Search

For any ring A$A$ and a small, pre‐additive, Hom‐finite, and locally bounded category Q$Q$ that has a Serre functor and satisfies the (strong) retraction property, we show that the category of additive functors Q→AMod$Q \rightarrow {}_{A} \operatorname{Mod}$ has a projective and an injective model structure. These model structures have the same trivial objects and weak equivalences, which in most cases can be naturally characterized in terms of certain (co)homology functors introduced in this paper. The associated homotopy category, which is triangulated, is called the Q$Q$‐shaped derived category of A$A$. The usual derived category of A$A$ is one example; more general examples arise by taking Q$Q$ to be the mesh category of a suitably nice stable translation quiver. This paper builds upon, and generalizes, works of Enochs, Estrada, and García‐Rozas (Math. Nachr. 281 (2008), no. 4, 525–540) and Dell'Ambrogio, Stevenson, and Šťovíček (Math. Z. 287 (2017), no. 3‐4, 1109–1155). [ABSTRACT FROM AUTHOR]

In this paper, we push forward the conjecture on a noncommutative version of the maximal ergodic theorem for positive contractions due to Ackoglu. That is, we establish the one‐sided maximal ergodic inequalities for a large subclass of positive operators on noncommutative Lp$L_p$‐spaces for a fixed 1

Given k⩾2$k\geqslant 2$ and two k$k$‐graphs (k$k$‐uniform hypergraphs) F$F$ and H$H$, an F$F$‐factor in H$H$ is a set of vertex‐disjoint copies of F$F$ that together cover the vertex set of H$H$. Lenz and Mubayi [J. Combin. Theory Ser. B, 2016] studied the F$F$‐factor problem in quasi‐random k$k$‐graphs with minimum degree Ω(nk−1)$\Omega (n^{k-1})$. They posed the problem of characterizing the k$k$‐graphs F$F$ such that every sufficiently large quasi‐random k$k$‐graph with constant edge density and minimum degree Ω(nk−1)$\Omega (n^{k-1})$ contains an F$F$‐factor, and, in particular, they showed that all linear k$k$‐graphs satisfy this property. In this paper we prove a general theorem on F$F$‐factors which reduces the F$F$‐factor problem of Lenz and Mubayi to a natural sub‐problem, that is, the F$F$‐cover problem. By using this result, we answer the question of Lenz and Mubayi for those F$F$ which are k$k$‐partite k$k$‐graphs, and for all 3‐graphs F$F$, separately. Our characterization result on 3‐graphs is motivated by the recent work of Reiher, Rödl, and Schacht [J. Lond. Math. Soc., 2018] that classifies the 3‐graphs with vanishing Turán density in quasi‐random k$k$‐graphs. [ABSTRACT FROM AUTHOR]

Harron and Snowden (J. reine angew. Math. 729 (2017), 151–170) counted the number of elliptic curves over Q$\mathbb {Q}$ up to height X$X$ with torsion group G$G$ for each possible torsion group G$G$ over Q$\mathbb {Q}$. In this paper, we generalize their result to all number fields and all level structures G$G$ such that the corresponding modular curve XG$X_G$ is a weighted projective line P(w0,w1)$\mathbb {P}(w_0,w_1)$ and the morphism XG→X(1)$X_G\rightarrow X(1)$ satisfies a certain condition. In particular, this includes all modular curves X1(m,n)$X_1(m,n)$ with coarse moduli space of genus 0. We prove our results by defining a size function on P(w0,w1)$\mathbb {P}(w_0,w_1)$ following unpublished work of Deng (Preprint, https://arxiv.org/abs/math/9812082), and working out how to count the number of points on P(w0,w1)$\mathbb {P}(w_0,w_1)$ up to size X$X$. [ABSTRACT FROM AUTHOR]

