Authors:KEVIN AGUYAR BRIX; TOKE MEIER CARLSEN Pages: 289 - 298 Abstract: A one-sided shift of finite type $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines on the one hand a Cuntz–Krieger algebra ${\mathcal{O}}_{A}$ with a distinguished abelian subalgebra ${\mathcal{D}}_{A}$ and a certain completely positive map $\unicode[STIX]{x1D70F}_{A}$ on ${\mathcal{O}}_{A}$. On the other hand, $(\mathsf{X}_{A},\unicode[STIX]{x1D70E}_{A})$ determines a groupoid ${\mathcal{G}}_{A}$ together with a certain homomorphism $\unicode[STIX]{x1D716}_{A}$ on ${\mathcal{G}}_{A}$. We show that each of these two sets of data completely characterizes the one-sided conjugacy class of $\mathsf{X}_{A}$. This strengthens a result of Cuntz and Krieger. We also exhibit an example of two irreducible shifts of finite type which are eventually conjugate but not conjugate. This provides a negative answer to a question of Matsumoto of whether eventual conjugacy implies conjugacy. PubDate: 2020-12-01T00:00:00.000Z DOI: 10.1017/S1446788719000168 Issue No:Vol. 109, No. 3 (2020)
Authors:DANIEL GONÇALVES; DANILO ROYER Pages: 299 - 319 Abstract: We realize Leavitt ultragraph path algebras as partial skew group rings. Using this realization we characterize artinian ultragraph path algebras and give simplicity criteria for these algebras. PubDate: 2020-12-01T00:00:00.000Z DOI: 10.1017/S144678871900020X Issue No:Vol. 109, No. 3 (2020)
Authors:GUOEN HU Pages: 320 - 339 Abstract: Let $T_{1}$, $T_{2}$ be two Calderón–Zygmund operators and $T_{1,b}$ be the commutator of $T_{1}$ with symbol $b\in \text{BMO}(\mathbb{R}^{n})$. In this paper, by establishing new bilinear sparse dominations and a new weighted estimate for bilinear sparse operators, we prove that the composite operator $T_{1}T_{2}$ satisfies the following estimate: for $\unicode[STIX]{x1D706}>0$ and weight $w\in A_{1}(\mathbb{R}^{n})$, $$\begin{eqnarray}\displaystyle & & \displaystyle w(\{x\in \mathbb{R}^{n}:\,|T_{1}T_{2}f(x)|>\unicode[STIX]{x1D706}\})\nonumber\\ \displaystyle & & \displaystyle \qquad \lesssim [w]_{A_{1}}[w]_{A_{\infty }}\log (\text{e}+[w]_{A_{\infty }})\int _{\mathbb{R}^{n}}\frac{|f(x)|}{\unicode[STIX]{x1D706}}\log \bigg(\text{e}+\frac{|f(x)|}{\unicode[STIX]{x1D706}}\bigg)w(x)\,dx,\nonumber\end{eqnarray}$$ while the composite operator $T_{1,b}T_{2}$ satisfies $$\begin{eqnarray}\displaystyle & & \displaystyle w(\{x\in \mathbb{R}^{n}:\,|T_{1,b}T_{2}f(x)|>\unicode[STIX]{x... PubDate: 2020-12-01T00:00:00.000Z DOI: 10.1017/S1446788719000107 Issue No:Vol. 109, No. 3 (2020)
Authors:E. I. KHUKHRO; P. SHUMYATSKY, G. TRAUSTASON Pages: 340 - 350 Abstract: Let $g$ be an element of a finite group $G$ and let $R_{n}(g)$ be the subgroup generated by all the right Engel values $[g,_{n}x]$ over $x\in G$. In the case when $G$ is soluble we prove that if, for some $n$, the Fitting height of $R_{n}(g)$ is equal to $k$, then $g$ belongs to the $(k+1)$th Fitting subgroup $F_{k+1}(G)$. For nonsoluble PubDate: 2020-12-01T00:00:00.000Z DOI: 10.1017/S1446788719000181 Issue No:Vol. 109, No. 3 (2020)
Authors:ALESSANDRO LANGUASCO; ALESSANDRO ZACCAGNINI Pages: 351 - 370 Abstract: We improve some results in our paper [A. Languasco and A. Zaccagnini, ‘Short intervals asymptotic formulae for binary problems with prime powers’, J. Théor. Nombres Bordeaux 30 (2018) 609–635] about the asymptotic formulae in short intervals for the average number of representations of integers of the forms $n=p_{1}^{\ell _{1}}+p_{2}^{\ell _{2}}$ and $n=p^{\ell _{1}}+m^{\ell _{2}}$, where $\ell _{1},\ell _{2}\geq 2$ are fixed integers, $p,p_{1},p_{2}$ are prime numbers and $m$ is an integer. We also remark that the techniques here used let us prove that a suitable asymptotic formula for the average number of representations of integers $n=\sum _{i=1}^{s}p_{i}^{\ell }$, where $s$, $\ell$ are two integers such that $2\leq s\leq \ell -1$, $\ell \geq 3$ and $p_{i}$, PubDate: 2020-12-01T00:00:00.000Z DOI: 10.1017/S1446788719000120 Issue No:Vol. 109, No. 3 (2020)
Authors:LAURENŢIU G. MAXIM Pages: 371 - 415 Abstract: Vanishing cycles, introduced over half a century ago, are a fundamental tool for studying the topology of complex hypersurface singularity germs, as well as the change in topology of a degenerating family of projective manifolds. More recently, vanishing cycles have found deep applications in enumerative geometry, representation theory, applied algebraic geometry, birational geometry, etc. In this survey, we introduce vanishing cycles from a topological perspective and discuss some of their applications. PubDate: 2020-12-01T00:00:00.000Z DOI: 10.1017/S1446788720000403 Issue No:Vol. 109, No. 3 (2020)
Authors:J. TYAGI; R. B. VERMA Pages: 416 - 430 Abstract: In this article, we establish a Lyapunov-type inequality for the following extremal Pucci’s equation: $$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}{\mathcal{M}}_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC}}^{+}(D^{2}u)+b(x)|Du|+a(x)u=0 & \text{in}~\unicode[STIX]{x1D6FA},\\ u=0 & \text{on}~\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$ where $\unicode[STIX]{x1D6FA}$ is a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq 2$. This work generalizes the well-known works on the Lyapunov inequality for extremal Pucci’s equations with gradient nonlinearity. PubDate: 2020-12-01T00:00:00.000Z DOI: 10.1017/S1446788719000569 Issue No:Vol. 109, No. 3 (2020)