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This paper considers a specific autonomous Schwarzian differential equation given by Sf=c(f−σ1)(f−σ2)/[(f−τ1)(f−τ2)]$$\begin{equation*} \hspace*{35pt} S_f=c(f-\sigma _1)(f-\sigma _2)/[(f-\tau _1)(f-\tau _2)] \end{equation*}$$and presents a complete characterization of its transcendental meromorphic solutions, thus confirming a conjecture posed by Liao and Wu recently. By combining our results with those obtained in a previous paper (Math. Z. 2 (2022), 1657–1672), we are able to explicitly construct all transcendental meromorphic solutions of the autonomous Schwarzian differential equations. Additionally, we identify all nonconstant rational solutions of the autonomous Schwarzian differential equations, with the exception of one special case. [ABSTRACT FROM AUTHOR]

We show that if G$G$ is a finitely generated group such that its profinite completion Ĝ$\widehat{G}$ is "far from being projective" (i.e., the kernel of the universal Frattini cover of Ĝ$\widehat{G}$ is not a small profinite group), then the class of existentially closed G$G$‐actions on fields is not elementary. Since any infinite, finitely generated, virtually free, and not free group is "far from being projective," the main result of this paper corrects an error in our paper, Beyarslan and Kowalski (Proc. London Math. Soc., (2) 118 (2019), 221–256), by showing the negation of Theorem 3.26 in that paper. [ABSTRACT FROM AUTHOR]

This is an overview on derived nonhomogeneous Koszul duality over a field, mostly based on the author's memoir L. Positselski, Memoirs of the American Math. Society 212 (2011), no. 996, vi+133. The paper is intended to serve as a pedagogical introduction and a summary of the covariant duality between DG‐algebras and curved DG‐coalgebras, as well as the triality between DG‐modules, CDG‐comodules, and CDG‐contramodules. Some personal reminiscences are included as a part of historical discussion. [ABSTRACT FROM AUTHOR]

In this paper, inspired by Brendle (Comm. Pure Appl. Math. 76 (2023), 2192) and Johne (arXiv:2103.08496, 2021), we prove a Sobolev inequality on manifolds with density and asymptotically nonnegative Bakry–Émery Ricci curvature. [ABSTRACT FROM AUTHOR]

This paper extends some results of Mattila (J. Fractal Geom. 66 (2021) 389–401 and Ann. Acad. Sci. Fenn. A Math. 42 (2017) 611–620), in particular, removing assumptions of positive lower density. We give conditions on a general family Pλ:Rn→Rm,λ∈Λ$P_{\lambda }:\mathbb {R}^n\rightarrow \mathbb {R}^m, \lambda \in \Lambda$, of orthogonal projections which guarantee that the Hausdorff dimension formula dimA∩Pλ−1{u}=s−m$\dim A\cap P_{\lambda }^{-1}\lbrace u\rbrace =s-m$ holds generically for measurable sets A⊂Rn$A\subset \mathbb {R}^{n}$ with positive and finite s$s$‐dimensional Hausdorff measure, s>m$s>m$. As an application we prove for Borel sets A,B⊂Rn$A,B\subset \mathbb {R}^{n}$ with positive s$s$‐ and t-dimensional$t{\text{-dimensional}}$ measures that if s+(n−1)t/n>n$s + (n-1)t/n > n$, then dimA∩(g(B)+z)⩾s+t−n$\dim A\cap (g(B)+z) \geqslant s+t - n$ for almost all rotations g$g$ and for positively many z∈Rn$z\in \mathbb {R}^{n}$. We shall also give an application to the estimates of the dimension of the set of exceptional rotations. [ABSTRACT FROM AUTHOR]

We establish higher dimensional versions of a recent theorem by Chen and Haynes [Int. J. Number Theory 19 (2023), 1405–1413] on the expected value of the smallest denominator of rational points in a randomly shifted interval of small length, and of the closely related 1977 Kruyswijk–Meijer conjecture recently proved by Balazard and Martin [Bull. Sci. Math. 187 (2023), Paper No. 103305]. We express the distribution of smallest denominators in terms of the void statistics of multidimensional Farey fractions and prove convergence of the distribution function and certain finite moments. The latter was previously unknown even in the one‐dimensional setting. We furthermore obtain a higher dimensional extension of Kargaev and Zhigljavsky's work on moments of the distance function for the Farey sequence [J. Number Theory 65 (1997), 130–149] as well as new results on pigeonhole statistics. [ABSTRACT FROM AUTHOR]

