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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]

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]

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]

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 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]

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]

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]

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]

A short and simple proof of necessity in the McCullough-Quiggin characterization of positive semi-definite kernels with the complete Pick property is presented. [ABSTRACT FROM AUTHOR]

Let Tƒ = ∑IεI (ƒ,hI+) hI−. Here, |εI| = 1 , and hJ is the Haar function defined on dyadic interval J. We show that, for instance, ||T||L2(w) → L2(w) ≲ [w]A2+. Above, we use the one-sided A2 characteristic for the weight w. This is an instance of a one-sided A2 conjecture. Our proof of this fact is difficult, as the very quick known proofs of the A2 theorem do not seem to apply in the one-sided setting. [ABSTRACT FROM AUTHOR]

In [Invent. Math. 207 (2017), pp. 1289-1299] it is claimed that Kobayashi isometries in strictly convex domains are holomorphic or antiholomorphic. Generally, following the idea of St. Antonakoudis of using the relation between Klein and PoincarÉ models, we give simplified and complete proof of this fact. [ABSTRACT FROM AUTHOR]

We provide a simpler proof of the Moser-Trudinger type inequality with logarithmic weight wβ = (-ln ∣x∣ )β(N-1), β ∈ [0, 1) in dimension N ≥ 2 recently established by Calanchi and Ruf. Our proof is based on a suitable change of functions on B and the classical Moser-Trudinger inequality on B. We also prove the existence of maximizers for this inequality when β ≥ 0 sufficiently small. [ABSTRACT FROM AUTHOR]

We introduce new Elmendorf constructions for equivariant categories and posets, and we prove that they are compatible with the classical topological one. Our constructions are more concrete than their model-categorical counterparts, and they give rise to new proofs of the Elmendorf theorems for equivariant categories and posets. [ABSTRACT FROM AUTHOR]

We prove a reducibility result for sl(2,R) quasi-periodic cocycles close to a constant elliptic matrix in ultra-differentiable classes, under an adapted arithmetic condition which extends the Brjuno-Rüssmann condition in the analytic case. The proof is based on an elementary property of the fibered rotation number and deals with ultra-differentiable functions with a weighted Fourier norm. We also show that a weaker arithmetic condition is necessary for reducibility, and that it can be compared to a sufficient arithmetic condition. [ABSTRACT FROM AUTHOR]

We present a CFSG-free proof of the fact that the degree of nilpotence of a finite nonnilpotent group is less than 1/2. [ABSTRACT FROM AUTHOR]

We give an elementary proof of a theorem that characterizes quasisymmetric maps of the unit circle in terms of shear coordinates on the Farey tesselation. The proof only uses the normal family argument for quasisymmetric maps and some elementary hyperbolic geometry. [ABSTRACT FROM AUTHOR]

A compact Kähler manifold with quasi-negative holomorphic sectional curvature must have ample canonical bundle. This was conjectured by Wu-Yau and is recently proved by the continuity method. In this note, we will give an alternative proof of this result by using the Kähler-Ricci flow. [ABSTRACT FROM AUTHOR]

We show that local weak solutions to parabolic systems of p-Laplace type are Hölder continuous in time with values in a spatial Lebesgue space and Hölder continuous on almost every time line. We provide an elementary and self-contained proof building on the local higher integrability result of Kinnunen and Lewis. [ABSTRACT FROM AUTHOR]

We show that there are only finitely many triples of integers 0 < a < b < c such that the product of any two of them is the value of a given polynomial with integer coefficients evaluated at an S -unit that is also a positive integer. The proof is based on a result of Corvaja and Zannier and thus is ultimately a consequence of the Schmidt subspace theorem. [ABSTRACT FROM AUTHOR]

We give a short, direct proof that the full holonomy group of a solution to the Ricci flow is invariant up to isomorphism using the invariance of the reduced holonomy under the flow. [ABSTRACT FROM AUTHOR]

We consider norms on a complex separable Hilbert space such that < a ξ,ξ > ≤ ||ξ||2 ≤ < b ξ,ξ> for positive invertible operators a and b that differ by an operator in the Schatten class. We prove that these norms have unitarizable isometry groups. As a result, if their isometry groups do not leave any finite dimensional subspace invariant, then the norms must be Hilbertian. The approach involves metric geometric arguments related to the canonical action on the non-positively curved space of positive invertible Schatten perturbations of the identity. Our proof of the main result uses a generalization of a unitarization theorem which follows from the Bruhat-Tits fixed point theorem. [ABSTRACT FROM AUTHOR]

