Volume 23, Issue 4

Let R be a commutative ring which contains the rationals as a subring and let g be a chain complex. Suppose given an sh-Lie algebra structure on g , that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra T ′ on the suspension of g and write the perturbed coalgebra as T ″. Suppose, furthermore, given a contraction of g onto a chain complex M . We show that the data determine an sh-Lie algebra structure on M , that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra S ′ on the suspension of M , a Lie algebra twisting cochain from the perturbed coalgebra S ″ to the loop Lie algebra L on the perturbed coalgebra T ″, and an extension of this Lie algebra twisting cochain to a contraction of chain complexes from the Cartan–Chevalley–Eilenberg coalgebra on L onto S ″ which is natural in the data. For the special case where M and g are connected we also construct an explicit extension of the perturbed retraction to an sh-Lie map. This approach includes a very general solution of the master equation.

We find an optimal gradient estimate in Orlicz spaces for a nonlinear elliptic equation in divergence form with discontinuous nonlinearity in a non-smooth domain.

Let ( m, b ) be a pair of natural numbers. For m even (resp. m odd and b ≥ 2) we show that if there is an m -dimensional non-formal compact oriented manifold with first Betti number b 1 = b , there is also a symplectic (resp. contact) manifold with these properties.

In this paper, we explore a general method to derive H p → L p boundedness from H p → H p boundedness of linear operators, an idea originated in the work of Han and Lu in dealing with the multiparameter flag singular integrals ([Discrete Littlewood-Paley-Stein theory and multi-parameter Hardy spaces associated with the flag singular integrals]). These linear operators include many singular integral operators in one parameter and multiparameter settings. In this paper, we will illustrate further that this method will enable us to prove the H p → L p boundedness on product spaces of homogeneous type in the sense of Coifman and Weiss ([Lecture Notes in Math. 242: 1971]) where maximal function characterization of Hardy spaces is not available. Moreover, we also provide a particularly easy argument in those settings such as one parameter or multiparameter Hardy spaces and where the maximal function characterization exists. The key idea is to prove ‖ƒ‖ L p ≤ C ‖ƒ ‖ H p for ƒ ∈ L q ∩ H p (1 < q < ∞, 0 < p ≤ 1). It is surprising that this simple result even in this classical setting has been absent in the literature.

We study the minimal surface equation in Minkowskian geometry in , which is a well-known quasilinear wave equation. The classical result of Lindblad, [Proc. Amer. Math. Soc. 132: 1095–1102, 2004], establishes global existence of small and smooth solutions (i.e. global regularity), provided the initial data is small, compactly supported and very smooth. In the present paper, we achieve more precise results. We show that, at least when n ≥ 4 (or n = 3, but with radial data), it is enough to assume the smallness of some scale invariant quantities, involving (unweighted) Sobolev norms only. In the 3D case, such a proof fails as a consequence of the well-known Strichartz inequality “missing endpoint” and one has instead slightly weaker results, which requires smallness of the data in certain W s,p , p < 2, spaces. In the 2D case, this fails as well, since the free solution of the 2D wave equation fails to be square integrable, but only belongs to L 2,∞ , another failure by an endpoint. Thus, an important question is left open: Can one prove global regularity for the 2D minimal surface equation, assuming smallness in unweighted Sobolev spaces only?

Let L be a first-order language with equality and let 𝔘 be an L -structure of cardinality κ . If ℵ 0 ≤ λ ≤ κ , then we say that 𝔘 is elementarily λ -homogeneous iff any two substructures of cardinality λ are elementarily equivalent, and λ -homogeneous iff any two substructures of cardinality λ are isomorphic. In this note, we classify the elementarily λ -homogeneous structures ( A , ƒ) where ƒ : A → A is a function and λ is a cardinal such that ℵ 0 ≤ λ ≤ | A |. As a corollary, we obtain a complete description of the Jónsson algebras ( A , ƒ), where ƒ : A → A .

An (affine) algebraic monoid is an affine variety over an algebraically closed field K endowed with a monoid structure such that the product map is an algebraic variety morphism. Let M be an irreducible algebraic monoid with G (⊊ M ) its unit group, ker( M ) its semigroup kernel, E (ker( M )) the set of minimal idempotents of M . Algebraically, ker( M ) is the minimum regular 𝒥-class in M ; geometrically, ker( M ) is a retract of M but also the unique closed orbit under the natural G × G -action on M . We study regularity conditions for M and the relationships between ker( M ) and R u ( G ). We also study an algebraic group embedding problem. The principal results are: (i) dim R u ( G ) = dim E (ker( M )) + dim R u ( H ) + dim R u ( G ( e )), where e ∈ E (ker( M )), H is a maximal subgroup of ker( M ) and G ( e ) ≔ { x ∈ G | xe = ex = e }. (ii) A connected linear algebraic group with nontrivial characters can be realized as a proper unit group of some irreducible normal regular algebraic monoid. (iii) When char ( K ) = 0, if M is regular, dim R u ( G ) = dim ker( M ) implies R u ( G ) ≅ ker( M ) as algebraic varieties.

We introduce a new type of convergence in probability theory, which we call “mod-Gaussian convergence”. It is directly inspired by theorems and conjectures, in random matrix theory and number theory, concerning moments of values of characteristic polynomials or zeta functions. We study this type of convergence in detail in the framework of infinitely divisible distributions, and exhibit some unconditional occurrences in number theory, in particular for families of L -functions over function fields in the Katz–Sarnak framework. A similar phenomenon of “mod-Poisson convergence” turns out to also appear in the classical Erdős–Kac Theorem.

We shall establish uniform large deviations for the pinned Brownian motions on a class of two dimensional hyperbolic Riemannian manifolds. It will be shown that the corresponding rate function is given via the infinitesimal generator of a new diffusion process obtained by a harmonic transform of the Brownian motion based on a ground state for the Laplacian.