Recently we found that the sufficient condition in the main result of our paper entitled "Heisenberg uniqueness pairs for the hyperbola, Bull. Lond. Math. Soc. 53 (2021), no. 1, 16–25" is incorrect. The purpose of this corrigendum is to point out the gap in the proof given in the above‐mentioned paper and supply a correct sufficient condition with proof. We prove the following result: let Γ$\Gamma$ be the hyperbola xy=1$xy=1$ in R2$\mathbb {R}^2$, and Λβ,θ$\Lambda _{\beta ,\theta }$ be the lattice‐cross defined by Λβ,θ=((Z+{θ})×{0})∪({0}×βZ)$\Lambda _{\beta ,\theta }=((\mathbb {Z}+\lbrace \theta \rbrace)\times \lbrace 0\rbrace)\cup (\lbrace 0\rbrace \times \beta \mathbb {Z})$, where θ∈R$\theta \in \mathbb {R}$, and β>0$\beta >0$. Then (Γ,Λβ,θ)$(\Gamma ,\Lambda _{\beta ,\theta })$ is a Heisenberg uniqueness pair for β⩽1$\beta \leqslant 1$. [ABSTRACT FROM AUTHOR]

We fill in a gap in the proof of the main theorem in the above‐mentioned paper. At the same time, we prove a slightly stronger version of the theorem needed for another paper. [ABSTRACT FROM AUTHOR]

This paper is concerned with the noncommutative analog of the normal subgroup theorem for certain groups. Inspired by Kalantar and Panagopoulos (arXiv:2108.02928, 2021, 16), we show that all Γ$\Gamma$‐invariant subalgebras of LΓ$L\Gamma$ and Cr∗(Γ)$C^*_r(\Gamma)$ are (Γ$\Gamma$‐)coamenable. The groups we work with satisfy a singularity phenomenon described by Bader et al. (Invent. Math. 229 (2022), 929–985). The setup of singularity allows us to obtain a description of Γ$\Gamma$‐invariant intermediate von Neumann subalgebras L∞(X,ξ)⊂M⊂L∞(X,ξ)⋊Γ$L^{\infty }(X,\xi)\subset \mathcal {M}\subset L^{\infty }(X,\xi)\rtimes \Gamma$ in terms of the normal subgroups of Γ$\Gamma$. [ABSTRACT FROM AUTHOR]

In 1987, Michael Wiegner in his seminal paper (J. Lond. Math. Soc. (2), 35 (1987) 303–313) provided an important result regarding the energy decay of Leray solutions u(·,t)$ \bm{u}(\cdot ,t)$ to the incompressible Navier–Stokes in Rn$ \mathbb {R}^{n}$: if the associated Stokes flows had their L2$L^{2}$ norms bounded by O(1+t)−α$ O(1 + t)^{-\;\!\alpha }$ for some 0<α⩽(n+2)/4$ 0 < \alpha \leqslant (n+2)/4$, then the same would be true of ∥u(·,t)∥L2(Rn)$ \Vert \hspace{0.56917pt} \bm{u}(\cdot ,t) \hspace{0.56917pt} \Vert _{L^{2}(\mathbb {R}^{n})}$. The converse also holds, as shown by Skalák (J. Math. Fluid Mech. 16 (2014) 431–446) and by our analysis below, which uses a more straightforward argument. As an application of these results, we discuss the genericity problem of algebraic decay estimates for Leray solutions of the unforced Navier–Stokes equations. In particular, we prove that Leray solutions with algebraic decay generically satisfy two‐sided bounds of the form (1+t)−α≲∥u(·,t)∥L2(Rn)≲(1+t)−α$(1+t)^{-\alpha }\lesssim \Vert \bm{u}(\cdot ,t)\Vert _{L^2(\mathbb {R}^n)} \lesssim (1+t)^{-\alpha }$. [ABSTRACT FROM AUTHOR]

