International Journal of Mathematics

Extending earlier work of the authors, this is the first in a series of papers devoted to the vertex-algebraic structure of principal subspaces of standard modules for twisted affine Kac–Moody algebras. In this part, we develop the necessary theory of principal subspaces for the affine Lie algebra , which we expect can be extended to higher rank algebras. As a "test case", we consider the principal subspace of the basic -module and explore its structure in depth.

In 1984, Libera and Zlotkiewicz proved that the inverse of the square-root transform of the Koebe function is the extremal function for the inverses of odd univalent functions. The purpose of this paper is to point out that this is not the case for the m-fold symmetric bi-starlike functions by demonstrating the unpredictability of the coefficients of such functions.

For every p ≥ 2, we give a characterization of positive definite functions on a free group with finitely many generators, which can be extended to positive linear functionals on the free group C*-algebra associated with the ideal ℓ_p. This is a generalization of Haagerup's characterization for the case of the reduced free group C*-algebra. As a consequence, the canonical quotient map between the associated C*-algebras is not injective, and they have a unique tracial state.

The author constructs the moduli of representations whose images generate the subalgebra of upper triangular matrices (up to inner automorphisms) of the full matrix ring for any groups and any monoids.

In this paper, we introduce the notion of modified K-stability (K-semistability) associated with the soliton field on a Fano manifold M. Then we prove that if the modified K-energy associated with the soliton vector field is bounded from below, M is modified K-semistable associated with the soliton field. This result generalizes the corresponding result in a paper of Ding and Tian.

We study the topology of the fibers of real analytic maps ℝⁿ → ℝ^p, n > p, in a neighborhood of a critical point. We first prove that every real analytic map-germ f : ℝⁿ → ℝ^p, p ≥ 1, with arbitrary critical set, has a Milnor–Lê type fibration away from the discriminant. Now assume also that f has the Thom a_f-property, and its zero-locus has positive dimension. Also consider another real analytic map-germ g : ℝⁿ → ℝ^k with an isolated critical point at the origin. We have Milnor–Lê type fibrations for f and for (f, g) : ℝⁿ → ℝ^p+k, and we prove for these the analogous of the classical Lê–Greuel formula, expressing the difference of the Euler characteristics of the fibers F_f and F_f,g in terms of an invariant associated to these maps. This invariant can be expressed in various ways: as the index of the gradient vector field of a map on F_f associated to g; as the number of critical points of on F_f; or in terms of polar multiplicities. When p = 1 and k = 1, this invariant can also be expressed algebraically, as the signature of a certain bilinear form. When the germs of f and (f, g) are both isolated complete intersection singularities, we exhibit an even deeper relation between the topology of the fibers F_f and F_f,g, and construct in this setting, an integer-valued invariant, that we call the curvatura integra that picks up the Euler characteristic of the fibers. This invariant, and its name, spring from Gauss' theorem, and its generalizations by Hopf and Kervaire, expressing the Euler characteristic of a manifold (with some conditions) as the degree of a certain map.

We describe primitive and prime ideals in the C*-algebra C*(E) of a graph E satisfying Condition (K), together with the topologies on each of these spaces. In particular, we find that primitive ideals correspond to the set of maximal tails disjoint union the set of finite-return vertices, and that prime ideals correspond to the set of clusters of maximal tails disjoint union the set of finite-return vertices.

Let be a convex polyhedron of dimension n. Denote and let be its closure. We prove that for n = 3 the semialgebraic sets and are polynomial images of ℝ³. The former techniques cannot be extended in general to represent the semialgebraic sets and as polynomial images of ℝⁿ if n ≥ 4.

Bondal's conjecture states that the Frobenius push-forward of the structure sheaf 𝒪_X generates the derived category D^b(X) for smooth projective toric varieties X. Bernardi and Tirabassi exhibit a full strong exceptional collection consisting of line bundles on smooth toric Fano 3-folds assuming Bondal's conjecture. In this paper, we prove Bondal's conjecture for smooth toric Fano 3-folds and improve upon their result using birational geometry.