Authors:HOSSEIN ABBASPOUR; FRIEDRICH WAGEMANN Pages: 145 - 174 Abstract: We construct a cycle in higher Hochschild homology associated to the two-dimensional torus which represents 2-holonomy of a nonabelian gerbe in the same way as the ordinary holonomy of a principal G-bundle gives rise to a cycle in ordinary Hochschild homology. This is done using the connection 1-form of Baez–Schreiber. A crucial ingredient in our work is the possibility to arrange that in the structure crossed module $\unicode[STIX]{x1D707}:\mathfrak{h}\rightarrow \mathfrak{g}$ of the principal 2-bundle, the Lie algebra $\mathfrak{h}$ is abelian, up to equivalence of crossed modules. PubDate: 2021-04-01T00:00:00.000Z DOI: 10.1017/S1446788719000314 Issue No:Vol. 110, No. 2 (2021)
Authors:DAVID ANGELES; JASON LO, COURTNEY M. VAN DER LINDEN Pages: 175 - 193 Abstract: We give a list of statements on the geometry of elliptic threefolds phrased only in the language of topology and homological algebra. Using only notions from topology and homological algebra, we recover existing results and prove new results on torsion pairs in the category of coherent sheaves on an elliptic threefold. PubDate: 2021-04-01T00:00:00.000Z DOI: 10.1017/S144678872000018X Issue No:Vol. 110, No. 2 (2021)
Authors:JAESEONG HEO; SOORAN KANG, YONGDO LIM Pages: 194 - 219 Abstract: The aim of this paper is to study the heat kernel and the jump kernel of the Dirichlet form associated to the ultrametric Cantor set $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ that is the infinite path space of the stationary $k$-Bratteli diagram ${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$, where $\unicode[STIX]{x1D6EC}$ is a finite strongly connected $k$-graph. The Dirichlet form which we are interested in is induced by an even spectral triple $(C_{\operatorname{Lip}}(\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}),\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}},{\mathcal{H}},D,\unicode[STIX]{x1D6E4})$ and is given by $$\begin{eqnarray}Q_{s}(f,g)=\frac{1}{2}\int _{\unicode[STIX]{x1D6EF}}\operatorname{Tr}(|D|^{-s}[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(f)]^{\ast }[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(g)])\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D719}),\end{eqnarray}$$ where $\unicode[STIX]{x1D6EF}$ is the space of choice functions on $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}\times \unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$. There are two ultrametrics, $d^{(s)}$ and PubDate: 2021-04-01T00:00:00.000Z DOI: 10.1017/S1446788719000429 Issue No:Vol. 110, No. 2 (2021)
Authors:HADI KHATIBZADEH; VAHID MOHEBBI Pages: 220 - 242 Abstract: As a continuation of previous work of the first author with Ranjbar [‘A variational inequality in complete CAT(0) spaces’, J. Fixed Point Theory Appl. 17 (2015), 557–574] on a special form of variational inequalities in Hadamard spaces, in this paper we study equilibrium problems in Hadamard spaces, which extend variational inequalities and many other problems in nonlinear analysis. In this paper, first we study the existence of solutions of equilibrium problems associated with pseudo-monotone bifunctions with suitable conditions on the bifunctions in Hadamard spaces. Then, to approximate an equilibrium point, we consider the proximal point algorithm for pseudo-monotone bifunctions. We prove existence of the sequence generated by the algorithm in several cases in Hadamard spaces. Next, we introduce the resolvent of a bifunction in Hadamard spaces. We prove convergence of the resolvent to an equilibrium point. We also prove $\triangle$-convergence of the sequence generated by the proximal point algorithm to an equilibrium point of the pseudo-monotone bifunction and also the strong convergence under additional assumptions on the bifunction. Finally, we study a regularization of Halpern type and prove the strong convergence of the generated sequence to an equilibrium point without any additional assumption on the pseudo-monotone bifunction. Some examples in fixed point theory and convex minimization are also presented. PubDate: 2021-04-01T00:00:00.000Z DOI: 10.1017/S1446788719000041 Issue No:Vol. 110, No. 2 (2021)
C1+VECTOR+FIELDS&rft.title=Journal+of+the+Australian+Mathematical+Society&rft.issn=1446-7887&rft.date=2021&rft.volume=110&rft.spage=243&rft.epage=259&rft.aulast=LEE&rft.aufirst=MANSEOB&rft.au=MANSEOB+LEE&rft.au=LE+HUY+TIEN&rft_id=info:doi/10.1017/S1446788720000415">CHAIN COMPONENTS WITH THE STABLE SHADOWING PROPERTY FOR C1 VECTOR FIELDS
Authors:MANSEOB LEE; LE HUY TIEN Pages: 243 - 259 Abstract: Let M be a closed n-dimensional smooth Riemannian manifold, and let X be a $C^1$-vector field of $M.$ Let $\gamma $ be a hyperbolic closed orbit of $X.$ In this paper, we show that X has the $C^1$-stably shadowing property on the chain component $C_X(\gamma )$ if and only if $C_X(\gamma )$ is the hyperbolic homoclinic class. PubDate: 2021-04-01T00:00:00.000Z DOI: 10.1017/S1446788720000415 Issue No:Vol. 110, No. 2 (2021)
$\widehat{{\mathcal{A}}}$&rft.title=Journal+of+the+Australian+Mathematical+Society&rft.issn=1446-7887&rft.date=2021&rft.volume=110&rft.spage=260&rft.epage=265&rft.aulast=MURAHARA&rft.aufirst=HIDEKI&rft.au=HIDEKI+MURAHARA&rft.au=TOMOKAZU+ONOZUKA&rft_id=info:doi/10.1017/S1446788719000533">DERIVATION RELATION FOR FINITE MULTIPLE ZETA VALUES IN $\widehat{{\mathcal{A}}}$
Authors:HIDEKI MURAHARA; TOMOKAZU ONOZUKA Pages: 260 - 265 Abstract: Ihara et al. proved the derivation relation for multiple zeta values. The first-named author obtained its counterpart for finite multiple zeta values in ${\mathcal{A}}$. In this paper, we present its generalization in $\widehat{{\mathcal{A}}}$. PubDate: 2021-04-01T00:00:00.000Z DOI: 10.1017/S1446788719000533 Issue No:Vol. 110, No. 2 (2021)
Authors:HANS-OLAV TYLLI; HENRIK WIRZENIUS Pages: 266 - 288 Abstract: We initiate a study of structural properties of the quotient algebra ${\mathcal{K}}(X)/{\mathcal{A}}(X)$ of the compact-by-approximable operators on Banach spaces $X$ failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from $c_{0}$ into ${\mathcal{K}}(Z)/{\mathcal{A}}(Z)$, where $Z$ belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a nonseparable space $c_{0}(\unicode[STIX]{x1D6E4})$ into ${\mathcal{K}}(Z_{FJ})/{\mathcal{A}}(Z_{FJ})$, where $Z_{FJ}$ is a universal compact factorisation space arising from the work of Johnson and Figiel. PubDate: 2021-04-01T00:00:00.000Z DOI: 10.1017/S1446788719000211 Issue No:Vol. 110, No. 2 (2021)