Given a collection of hypergraphs H=(H1,...,Hm)${{\bf H}}=(H_1, \ldots , H_m)$ with the same vertex set, an m$m$‐edge graph F⊂∪i∈[m]Hi$F\subset \cup _{i\in [m]}H_i$ is a transversal if there is a bijection ϕ:E(F)→[m]$\phi :E(F)\rightarrow [m]$ such that e∈E(Hϕ(e))$e\in E(H_{\phi (e)})$ for each e∈E(F)$e\in E(F)$. How large does the minimum degree of each Hi$H_i$ need to be so that H${{\bf H}}$ necessarily contains a copy of F$F$ that is a transversal? Each Hi$H_i$ in the collection could be the same hypergraph, hence the minimum degree of each Hi$H_i$ needs to be large enough to ensure that F⊆Hi$F\subseteq H_i$. Since its general introduction by Joos and Kim (Bull. Lond. Math. Soc. 52 (2020) 498–504), a growing body of work has shown that in many cases this lower bound is tight. In this paper, we give a unified approach to this problem by providing a widely applicable sufficient condition for this lower bound to be asymptotically tight. This is general enough to recover many previous results in the area and obtain novel transversal variants of several classical Dirac‐type results for (powers of) Hamilton cycles. For example, we derive that any collection of rn$rn$ graphs on an n$n$‐vertex set, each with minimum degree at least (r/(r+1)+o(1))n$(r/(r+1)+o(1))n$, contains a transversal copy of the r$r$th power of a Hamilton cycle. This can be viewed as a rainbow version of the Pósa–Seymour conjecture. [ABSTRACT FROM AUTHOR]

Let L=Δ−⟨∇ϕ,∇·⟩$L=\Delta -\langle \nabla \phi , \nabla \cdot \rangle$ be a symmetric diffusion operator with an invariant measure μ(dx)=e−ϕ(x)m(dx)$\mu (\mathrm{d}x)=e^{-\phi (x)}\mathfrak {m} (\mathrm{d}x)$ on a complete non‐compact smooth Riemannian manifold (M,g)$(M,g)$ with its volume element m=volg$\mathfrak {m} =\text{\rm vol}_g$, and ϕ∈C2(M)$\phi \in C^2(M)$ a potential function. In this paper, we prove a Laplacian comparison theorem on weighted complete Riemannian manifolds with CD(K,m)${\rm CD}(K, m)$‐condition for m⩽1$m\leqslant 1$ and a continuous function K$K$. As consequences, we give the optimal conditions on m$m$‐Bakry–Émery Ricci tensor for m⩽1$m\leqslant 1$ such that the (weighted) Myers' theorem, Bishop–Gromov volume comparison theorem, stochastic completeness and Feller property of L$L$‐diffusion processes hold on weighted complete Riemannian manifolds. Some of these results were well studied for m$m$‐Bakry–Émery Ricci curvature for m⩾n$m\geqslant n$ (Li, J. Math. Pures Appl. (9) 84 (2005), 1295–1361; Lott, Comment. Math. Helv. 78 (2003), 865–883; Qian, Q. J. Math. 48 (1987), 235–242; Wei and Wylie, J. Differential Geom. 83 (2009), 377–405) or m=1$m=1$ (Wylie, Trans. Amer. Math. Soc. 369 (2017), 6661–6681; Wylie and D. Yeroshkin, Preprint). When m<1$m<1$, our results are new in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider local smoothing estimates for the fractional Schrödinger operator eit(−Δ)α/2$e^{it(-\Delta)^{\alpha /2}}$ with α>1$\alpha >1$. Using the k$k$‐broad 'norm' estimates of Guth–Hickman–Iliopoulou (Acta Math. 223 (2019), 251–376), we improve the previously best‐known results of local smoothing estimates of Guo–Roos–Yung (arXiv:1710.10988 (2017)) and Rogers–Seeger (J. Für Die Reine und Ang. Math. (Crelles Journal), 640 (2010), 47–66). [ABSTRACT FROM AUTHOR]

Let σ=(σ1,σ2,⋯,σn)∈Sn−1 and dσ denote the normalized Lebesgue measure on Sn−1,n⩾2. For functions f1,f2,⋯,fn defined on R, consider the multilinear operator given by T(f1,f2,⋯,fn)(x)=∫Sn−1∏j=1nfj(x−σj)dσ,x∈R.In this paper, we obtain necessary and sufficient conditions on exponents p1,p2,⋯,pn and r for which the operator T is bounded from ∏j=1nLpj(R)→Lr(R), where 1⩽pj,r⩽∞,j=1,2,⋯,n. This generalizes the results obtained in (Bak and Shim, J. Funct. Anal. 157 (1998) 534–553; Oberlin, Trans. Amer. Math. Soc. 310 (1988) 821–835). [ABSTRACT FROM AUTHOR]

This note is a corrigendum to the author's paper "On a Problem of Lang for Matrix Polynomials, Bull. London Math. Soc. 54 (2022), 1552–1567". [ABSTRACT FROM AUTHOR]