Recently, Glasner, Tsankov, Weiss, and Zucker showed that if Γ is an infinite discrete group, then every minimal Γ-flow is disjoint from the Bernoulli shift 2Γ. Their proof is somewhat involved; in particular, it invokes separate arguments for different classes of groups. In this note, we give a short and self-contained proof of their result using purely combinatorial methods applicable to all groups at once. Our proof relies on the Lovász Local Lemma, an important tool in probabilistic combinatorics that has recently found several applications in the study of dynamical systems. [ABSTRACT FROM AUTHOR]

We show that a pointwise finite-dimensional persistence module indexed over a small category decomposes into a direct sum of indecomposables with local endomorphism rings. As an application of this result we give new, short proofs of fundamental structure theorems for persistence modules. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ Rn be a bounded C1 domain and p > 1. For α > 0, define the quantity Λ (α) = infu∈W1,p(Ω),u≠0(∫Ω|∇u|pdx − α ∫∂Ω|u|pds/ ∫Ω|u|pdx with ds being the hypersurface measure, which is the lowest eigenvalue of the p-Laplacian in Ω with a non-linear α-dependent Robin boundary condition. We show the asymptotics Λ (α) = (1−p)αp/(p-1) + o(αp/(p−1)) as v tends to + ∞. The result was only known for the linear case p = 2 or under stronger smoothness assumptions. Our proof is much shorter and is based on completely different and elementary arguments, and it allows for an improved remainder estimate for C1,λ domains. [ABSTRACT FROM AUTHOR]

We provide a simpler proof and slight strengthening of Morrey's famous lemma on \varepsilon -conformal mappings. Our result more generally applies to Sobolev maps with values in a complete metric space, and we obtain applications to the existence of area minimizing surfaces of higher genus in metric spaces. Unlike Morrey's proof, which relies on the measurable Riemann mapping theorem, we only need the existence of smooth isothermal coordinates established by Korn and Lichtenstein. [ABSTRACT FROM AUTHOR]

We give a short proof of the Gagliardo-Nirenberg inequality with BMO term as well as the classical Gagliardo-Nirenberg inequality, applying Hedberg's method, which was used for the Riesz potential, to Muramatu's integral formula. Compared with the proof given by Strzelecki [Bull. London Math. Soc. 38 (2006), pp. 294-300], we do not need the duality of the Hardy space and the BMO space. [ABSTRACT FROM AUTHOR]

We give a new proof of an index theorem for fiber bundles of compact topological manifolds due to Dwyer, Weiss, and Williams, which asserts that the parametrized A-theory characteristic of such a fiber bundle factors canonically through the assembly map of A-theory. Furthermore our main result shows a refinement of this statement by providing such a factorization for an extended A-theory characteristic, defined on the parametrized topological cobordism category. The proof uses a convenient framework for bivariant theories and recent results of Gomez-Lopez and Kupers on the homotopy type of the topological cobordism category. We conjecture that this lift of the extended A-theory characteristic becomes highly connected as the manifold dimension increases. [ABSTRACT FROM AUTHOR]

Let k be a field of characteristic zero, let R be the ring of formal power series in n variables over k, and let D(R,k) be the ring of k-linear differential operators on R. If M is a finitely generated D(R,k)-module, then d(M) ≥ n where d(M) is the dimension of M. This inequality is called the Bernstein inequality. We provide a short proof. [ABSTRACT FROM AUTHOR]

Let M be a factor acting on a complex, separable Hilbert space H. An operator a ∈ M is said to be irreducible in M if W*(a), the von Neumann subalgebra generated by a in M, is an irreducible subfactor of M, i.e., W*(a)'∩ M = C I. In this note, we prove that each operator a ∈ M is a sum of two irreducible operators in M, which can be viewed as a natural generalization of a theorem in [Proc. Amer. Math. Soc. 21 (1969), pp. 251-252], with a completely different proof. [ABSTRACT FROM AUTHOR]

