International Journal of Mathematics

We give a simple characterization of the parabolic geodesics introduced by Čap, Slovák and Žádník for all parabolic geometries. This goes through the definition of a natural connection on the space of Weyl structures. We then show that parabolic geodesics can be characterized as the following data: a curve on the manifold and a Weyl structure along the curve, so that the curve is a geodesic for its companion Weyl structure and the Weyl structure is parallel along the curve and in the direction of the tangent vector of the curve.

Glimm's theorem says that a UHF algebra is almost embedded in a separable C*-algebra not of type I. Applying his methods we obtain a covariant version of his result; a UHF algebra with a product type automorphism is covariantly embedded in such a C*-algebra equipped with an automorphism with full Connes spectrum.

We study equivariant resolutions and local cohomologies of toric sheaves for affine toric varieties, where our focus is on the construction of new examples of indecomposable maximal Cohen–Macaulay modules of higher rank. A result of Klyachko states that the category of reflexive toric sheaves is equivalent to the category of vector spaces together with a certain family of filtrations. Within this setting, we develop machinery which facilitates the construction of minimal free resolutions for the smooth case as well as resolutions which are acyclic with respect to local cohomology functors for the general case. We give two main applications. First, over the polynomial ring, we determine in explicit combinatorial terms the ℤⁿ-graded Betti numbers and local cohomology of reflexive modules whose associated filtrations form a hyperplane arrangement. Second, for the nonsmooth, simplicial case in dimension d ≥ 3, we construct new examples of indecomposable maximal Cohen–Macaulay modules of rank d – 1.

First, we establish the Stein–Weiss inequality for the B-Riesz potential generated by the Riemann–Liouville operator. Next, we prove the Pitt's and Beckner logarithmic inequalities related to the connected Fourier transform.

We introduce a class of automorphisms of compact quantum groups (CQGs) which may be thought of as inner automorphisms and explore the behavior of normal subgroups of CQGs under these automorphisms. We also define the notion of center of a CQG and compute the center for several examples. We briefly touch upon the commutator subgroup of a CQG and discuss how its relation with the center can be different from the classical case.

Using the method of Witten deformation, we express the basic index of a transversal Dirac operator over a Riemannian foliation as the sum of integers associated to the critical leaf closures of a given foliated bundle map.

We prove that for any closed parallelizable n-manifold Mⁿ, if the dimension n ≠ 7, or if n = 7 and the Kervaire semi-characteristic χ_½(M⁷) is zero, then Mⁿ can be embedded in the Euclidean space ℝ²ⁿ with a certain symplectic structure as a Lagrangian submanifold. By the results of Gromov and Fukaya, our result gives rise to symplectic structures of ℝ²ⁿ(n ≥ 3) which are not conformally equivalent to open domains in standard ones.

We study the Seiberg–Witten (SW) equations on surfaces of logarithmic general type. First, we show how to construct irreducible solutions of the SW equations for any metric which is "asymptotic" to a Poincaré type metric at infinity. Then we compute a lower bound for the L²-norm of scalar curvature on these spaces and give non-existence results for Einstein metrics on blow-ups.

Let be a semi-finite von Neumann algebra equipped with a faithful semi-finite normal trace τ, and f(t) be a convex function with f(0) = 0. The trace Jensen inequality in our previous work states τ(f(a*xa)) ≤ τ(a*f(x)a) (as long as the both sides are well-defined) for a contraction and a semi-bounded τ-measurable operator x. Validity of this inequality for (not necessarily semi-bounded) self-adjoint τ-measurable operators is investigated.

In Theorem 3.3 of [I. Biswas and P. Chatterjee, On the exactness of Kostant–Kirillov form and the second cohomology of nilpotent orbits, Int. J. Math.23(8) (2012)], the second cohomology of a quotient of a compact semisimple real Lie group was computed. In this addendum to the above paper, we give a simple topological proof of this theorem.