In this paper, we provide an algorithm to estimate from below the dimension of self‐similar measures with overlaps. As an application, we show that for any β∈(1,2)$ \beta \in (1,2)$, the dimension of the Bernoulli convolution μβ$ \mu _{\beta }$ satisfies dim(μβ)⩾0.98040856,\begin{equation*}\hskip7pc \dim (\mu _{\beta })\geqslant 0.98040856,\hskip-7pc\vspace*{-6pt} \end{equation*}which improves a previous uniform lower bound 0.82 obtained by Hare and Sidorov (Exp. Math. 27 (2018), no. 4, 414–418). This new uniform lower bound is very close to the known numerical approximation 0.98040931953±10−11$ 0.98040931953\pm 10^{-11}$ for dimμβ3$\dim \mu _{\beta _3}$, where β3≈1.839286755214161$ \beta _{3} \approx 1.839286755214161$ is the largest root of the polynomial x3−x2−x−1$ x^{3}-x^{2}-x-1$. Moreover, the infimum infβ∈(1,2)dim(μβ)$\inf _{\beta \in (1,2)}\dim (\mu _{\beta })$ is attained at a parameter β∗$\beta _*$ in a small interval (β3−10−8,β3+10−8).\begin{equation*}\hskip7pc (\beta _{3} -10^{-8}, \beta _{3} + 10^{-8}).\hskip-7pc \end{equation*}When β$\beta$ is a Pisot number, we express dim(μβ)$\dim (\mu _{\beta })$ in terms of the measure‐theoretic entropy of the equilibrium measure for certain matrix pressure function, and present an algorithm to estimate dim(μβ)$\dim (\mu _\beta)$ from above as well. [ABSTRACT FROM AUTHOR]

In a short fragment published for the first time with his lost notebook in 1988 (Ramanujan, The lost notebook and other unpublished papers (Narosa, New Delhi, 1988)), Ramanujan offered two beautiful identities, associated, respectively, with the classical circle and divisor problems. In fact, they are analogues, with an additional variable, but not generalizations, of classical identities associated with these two famous problems. After Ramanujan's death in 1920, the lost notebook and fragments of papers of Ramanujan were sent to Hardy. We do not have any official record of what was included in this mailing, but it is likely that the aforementioned fragment was included in this parcel. If so, then it is possible that Ramanujan wrote it at the end of his life in either 1919 or 1920. On the other hand, from a paper that Hardy published in 1915 (Hardy, Quart. J. Math. (Oxford) 46 (1915) 263–283) on the circle problem, we are aware that by early in his stay in England, Ramanujan had a strong interest in these problems, and so the fragment may emanate from this period. Two of the present authors and Kim published a proof of the identity connected with the circle problem in 2013 (Berndt, Kim and Zaharescu, Adv. Math. 236 (2013) 24–59). In this paper, a proof of the second identity is given for the first time. [ABSTRACT FROM AUTHOR]

In this paper, we establish a weak Serrin‐type blowup criterion for the Cauchy problem of the three‐dimensional (3D) compressible barotropic Navier–Stokes equations in the whole space. It shows that the strong or smooth solution exists globally if the velocity satisfies the weak Serrin's condition and the L∞(0,T;Lα0)‐norm of the density is bounded, where α0 is positive constant. Therefore, if the weak Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanish or vacuum appears in the non‐vacuum region or even milder singularities) to form before the density becomes unbounded. Furthermore, the initial data can be arbitrarily large and contain vacuum states. The proof is based on the new a priori estimates for 3D compressible Navier–Stokes equations. In particular, this extends the corresponding Huang et al.'s results (SIAM J. Math. Anal. 43 (2011) 1872–1886). [ABSTRACT FROM AUTHOR]

Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. These two metrics are in general nonequivalent up to bilipschitz homeomorphism. We give a necessary and sufficient condition for a normal surface singularity to be Lipschitz normally embedded (LNE), that is, to have bilipschitz equivalent outer and inner metrics. In a partner paper (Neumann, Pedersen and Pichon, J. London Math. Soc. (2020), https://doi.org/10.1112/jlms.12280), we apply it to prove that rational surface singularities are LNE if and only if they are minimal. [ABSTRACT FROM AUTHOR]

Given an iterated function system (IFS) of contractive similitudes, the theory of Gromov hyperbolic graph on the IFS has been established recently. In the paper, we introduce a notion of simple augmented tree which is a Gromov hyperbolic graph. By generalizing a combinatorial device of rearrangeable matrix, we show that there exists a near‐isometry between the simple augmented tree and the symbolic space of the IFS, so that their hyperbolic boundaries are Lipschitz equivalent. We then apply this to consider the Lipschitz equivalence of self‐similar sets with or without assuming the open set condition. Moreover, we also provide a criterion for a self‐similar set to be a Cantor‐type set which completely answers an open question raised in [Luo and Lau, Adv. Math. 235 (2013) 555–579]. Our study extends the previous works. [ABSTRACT FROM AUTHOR]