Balanced metrics on Cartan and Cartan–Hartogs domains

This paper consists of two results dealing with balanced metrics (in Donaldson terminology) on noncompact complex manifolds. In the first one we describe all balanced metrics on Cartan domains. In the second one we show that the only Cartan–Hartogs domain which admits a balanced metric is the complex hyperbolic space. By combining these results with those obtained in Loi and Zedda (Mathematische Annalen, 2011, to appear) we also provide the first example of complete, Kähler-Einstein and projectively induced metric g such that α g is not balanced for all α > 0.     

Maurer–Cartan Moduli and Theorems of Riemann–Hilbert Type

Applied Categorical Structures - We study Maurer–Cartan moduli spaces of dg algebras and associated dg categories and show that, while not quasi-isomorphism invariants, they are invariants of...     

Frobenius–Ehresmann structures and Cartan geometries in positive characteristic

The aim of the present paper is to lay the foundation for a theory of Ehresmann structures in positive characteristic, generalizing the Frobenius-projective and Frobenius-affine structures defined in the previous work. This theory deals with atlases of étale coordinate charts on varieties modeled on homogeneous spaces of algebraic groups, which we call Frobenius–Ehresmann structures. These structures are compared with Cartan geometries in positive characteristic, as well as with higher-dimensional generalizations of dormant indigenous bundles. In particular, we investigate the conditions under which these geometric structures are equivalent to each other. Also, we consider the classification problem of Frobenius–Ehresmann structures on algebraic curves. The latter half of the present paper discusses the deformation theory of indigenous bundles in the algebraic setting. The tangent and obstruction spaces of various deformation functors are computed in terms of the hypercohomology groups of certain complexes. As a consequence, we formulate and prove the Ehresmann–Weil–Thurston principle for Frobenius–Ehresmann structures. This fact asserts that deformations of a variety equipped with a Frobenius–Ehresmann structure are completely determined by their monodromy crystals.     

Cartan–Laptev method in the theory of multidimensional three-webs

We show how the Cartan–Laptev method that generalizes Elie Cartan’s method of external forms and moving frames is applied to the study of closed G-structures defined by multidimensional three-webs formed on a C ^s -smooth manifold of dimension 2r, r ≥ 1, s ≥ 3, by a triple of foliations of codimension r. We say that a tensor T belonging to a differential-geometric object of order s of a three-web W is closed if it can be expressed in terms of components of objects of lower order s. We find all closed tensors of a three-web and the geometric sense of one of relations connecting three-web tensors. We also point out some sufficient conditions for the web to have a closed G-structure. It follows from our results that the G-structure associated with a hexagonal three-web W is a closed G-structure of class 4. It is proved that basic tensors of a three-web W belonging to a differential-geometric object of order s of the web can be expressed in terms of an s-jet of the canonical expansion of its coordinate loop, and conversely. This implies that the canonical expansion of every coordinate loop of a three-web W with closed G-structure of class s is completely defined by an s-jet of this expansion. We also consider webs with one-digit identities of kth order in their coordinate loops and find the conditions for these webs to have the closed G-structure.     

Remarks on regularity and uniqueness of the Dirac–Klein–Gordon equations in one space dimension

In recent work, the authors extended the local and global well-posedness theory for the 1D Dirac–Klein–Gordon equations, but the uniqueness of the solutions was only known in the contraction spaces (of Bourgain–Klainerman–Machedon type). Here we prove some unconditional uniqueness results [that is, uniqueness in the larger space C([0,T];X ₀), where X ₀ denotes the data space]. We also prove a result about persistence of higher regularity, which is stronger than the standard version obtained from the contraction argument, since our result allows to independently increase the regularity of the spinor and scalar fields, whereas in the standard result they must be increased by the same amount.     

Combinatorial proofs of the Newton–Girard and Chapman–Costas-Santos identities

In this paper we give combinatorial proofs of some well known identities and obtain some generalizations. We give a visual proof of a result of Chapman and Costas-Santos regarding the determinant of sum of matrices. Also we find a new identity expressing the permanent of sum of matrices. Besides, we give a graph theoretic proof of the Newton–Girard identity in a generalized form.     

