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In 1975 Barwise and Schlipf published a landmark paper whose main theorem asserts that a nonstandard model M of PA (Peano arithmetic) is recursively saturated iff M has an expansion that satisfies the subsystem Δ11−CA0 of second order arithmetic. In this paper we identify a crucial error in the Barwise-Schlipf proof of the right-to-left direction of the theorem, and we offer a correct proof of the problematic direction. [ABSTRACT FROM AUTHOR]

A misuse of terminology has occurred in the statement and proof of Theorem 4.1 in our paper [Proc. Amer. Math. Soc. 143 (2015), pp. 1981–1989] which caused some justifiable misinterpretation of this result. To recover this result we provide a new definition and give the statement of our result in terms of this definition. The proof of the new version is an improvement of the old proof. The effect of the new definition on further relevant results in our paper has been adopted in a remark. [ABSTRACT FROM AUTHOR]

In this paper we will give a simple stopping time proof in the Rd setting of the matrix weighted Triebel-Lizorkin bounds proved by F. Nazarov–S. Treil and A. Volberg, respectively. Furthermore, we provide explicit matrix Ap characteristic dependence and also discuss some interesting open questions. [ABSTRACT FROM AUTHOR]

In this paper, we extend the work by the first author and Q. Chen [Duke Math. J. 162 (2013), pp. 1149-1169] to classify n-dimensional (n ≥ 5) complete D-flat gradient steady Ricci solitons. More precisely, we prove that any n-dimensional complete noncompact gradient steady Ricci soliton with vanishing D-tensor is either Ricci-flat, or isometric to the Bryant soliton. Furthermore, the proof extends to the shrinking case and the expanding case as well. [ABSTRACT FROM AUTHOR]

In this paper we enumerate the centrally symmetric lozenge tilings of a hexagon with a shamrock removed from its center. Our proof is based on a variant of Kuo's graphical condensation method in which only three of the four involved vertices are on the same face. As a special case, we obtain a new proof of the enumeration of the self-complementary plane partitions. [ABSTRACT FROM AUTHOR]

We present a short proof of Klartag's central limit theorem for convex bodies, using only the most classical facts about log-concave functions. An appendix is included where we give the proof that the thin shell implies CLT. The paper is accessible to anyone. [ABSTRACT FROM AUTHOR]

This paper provides certain computations with transfer associated with projective bundles of \mathrm {Spin} vector bundles. One aspect is to revise the proof of the main result of [Trans. Amer. Math. Soc.349 (1997), pp. 4385-4399] which says that all fourfold products of the Ray classes are zero in symplectic cobordism. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundedness of weak solutions to quasilinear parabolic equations of the form ut − div A(x,t,∇ u) = 0, where the nonlinearity A(x,t,∇ u) is modelled after the well-studied p-Laplace operator. The question of boundedness has received a lot of attention over the past several decades with the existing literature showing that weak solutions in either 2N/N+2 < p < 2, p = 2, or 2 < p, are bounded. The proof is essentially split into three cases mainly because the estimates that have been obtained in the past always included an exponent of the form 1/p−2 or 1/2−p, which blows up as p → 2. In this note, we prove the boundedness of weak solutions in the full range 2N/N+2 < p < ∞ without having to consider the singular and degenerate cases separately. Subsequently, in a slightly smaller regime of 2N/N+1 < p < ∞, we also prove an improved boundedness estimate. [ABSTRACT FROM AUTHOR]

In this paper we give a simple proof of the existence and plenitude of Runge tubes in Cn (n > 1) and, more generally, in Stein manifolds with the density property. We show in particular that for any algebraic submanifold X of codimension at least two in a complex Euclidean space Cn, the normal bundle of X admits a holomorphic embedding onto a Runge domain in Cn which agrees with the inclusion map X → Cn on the zero section. [ABSTRACT FROM AUTHOR]

In this paper, we give a new and quick proof for the Hartman-Grobman theorem for random dynamical systems. This approach does not involve any previously proved existence of the stable and unstable manifolds. [ABSTRACT FROM AUTHOR]

In this paper, we give a new proof for the Daniel-Stone theorem for random probability measures without using the result of the classical Daniel-Stone theorem. [ABSTRACT FROM AUTHOR]

M. K. Dabkowski and J. H. Przytycki introduced the nth Burnside group of a link, which is an invariant preserved by n-moves. Using this invariant, for an odd prime p, they proved that there are links which cannot be reduced to trivial links via p-moves. It is generally difficult to determine if pth Burnside groups associated to a link and the corresponding trivial link are isomorphic. In this paper, we give a necessary condition for the existence of such an isomorphism. Using this condition we give a simple proof for their results that concern p-move reducibility of links. [ABSTRACT FROM AUTHOR]

This erratum concerns Lemma 3.1 in the original paper [Proc. Amer. Math. Soc. 147 (2019), pp. 1657-1669]. The statement of that lemma needs to be slightly changed since the statement of Claim 3.5 must be changed. The proof of the lemma is mostly still kept as the original one with some lines modified. This change does not effect the proof of the main results. [ABSTRACT FROM AUTHOR]

In our paper "On solvable compact Clifford-Klein forms" [Proc. Amer. Math. Soc. 145 (2017), pp. 1819-1832] we have found a small gap in the proof of the main theorem. In this note we show that the result is correct and that the proof goes through after a slight modification. [ABSTRACT FROM AUTHOR]