Isolated Singularities of Polyharmonic Operator in Even Dimension

We consider the equation $\Delta^2 u=g(x,u) \geq 0$ in the sense of distribution in $\Omega'=\Omega\setminus \{0\}$ where $u$ and $-\Delta u\geq 0.$ Then it is known that $u$ solves $\Delta^2 u=g(x,u)+\alpha \delta_0-\beta \Delta \delta_0,$ for some non-negative constants $\alpha$ and $\beta.$ In this paper we study the existence of singular solutions to $\Delta^2 u= a(x) f(u)+\alpha \delta_0-\beta \Delta \delta_0$ in a domain $\Omega\subset \mathbb{R}^4,$ $a$ is a non-negative measurable function in some Lebesgue space. If $\Delta^2 u=a(x)f(u)$ in $\Omega',$ then we find the growth of the nonlinearity $f$ that determines $\alpha$ and $\beta$ to be $0.$ In case when $\alpha=\beta =0,$ we will establish regularity results when $f(t)\leq C e^{\gamma t},$ for some $C, \gamma>0.$ This paper extends the work of Soranzo (1997) where the author finds the barrier function in higher dimensions $(N\geq 5)$ with a specific weight function $a(x)=|x|^\sigma.$ Later we discuss its analogous generalization for the polyharmonic operator

Cohomology of Lie algebroid over Algebraic spaces

We consider Lie algebroid over an algebraic space as a quasicoherent sheaf of Lie-Rinehart algebras. We compute algebraic (analytic) de Rham cohomologies for some free divisors and the associated logarithmic de Rham cohomologies as well. We express hypercohomology for a locally free Lie algebroid (of finite or infinte rank) as derived functor and simplify it via $\check{C}$ech cohomology. Furthermore, we define the Hochschild hypercohomology of a sheaf of generalized bialgebras and study the special cases, namely Hochschild hypercohomology of universal enveloping algebroid and jet algebroid of a Lie algebroid. We present a version of Hochschild-Kostant-Rosenberg (HKR) theorem for a locally free Lie algebroid as well as its dual version.Comment: Major revision has been don

Hermitian Lie algebroids over analytic spaces

We consider some aspects of complex Riemannian geometry for complex algebraic varieties and study Hermitian metrics on analytic spaces. Then we define Hermitian metrics on a holomorphic Lie algebroid and consider the associated characteristic foliation with canonically induced inner product. Moreover, we consider an example of a Hermitian Lie algebroid $\mathcal{L}$ and describe the induced inner product on a special $\mathcal{L}$-invariant subspace. Later, we consider hypercohomologies associated with leaf spaces, leaves and some $\mathcal{L}$-invariant subspaces for the characteristic foliation $\mathfrak{a}(\mathcal{L})$ of a holomorphic Lie algebroid $\mathfrak{a}: \mathcal{L} \rightarrow \mathcal{T}_X$ over a Hermitian manifold $X$

Shakespeare, Macbeth and the Hindu Nationalism of Nineteenth-Century Bengal

The essay examines a Bengali adaptation of Macbeth, namely Rudrapal Natak (published 1874) by Haralal Ray, juxtaposing it with differently accented commentaries on the play arising from the English-educated elites of 19th Bengal, and relating the play to the complex phenomenon of Hindu nationalism. This play remarkably translocates the mythos and ethos of Shakespeare’s original onto a Hindu field of signifiers, reformulating Shakespeare’s Witches as bhairavis (female hermits of a Tantric cult) who indulge unchallenged in ghastly rituals. It also tries to associate the gratuitous violence of the play with the fanciful yearning for a martial ideal of nation-building that formed a strand of the Hindu revivalist imaginary. If the depiction of the Witch-figures in Rudrapal undercuts the evocation of a monolithic and urbane Hindu sensibility that would be consistent with colonial modernity, the celebration of their violence may be read as an effort to emphasize the inclusivity (as well as autonomy) of the Hindu tradition and to defy the homogenizing expectations of Western enlightenmen