We introduce the affine ensemble, a class of determinantal point processes (DPP) in half-plane $\mathbb{C}^{+}$ associated with $ax + b$ (affine) group, depending on an admissible Hardy function $\psi$. obtain asymptotic behavior variance, exact value constant, and non-asymptotic upper lower bounds for variance compact set $\Omega \subset \mathbb{C}^{+}$. As special case one recovers DPP relate...

We develop techniques of mimicking the Frobenius action in study universal homeomorphisms mixed characteristic. As a consequence, we show characteristic Keel's base point free theorem obtaining applications towards Minimal Model Program, generalise Kollár's on existence quotients by finite equivalence relations to characteristic, and provide new proof affine group schemes.

We study finite dimensional representations over some Noetherian algebras a field of characteristic zero. More precisely, we give necessary and sufficient conditions for the category locally to be closed under taking injective hulls extend results known group rings enveloping Ore extensions, Hopf crossed products, affine low Gelfand-Kirillov dimension.

2. (Jordan) A transitive permutation group of degree n > 1 contains a derangement. In fact (Cameron and Cohen) the proportion of derangements in a transitive group G is at least 1/n. Equality holds if and only if G is sharply 2transitive, and hence is the affine group {x 7→ ax + b : a, b ∈ F, a 6= 0} over a nearfield F. The finite nearfields were determined by Zassenhaus. They all have prime po...

The central objects of study in this thesis are affine difference algebraic groups. Similar to the case of affine algebraic groups, these groups can all be realized as subgroups of some general linear group defined by algebraic difference equations. However, the defining equations here are not simply polynomials in the matrix entries but difference polynomials, i.e., the defining equations invo...

in this paper we find systems of subgroups of a finite‎ ‎group‎, ‎which $bbb p$nobreakdash-hspace{0pt}subnormality guarantees supersolvability‎ ‎of the whole group‎.

This paper is the first in a series that describe a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group (in this paper for simplicity we consider only untwisted and simply connected case). The usual geometric Satake isomorphism for a reductive group G identifies the tensor category Rep(G) of finitedimensional representations of the Langlands dual group G with th...

Let $${\mathfrak {g}}$$ be a complex classical simple Lie algebra and let O nilpotent orbit of . The fundamental group $$\pi _1(O)$$ is finite. Take the universal covering ^0: X^0 \rightarrow O$$ Then ^0$$ extends to finite cover : X {\bar{O}}$$ By using Kirillov–Kostant form $$\omega _{KK}$$ on O, normal affine variety becomes conical symplectic variety. In this article we give an explicit con...

Planar functions were introduced by Dembowski and Ostrom ([4]) to describe projective planes possessing a collineation group with particular properties. Several classes of planar functions over a finite field are described, including a class whose associated affine planes are not translation planes or dual translation planes. This resolves in the negative a question posed in [4]. These planar f...

Let V0, (, ) be the real Euclidean space spanned by the root system R0 of g and let V be the space of affine linear functions on V0. We shall identify V with Rδ ⊕ V0 via the pairing (rδ + x, y) = r + (x, y) for r ∈ R, x, y ∈ V0. The dual affine root system is R = {mδ + α | m ∈ Z, α ∈ R0} ⊆ V where α ∨ means 2α (α,α) as usual. Fix a positive subsystem R + 0 ⊆ R0 with base {α1, · · · , αn} and le...