J. L. Lions’ problem on maximal regularity

This is a survey on recent progress concerning maximal regularity of non-autonomous equations governed by time-dependent forms on a Hilbert space. It also contains two new results showing the limits of the theory.     

On second-order elliptic operators with complete first-order boundary degeneration and strong outward drift

Second order elliptic operators and their parabolic counterparts are studied in the case of complete boundary degeneration of the leading order coefficients in the presence of a strong outward pointing drift. It is shown that the problem generates a positive analytic \(C_0\)-semigroup with maximal \(L_p\)-regularity in \(L_q\)-spaces. This result is based on hard analysis estimates for integral operators in combination with modern functional analytic tools like \({\mathcal R}\)-boundedness and the operator-valued functional calculus.     

$$\varvec{L^p}$$-norm estimate of the Bergman projection on Hartogs triangle

Let \({\mathcal {S}}\) denote the set of positive integers that may appear as the strong symmetric genus of a finite abelian group. We obtain a set of (simple) necessary and sufficient conditions for an integer g to belong to \({\mathcal {S}}\). We also prove that the set \({\mathcal {S}}\) has an asymptotic density and approximate its value.     

Injective restriction of characters

Suppose that G is a finite group and let H be a proper subgroup of G. Then the restriction map from the characters of G to the characters of H is never injective.     

Strict comparison theorems under sublinear expectations

In this paper, we consider strict comparison theorems in the framework of G-expectation, which is a type of sublinear expectation associated with fully nonlinear parabolic partial differential equations. In particular, we first apply Krylov–Safonov estimates to establish the strict comparison theorem for functions from the Lipschitz class \(Lip(\Omega )\). Then we prove generalized strict comparison theorems on the enlarged space \(L_G^1(\Omega )\), which is the Banach completion of \(Lip(\Omega )\) under the G-expectation.     

Maxima of stable random fields,nonsingular actions and finitely generated abelian groups: A survey

This is a self-contained introduction to the applications of ergodic theory of nonsingular (also known as quasi-invariant) group actions and the structure theorem for finitely generated abelian groups on the extreme values of stationary symmetric stable random fields indexed by ? ^d . It is based on a mini course given in the http://www.isid.ac.in/ lps16/PastWebsites/LPS8/index.html Eighth Lectures on Probability and Stochastic Processes (held in the Bangalore Centre of Indian Statistical Institute during December 6–10, 2013) except that a few recent references have been added in the concluding part. This article is a survey of existing work and the proofs are therefore skipped or briefly outlined.     

A non-cooperative interpretation of the kernel

The purpose of the paper is to propose a bargaining game to interpret the kernel non-cooperatively. Based on the idea of the Davis-Maschler reduced game, a bilateral bargaining procedure is provided in our game model. We show that the set of all subgame perfect equilibrium outcomes of our non-cooperative game coincides with the kernel for transferable utility games.     

Prime and primitive algebras with prescribed growth types

Bartholdi and Smoktunowicz constructed in 2014 finitely generated monomial algebras with prescribed sufficiently fast growth types. We show that their construction need not result in a prime algebra, but it can be modified to provide prime algebras without further limitations on the growth type.Moreover, using a construction of an inverse system of monomial ideals which arise from this construction, we are able to further construct finitely generated primitive algebras without further limitations on the growth type.Then, inspired by Zelmanov’s example in 1979, we show how our prime algebras can be constructed such that they contain non-zero locally nilpotent ideals; this is the very opposite of the primitive constructions.     

An upper bound on the Chebotarev invariant of a finite group

A subset {g ₁,..., g _d } of a finite group G invariably generates \(\left\{ {g_1^{{x_1}}, \ldots ,g_d^{{x_d}}} \right\}\) generates G for every choice of x _i ∈ G. The Chebotarev invariant C(G) of G is the expected value of the random variable n that is minimal subject to the requirement that n randomly chosen elements of G invariably generate G. The first author recently showed that \(C\left( G \right) \leqslant \beta \sqrt {\left| G \right|} \) for some absolute constant β. In this paper we show that, when G is soluble, then β is at most 5/3. We also show that this is best possible. Furthermore, we show that, in general, for each ε > 0 there exists a constant c _ε such that \(C\left( G \right) \leqslant \left( {1 + \in } \right)\sqrt {\left| G \right|} + {c_ \in }\).     

Universal and homogeneous structures on the Urysohn and Gurarij spaces

Using Fraïssé theoretic methods we enrich the Urysohn universal space by universal and homogeneous closed relations, retractions, closed subsets of the product of the Urysohn space itself and some fixed compact metric space, L-Lipschitz map to a fixed Polish metric space. The latter lifts to a universal linear operator of norm L on the Lispchitz-free space of the Urysohn space.Moreover, we enrich the Gurarij space by a universal and homogeneous closed subspace and norm one projection onto a 1-complemented subspace. We construct the Gurarij space by the classical Fraïssé theoretic approach.