We show that the graphs of nonseparating curves for oriented finite ctype surfaces are uniformly hyperbolic. Our proof follows the proof of uniform hyperbolicity of the graphs of curves for closed surfaces due to Przytycki-Sisto, while introducing new arguments using homology to certify that certain curves are nonseparating. As demonstrated by Aramayona-Valdez, this also proves that the graph of nonseparating curves for any oriented infinite type surface with finite positive genus is hyperbolic. [ABSTRACT FROM AUTHOR]

We give a short proof of the Hou-Sun identity using a q-hypergeometric summation. Furthermore, q-analogues of several other π-formulas are also established according to the q-hypergeometric summations and transformations and the telescoping method. [ABSTRACT FROM AUTHOR]

We prove a local regularity result for distributional solutions of Poisson's equation with Lp data. We use a very short argument based on Weyl's lemma and the Riesz-Fréchet representation theorem. [ABSTRACT FROM AUTHOR]

We prove that the class of finite two-graphs has the extension property for partial automorphisms (EPPA, or Hrushovski property), thereby answering a question of Macpherson. In other words, we show that the class of graphs has the extension property for switching automorphisms. We present a short, self-contained, purely combinatorial proof which also proves EPPA for the class of integer-valued antipodal metric spaces of diameter 3, answering a question of Aranda et al. The class of two-graphs is an important new example which behaves differently from all the other known classes with EPPA: Two-graphs do not have the amalgamation property with automorphisms (APA), their Ramsey expansion has to add a graph, it is not known if they have coherent EPPA, and even EPPA itself cannot be proved using the Herwig-Lascar theorem. [ABSTRACT FROM AUTHOR]

The symplectization of an overtwisted contact (R3,ξot) is shown to be an exotic symplectic R4. The technique of proof is also used to produce exotic symplectic R2n using a GPS-structure and applies to symplectizations of open contact manifolds. [ABSTRACT FROM AUTHOR]

We prove that thick groups (and more generally thick graphs) have trivial Floyd boundary. This shows that a wide class of finitely generated groups that are non-relatively hyperbolic have trivial Floyd boundary. In addition to giving new examples, our result provides a common proof and framework for many of the known results in the literature. [ABSTRACT FROM AUTHOR]

We show that a real Kähler submanifold in codimension 6 is essentially a holomorphic submanifold of another real Kähler submanifold in lower codimension if the second fundamental form is not sufficiently degenerated. We also give a shorter proof of this result when the real Kähler submanifold is minimal, using recent results about isometric rigidity. [ABSTRACT FROM AUTHOR]

Let Λ be a quasi-tilted algebra. If Λ is representation-finite, it was shown by Happel, Reiten, and Smalø that Λ is tilted. We provide a new, short proof of this result. [ABSTRACT FROM AUTHOR]

Let A be a hyperplane arrangement in Cn. We prove in an elementary way that the number of decomposition factors as a perverse sheaf of the direct image Rj*CŨ [n] of the constant sheaf on the complement Ũ to the arrangement is given by the Poincar'e polynomial of the arrangement. Furthermore, we describe the decomposition factors of Rj*CŨ [n] as certain local cohomology sheaves and give their multiplicity. These results are implicitly contained, with different proofs, in Looijenga [Contemp. Math., 150 (1993), pp. 205-228], Budur and Saito [Math. Ann., 347 (2010), no. 3, 545-579], Petersen [Geom. Topol., 21 (2017), no. 4, 2527-2555], and Oaku [Length and multiplicity of the local cohomology with support in a hyperplane arrangement, arXiv:1509.01813v1]. [ABSTRACT FROM AUTHOR]

A celebrated theorem due to Bannai-Bannai-Stanton says that if A is a set of points in Rd, which determines s distinct distances, then |A| ≤ (d + s s). In this note, we give a new simple proof of this result by combining Sylvester's Law of Inertia for quadratic forms with the proof of the so-called Croot-Lev-Pach Lemma from additive combinatorics. [ABSTRACT FROM AUTHOR]

This note contains a corrected proof of the main result (which remains unchanged) from [Proc. Amer. Math. Soc. 138 (2010), pp. 1405-1413]. It was recently observed that an argument has a gap, which now requires a more careful choice of the reduction map. [ABSTRACT FROM AUTHOR]