For some special window functions ψα∈H2(C+)$\psi _{\alpha } \in H^2(\mathbb {C}^+)$, we prove that, over all sets Δ⊂C+$\Delta \subset \mathbb {C}^+$ of fixed hyperbolic measure ν(Δ)$\nu (\Delta)$, those for which the Wavelet transform Wψα$W_{\psi _{\alpha }}$ with window ψα$\psi _{\alpha }$ concentrates optimally are exactly the discs with respect to the pseudo‐hyperbolic metric of the upper half space. This answers a question raised by Abreu and Dörfler in Abreu and Dörfler (Inverse Problems 28 (2012) 16). Our techniques make use of a framework recently developed by Nicola and Tilli in Nicola and Tilli (Invent. Math. 230 (2022) 1–30), but in the hyperbolic context induced by the dilation symmetry of the Wavelet transform. This leads us naturally to use a hyperbolic rearrangement function, as well as the hyperbolic isoperimetric inequality, in our analysis. [ABSTRACT FROM AUTHOR]

In this article, motivated by the study of symplectic structures on manifolds with boundary and the systematic study of b$b$‐symplectic manifolds started in Guillemin, Miranda, and Pires Adv. Math. 264 (2014), 864–896, we prove a slice theorem for Lie group actions on b$b$‐symplectic manifolds. [ABSTRACT FROM AUTHOR]

Assume that HT ht is a bounded HT ht -bimodule, with index of nilpotency HT ht and projective dimension HT ht . The Theorem and its proof read as follows: 6.5 Theorem Let HT ht be an extension of HT ht -algebras and let HT ht be an HT ht -bimodule. [Extracted from the article]

We investigate a construction providing pairs of Calabi–Yau varieties described as zero loci of pushforwards of a hyperplane section on a roof as described in Kanemitsu (Preprint, arXiv:1812.05392, 2018). We discuss the implications of such construction at the level of Hodge equivalence, derived equivalence and L${\mathbb {L}}$‐equivalence. For the case of K3 surfaces, we provide alternative interpretations for the Fourier–Mukai duality in the family of K3 surfaces of degree 12 of (New trends in algebraic geometry (Warwick, 1996), Lond. Math. Soc. Lecture Note Ser., vol. 264, Cambridge Univ. Press, Cambridge, 1999, pp. 311–326). In all these constructions, the derived equivalence lifts to an equivalence of matrix factorizations categories. [ABSTRACT FROM AUTHOR]

Oeljeklaus–Toma (OT) manifolds are higher dimensional analogues of Inoue‐Bombieri surfaces and their construction is associated to a finite extension K$K$ of Q$\mathbb {Q}$ and a subgroup of units U$U$. We characterize the existence of pluriclosed metrics (also known as strongly Kähler with torsion (SKT) metrics) on any OT manifold X(K,U)$X(K, U)$ purely in terms of number‐theoretical conditions, yielding restrictions on the third Betti number b3$b_3$ and the Dolbeault cohomology group H∂¯2,1$H^{2,1}_{\overline{\partial }}$. Combined with the main result in (Dubickas, Results Math. 76 (2021), 78), these numerical conditions render explicit examples of pluriclosed OT manifolds in arbitrary complex dimension. We prove that in complex dimension 4 and type (2,2)$(2, 2)$, the existence of a pluriclosed metric on X(K,U)$X(K, U)$ is entirely topological, namely, it is equivalent to b3=2$b_3=2$. Moreover, we provide an explicit example of an OT manifold of complex dimension 4 carrying a pluriclosed metric. Finally, we show that no OT manifold admits balanced metrics, but all of them carry instead locally conformally balanced metrics. [ABSTRACT FROM AUTHOR]

We give a criterion that simplifies the checking of the inductive Alperin weight condition for the remaining open cases of simple groups of Lie type (see Theorem 3.3). It is strongly related in form to the criterion of the second author for the inductive McKay conditions (see Späth, Bull. Lond. Math. Soc. 44 (2012), no. 3, 426–438, 2.12) that has proved very useful. The proof follows from a Clifford theory for weights intrinsically present in the proof of reduction theorems of the Alperin weight conjecture given by Navarro–Tiep and the second author. We also give a related criterion for the inductive blockwise Alperin weight condition (Theorem 4.5). [ABSTRACT FROM AUTHOR]