Cotorsion Pairs and Cartan–Eilenberg Categories

Cartan–Eilenberg categories were recently introduced by Guillén Santos, Navarro Aznar, Pascual and Roig. In the present paper, we give a method of constructing Cartan–Eilenberg categories for abelian categories, based on cotorsion pairs. In particular, we recover the left Cartan–Eilenberg structure for (un)bounded below chain complexes of modules (Pascual, Collect Math 63(2):203–216, 2012) and extend it to more general categories, including the category of (quasi-coherent)sheaves over a projective scheme.     

Codismantlability and projective dimension of the Stanley–Reisner ring of special hypergraphs

In this paper, we generalize the concept of codismantlable graphs to hypergraphs and show that some special vertex decomposable hypergraphs are codismantlable. Then we generalize the concept of bouquet in graphs to hypergraphs to extend some combinatorial invariants of graphs about disjointness of a set of bouquets. We use these invariants to characterize the projective dimension of Stanley–Reisner ring of special hypergraphs in some sense.     

Box dimension of Neimark–Sacker bifurcation

In this article we show how a change of a box dimension of orbits of two-dimensional discrete dynamical systems is connected to their bifurcations in a non-hyperbolic fixed point. This connection is already shown in the case of one-dimensional discrete dynamical systems and Hopf bifurcation for continuous systems. Namely, at the bifurcation point the box dimension changes from zero to a certain positive value which is connected to the appropriate bifurcation. We study a two-dimensional discrete dynamical system with only one multiplier on the unit circle, and show a result for the box dimension of an orbit on the centre manifold. We also consider a planar discrete system undergoing a Neimark–Sacker bifurcation. It is shown that box dimension depends on the order of non-degeneracy at the non-hyperbolic fixed point and on the angle–displacement map. As it was expected, we prove that the box dimension is different in the rational and irrational case.     

Characterization of Weyl operator in terms of Mehler–Fock transform

In this paper, we define the windowed-Mehler–Fock transform and introduce the corresponding Weyl transform. Further, we examine the boundedness of windowed-Mehler–Fock transform in Lebesgue space and establish some of its fundamental properties. Also, we give the criteria of boundedness and compactness of Weyl transform in Lebesgue space.     

A refinement of the Brascamp–Lieb–Poincaré inequality in one dimension

In this short note, we give a refinement of the Brascamp–Lieb inequality in the style of the Houdré–Kagan extension for the Poincaré inequality in one dimension. This is inspired by works by Helffer and by Ledoux.     

A wavelet cross-spectral analysis of solar–ENSO–rainfall connections in the Indian monsoons

The possible connections between the El Nino-Southern Oscillation (ENSO) phenomenon and Indian monsoon rainfall have been widely discussed in the meteorological literature. We show strong statistical evidence here for connections of ENSO with solar activity. This is particularly evident in a comparison between the two contrasting test periods of 1878–1913 and 1933–1964, representing three complete cycles of lowest and highest solar activity respectively since 1850. Wavelet statistical analysis reveals that the link between solar activity and ENSO is generally stronger than that between ENSO and rainfall but only slightly weaker than that between solar activity and rainfall. Over the two test periods an increase in solar activity is associated with a decrease in ENSO indices and an increase in the monsoon rainfall in the 8–16 y period band. In the 2–7 y period band the effects vary with region. The net effect of solar processes on rainfall thus appears to be the result of counteracting or cooperating influences on shorter (about 5–6 y) and longer (about 11–12 y) time scales, the latter on the whole dominating over the former. The present analysis thus suggests that the influence of solar processes on Indian rainfall operates in part indirectly through ENSO, but on more than one time scale.     

Characterization of Day–James Spaces and Generalized Day–James Spaces

In this paper, we introduce the definition of generalized Day–James space on R~n(n ≥2) and give a characterization of it, which extend some known results. In addition, we provide a sufficient and necessary condition for Day–James space, which reappeared Day's construction for any two-dimensional normed space to make Birkhoff orthogonality symmetry.