Every finite group $G$ acts on some nonorientable unbordered surfaces. The minimal topological genus of those surfaces is called the symmetric crosscap number of $G$. It is known that 3 is not the symmetric crosscap number of any group but it remains unknown whether there are other such values, called gaps. In this paper we obtain group presentations which allow one to find the actions realizing the symmetric crosscap number of groups of each group of order less than or equal to 63.

Let $c\unicode[STIX]{x1D719}_{k}(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. Recently, new Ramanujan-type congruences associated with $c\unicode[STIX]{x1D719}_{4}(n)$ were discovered. In this article, we discuss two approaches in proving such congruences using the theory of modular forms. Our methods allow us to prove congruences such as $c\unicode[STIX]{x1D719}_{4}(14n+6)\equiv 0\;\text{mod}\;7$ and Seller’s congruence $c\unicode[STIX]{x1D719}_{4}(10n+6)\equiv 0\;\text{mod}\;5$.

In this paper, we consider contact metric three-manifolds $(M;\unicode[STIX]{x1D702},g,\unicode[STIX]{x1D711},\unicode[STIX]{x1D709})$ which satisfy the condition $\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x1D709}}h=\unicode[STIX]{x1D707}h\unicode[STIX]{x1D711}+\unicode[STIX]{x1D708}h$ for some smooth functions $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D708}$, where $2h=\unicode[STIX]{x00A3}_{\unicode[STIX]{x1D709}}\unicode[STIX]{x1D711}$. We prove that if $M$ is conformally flat and if $\unicode[STIX]{x1D707}$ is constant, then $M$ is either a flat manifold or a Sasakian manifold of constant curvature $+1$. We cannot extend this result for a smooth function $\unicode[STIX]{x1D707}$. Indeed, we give such an example of a conformally flat contact three-manifold which is not of constant curvature.

We study the problem of extending a state on an abelian $C^{\ast }$-subalgebra to a tracial state on the ambient $C^{\ast }$-algebra. We propose an approach that is well suited to the case of regular inclusions, in which there is a large supply of normalizers of the subalgebra. Conditional expectations onto the subalgebra give natural extensions of a state to the ambient $C^{\ast }$-algebra; we prove that these extensions are tracial states if and only if certain invariance properties of both the state and conditional expectations are satisfied. In the example of a groupoid $C^{\ast }$-algebra, these invariance properties correspond to invariance of associated measures on the unit space under the action of bisections. Using our framework, we are able to completely describe the tracial state space of a Cuntz–Krieger graph algebra. Along the way we introduce certain operations called graph tightenings, which both streamline our description and provide connections to related finiteness questions in graph $C^{\ast }$-algebras. Our investigation has close connections with the so-called unique state extension property and its variants.

We attempt to discuss a new circle problem. Let $\unicode[STIX]{x1D701}(s)$ denote the Riemann zeta-function $\sum _{n=1}^{\infty }n^{-s}$ ($\text{Re}\,s>1$) and $L(s,\unicode[STIX]{x1D712}_{4})$ the Dirichlet $L$-function $\sum _{n=1}^{\infty }\unicode[STIX]{x1D712}_{4}(n)n^{-s}$ ($\text{Re}\,s>1$) with the primitive Dirichlet character mod 4. We shall define an arithmetical function $R_{(1,1)}(n)$ by the coefficient of the Dirichlet series $\unicode[STIX]{x1D701}^{\prime }(s)L^{\prime }(s,\unicode[STIX]{x1D712}_{4})=\sum _{n=1}^{\infty }R_{(1,1)}(n)n^{-s}$$(\text{Re}\,s>1)$. This is an analogue of $r(n)/4=\sum _{d|n}\unicode[STIX]{x1D712}_{4}(d)$. In the circle problem, there are many researches of estimations and related topics on the error term in the asymptotic formula for $\sum _{n\leq x}r(n)$. As a new problem, we deduce a ‘truncated Voronoï formula’ for the error term in the asymptotic formula for $\sum _{n\leq x}R_{(1,1)}(n)$. As a direct application, we show the mean square for the error term in our new problem.

Let $(X,d,\unicode[STIX]{x1D707})$ be a metric measure space endowed with a distance $d$ and a nonnegative, Borel, doubling measure $\unicode[STIX]{x1D707}$. Let $L$ be a nonnegative self-adjoint operator on $L^{2}(X)$. Assume that the (heat) kernel associated to the semigroup $e^{-tL}$ satisfies a Gaussian upper bound. In this paper, we prove that for any $p\in (0,\infty )$ and $w\in A_{\infty }$, the weighted Hardy space $H_{L,S,w}^{p}(X)$ associated with $L$ in terms of the Lusin (area) function and the weighted Hardy space $H_{L,G,w}^{p}(X)$ associated with $L$ in terms of the Littlewood–Paley function coincide and their norms are equivalent. This improves a recent result of Duong et al. [‘A Littlewood–Paley type decomposition and weighted Hardy spaces associated with operators’, J. Geom. Anal.26 (2016), 1617–1646], who proved that $H_{L,S,w}^{p}(X)=H_{L,G,w}^{p}(X)$ for $p\in (0,1]$ and $w\in A_{\infty }$ by imposing an extra assumption of a Moser-type boundedness condition on $L$. Our result is new even in the unweighted setting, that is, when $w\equiv 1$.

Let $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a nonhomogeneous metric measure space satisfying the so-called upper doubling and the geometric doubling conditions. In this paper, the authors give the natural definition of the generalized Morrey spaces on $({\mathcal{X}},d,\unicode[STIX]{x1D707})$, and then investigate some properties of the maximal operator, the fractional integral operator and its commutator, and the Marcinkiewicz integral operator.

The Birkhoff orthogonality has been recently intensively studied in connection with the geometry of Banach spaces and operator theory. The main aim of this paper is to characterize the Birkhoff orthogonality in ${\mathcal{L}}(X;Y)$ under the assumption that ${\mathcal{K}}(X;Y)$ is an $M$-ideal in ${\mathcal{L}}(X;Y)$. Moreover, we survey the known results, as well as giving some new and more general ones. Furthermore, we characterize an approximate Birkhoff orthogonality in ${\mathcal{K}}(X;Y)$.