Let X$X$ be a smooth projective complex variety and let D=D1+⋯+Dl$D=D_1+\cdots +D_l$ be a reduced normal crossing divisor on X$X$ with each component Dj$D_j$ smooth, irreducible and numerically effective. The log–local principle put forward in van Garrel et al. (Adv. Math. 350 (2019) 860–876) conjectures that the genus 0 log Gromov–Witten theory of maximal tangency of (X,D)$(X,D)$ is equivalent to the genus 0 local Gromov–Witten theory of X$X$ twisted by ⨁j=1lO(−Dj)$\bigoplus _{j=1}^{l}\mathcal {O}(-D_j)$. We prove that an extension of the log–local principle holds for X$X$ a (not necessarily smooth) Q$\mathbb {Q}$‐factorial projective toric variety, D$D$ the toric boundary, and descendant point insertions. [ABSTRACT FROM AUTHOR]

Let S(t) denote the argument of the Riemann zeta‐function, defined as S(t)=1πImlogζ(1/2+it).Assuming the Riemann hypothesis, we prove that S(t)=Ω±logtlogloglogtloglogt.This improves the classical Ω‐results of Montgomery (Theorem 2; Comment. Math. Helv. 52 (1977) 511–518) and matches with the Ω‐result obtained by Bondarenko and Seip (Theorem 2; Math. Ann. 372 (2018) 999–1015). [ABSTRACT FROM AUTHOR]

A matrix A is totally positive (or non‐negative) of order k, denoted TPk (or TNk), if all minors of size ⩽k are positive (or non‐negative). It is well known that such matrices are characterized by the variation diminishing property together with the sign non‐reversal property. We do away with the former, and show that A is TPk if and only if every submatrix formed from at most k consecutive rows and columns has the sign non‐reversal property. In fact, this can be strengthened to only consider test vectors in Rk with alternating signs. We also show a similar characterization for all TNk matrices — more strongly, both of these characterizations use a single vector (with alternating signs) for each square submatrix. These characterizations are novel, and similar in spirit to the fundamental results characterizing TP matrices by Gantmacher–Krein (Compos. Math. 4 (1937) 445–476) and P‐matrices by Gale–Nikaido (Math. Ann. 159 (1965) 81–93). As an application, we study the interval hull I(A,B) of two m×n matrices A=(aij) and B=(bij). This is the collection of C∈Rm×n such that each cij is between aij and bij. Using the sign non‐reversal property, we identify a two‐element subset of I(A,B) that detects the TPk property for all of I(A,B) for arbitrary k⩾1. In particular, this provides a test for total positivity (of any order), simultaneously for an entire class of rectangular matrices. In parallel, we also provide a finite set to test the total non‐negativity (of any order) of an interval hull I(A,B). [ABSTRACT FROM AUTHOR]

Let D be the unit disk. Kutzschebauch and Studer (Bull. Lond. Math. Soc. 51 (2019) 995–1004) recently proved that, for each continuous map A:D¯→SL(2,C), which is holomorphic in D, there exist continuous maps E,F:D¯→sl(2,C), which are holomorphic in D, such that A=eEeF. Also they asked if this extends to arbitrary compact bordered Riemann surfaces. We prove that this is possible. [ABSTRACT FROM AUTHOR]

These cubes HT ht cover HT ht (up to at most a zero-measure set). Lemma 1 follows as before with only minor modifications due to extending HT ht by 0 to all of HT ht . Another option is to use Nash's embedding theorem and consider the HT ht -dimensional compact manifold HT ht as isometrically embedded in HT ht . [Extracted from the article]

We continue a study of separable equivalence from (Kadison, Comptes Rendus Math. Reports Acad. Sci. Canada, 15 (1993) 223–228; Hokkaido Math. J. 24 (1995) 527–549). We prove that symmetric separable equivalent rings A and B are linked by a Frobenius bimodule APB such that A is P‐separable over B. Separably equivalent rings are linked by a biseparable bimodule P. In addition, the ring monomorphism A↪EndPB is split, separable Frobenius. It is observed that left and right finite projective bimodules over symmetric algebras are Frobenius bimodules; twisted by the Nakayama automorphisms if over Frobenius algebras. [ABSTRACT FROM AUTHOR]

We give sufficient conditions for a satellite knot to admit an L-space surgery, and use this result to give new infinite families of patterns that produce satellite L-space knots. [ABSTRACT FROM AUTHOR]

We prove that the range of exponents in the general L2 Fourier restriction theorem due to Mockenhaupt, Mitsis, Bak, and Seeger is sharp for a large class of measures on Rd. This extends to higher dimensions the sharpness result of Hambrook and Łaba. [ABSTRACT FROM AUTHOR]