent blowing-up process. When the colliding brane approaches the stack of branes as O(s2), we obtain additional conifold singularity at the enhanced point, which completes the full Dynkin diagram of the enhanced group as was found previously.

On relationship between the parameters characterizing nonlinearity and nonhomomorphy of vector spaces transformation

We find a relation between the W-intersection matrix (which characterizes the degree of “nonhomomorphy”) of a transformation and the difference distribution table and the correlation matrix (which characterize the degree nonlinearity of a transformation). An upper estimate for the dimension of a subspace invariant under almost bent functions is put forward. A formula for evaluation of the W-intersection matrix of a composition of two transformations is obtained.

Approximate answering of queries involving polyline–polyline topological relationships

In geographic information systems, pictorial query languages are visual languages which make easier the user to express queries by free-hand drawing. In this perspective, this article proposes an approach to provide approximate answers to pictorial queries that do not match with the content of the database, that is, the results are null. It addresses the polyline–polyline topological relationships and is based on an algorithm, called Approximate Answer Computation algorithm, which exploits the notions of Operator Conceptual Neighborhood graph and 16-intersection matrix. The operator conceptual neighborhood graph represents the conceptual topological neighborhood between Symbolic Graphical Objects and is used for relaxing constraints of queries. The nodes of the operator conceptual neighborhood graph are labeled with geo-operators whose semantics has been formalized. The 16-intersection matrix provides enriched query details with respect to the well-known Dimensionally Extended 9-Intersection Model proposed in the literature. A set of minimal 16-intersection matrices associated with each node of the operator conceptual neighborhood graph, upon the external space connectivity condition, is defined and the proof of its minimality is provided. The main idea behind each introduced notion is illustrated using a running example throughout this article.

Ribbon graphs and bialgebra of Lagrangian subspaces

To each ribbon graph we assign a so-called [Formula: see text]-space, which is a Lagrangian subspace in an even-dimensional vector space with the standard symplectic form. This invariant generalizes the notion of the intersection matrix of a chord diagram. Moreover, the actions of Morse perestroikas (or taking a partial dual) and Vassiliev moves on ribbon graphs are reinterpreted nicely in the language of [Formula: see text]-spaces, becoming changes of bases in this vector space. Finally, we define a bialgebra structure on the span of [Formula: see text]-spaces, which is analogous to the 4-bialgebra structure on chord diagrams.

On a class of generalized Kac–Moody algebras arising from 2-fold affinizations

We study a class of generalized Kac–Moody algebras which are fixed point subalgebras of the covering Kac–Moody algebras of some generalized intersection matrix (GIM) algebras of Slodowy.