Related papers: Big denominators and analytic normal forms with an…

Free analysis is a quantization of the usual function theory much like operator space theory is a quantization of classical functional analysis. Basic objects of free analysis are noncommutative functions. These are maps on tuples of…

For a germ $(X,0)$ of a normal complex analytic surface, let $E:=H^0({}^p_+IC_X\mathbb Z)_0$, where ${}^pIC_X\mathbb Z$ and ${}^p_+IC_X\mathbb Z$ denote the ordinary and dual middle-perversity intersection complexes with integral…

We give normal forms for generic k-dimensional parametric families $(Z_\varepsilon)_\varepsilon$ of germs of holomorphic vector fields near $0\in\mathbb{C}^2$ unfolding a saddle-node singularity $Z_0$, under the condition that there exists…

Let M be a real analytic manifold modeled on a locally convex space and K be a non-empty compact subset of M. We show that if an open neighborhood of K in M admits a complexification which is a regular topological space, then the germ of…

We discuss linear algebra of infinite-dimensional vector spaces in terms of algebraic (Hamel) bases. As an application we prove the surjectivity of a large class of linear partial differential operators with smooth ($\mathcal…

This article develops several main results for a general theory of homological algebra in categories such as the category of sheaves of idempotent modules over a topos. In the analogy with the development of homological algebra for abelian…

In non-archimedean setting, we establish a Lehto--Virtanen-type theorem for a morphism from the punctured Berkovich closed unit disk $\overline{\mathsf{D}}\setminus\{0\}$ in the Berkovich affine line to the Berkovich projective line…

Through abelian categories, homological lemmas for modules admit a self-dual treatment, where half of the proof of a lemma is sufficient to prove the full lemma. In this paper, we show how the context of a `noetherian form', recently…

In this paper, we first introduce stable functors with respect to a preenveloping/precovering subcategory and investigate some of their properties. Using that we then introduce and study a relative complete cohomology theory in abelian…

In [10], Halmos proved an interesting result that the set of irreducible operators is dense in $\mathcal B(\mathcal H)$ in the sense of Hilbert-Schmidt approximation. In a von Neumann algebra $\mathcal M$ with separable predual, an operator…

Building upon the work of Pavel in [P. Kolesnikov, Journal of Mathematical Physics, 56, 7 (2015)], we first present the cohomology of averaging operators on the Lie conformal algebras and use it to develop the cohomology of averaging Lie…

In math.RT/0302174 we developed a framework to study representations of groups of the form $G((t))$, where $G$ is an algebraic group over a local field $K$. The main feature of this theory is that natural representations of groups of this…

In the first half of twentieth century the theory of complex analytic functions and of their zerosets was fully developed. The definition of holomorphic function has a local nature. Germs of holomorphic functions form a distinguished…

In this paper, we treat $\mathscr{D}$-modules on the basic affine space $G/U$ and their global sections for a semisimple complex algebraic group $G$. Our aim is to prepare basic results about large non-irreducible modules for the branching…

We review the shape theory of $\infty$-topoi, and relate it with the usual cohomology of locally constant sheaves. Additionally, a new localization of profinite spaces is defined which allows us to extend the \'etale realization functor of…

We consider differential operators over a noncommutative algebra $A$ generated by vector fields. These are shown to form a unital associative algebra of differential operators, and act on $A$-modules $E$ with covariant derivative. We use…

In this article, we study the regularity of solutions to inhomogeneous time-fractional evolution equations involving anisotropic non-local operators in mixed-norm Sobolev spaces of variable order, with non-trivial initial conditions. The…

For a class of reducible Hamiltonian partial differential equations (PDEs) with arbitrary spatial dimensions, quantified by a quadratic polynomial with time-dependent coefficients, we present a comprehensive classification of long-term…

In this paper, the notion of local algebraic fundamental groups of normal complex analytic singularities are generalized to certain profinite groups called $D$-local algebraic fundamental groups which turns out to be useful even for the…

We study the algebra of regular functions on the big cell of the Gauss decomposition of a simple complex Lie group G. We prove that it is spanned by the matrix elements of big projective modules in the BGG category O, and admits a…