In 2011 Kenessey and Wengenroth gave a full description of closed range composition operators $C_\psi:C^{\infty}(\mathbb{R}^d)\toC^{\infty}(\mathbb{R})$, $F\mapsto F\circ\psi$, corresponding to smooth injective symbols $\psi:\mathbb{R}\to\mathbb{R}^d$. In 2012 Przestacki gave a sufficient condition for the range of $C_\psi$ to be closed in case if $\psi:\mathbb{R}\to\mathbb{R}$ is a smooth not necessarily injective symbol. Using their ideas we give a sufficient condition ensuring that the range of $C_\psi$ is closed when $\psi:\mathbb{R}\to\mathbb{R}^d$ is a smooth not necessarily injective function.

The notion of Johnson pseudo-contractibility for Banach algebras is introduced. We investigate this notion for Banach algebras defined on locally compact groups. For a compact metric space $X$ and $\alpha>0$, we show that the Lipschitz algebra $Lip_{\alpha}(X)$ is Johnson pseudo-contractible if and only if $X$ is finite. We give some examples to distinguish our new notion with the classical ones.

We investigate the existence of closed $G_2$-structures which are solitons for the Laplacian flow on nilpotent Lie groups. We obtain that seven of the twelve Lie algebras admitting a closed $G_2$-structure do admit a Laplacian soliton. Moreover, one of them admits a continuous family of Laplacian solitons which are pairwise non-homothetic and the Laplacian flow evolution on four of the Lie groups is not diagonal.

Given $t$ distinct colors, we order the $t$ subsets of $t-1$ colors in some arbitrary manner. Let $G_1, G_2, \ldots , G_t$ be graphs. The $(t-1)$-chromatic Ramsey number, denoted by $r_{t-1}^t(G_1, G_2, \ldots, G_t)$, is defined to be the least number $n$ such that if the edges of the complete graph $K_n$ are colored in any fashion with $t$ colors, then for some $i$ the subgraph whose edges are colored with the $i$th subset of colors contains a $G_i$. In this paper, we find the value of $r_4^5(G_1, \ldots, G_5)$ when each $G_i$ is a path.

In this paper, we characterize disk-cyclic and codisk-cyclic weighted pseudo-shifts on Banach sequence spaces, and consider the bilateral operator weighted shifts on $\ell^2(\mathbb{Z,\mathcal{K}})$ as a special case. Moreover, we present a counter-example to show that a result in [Y. X. Liang and Z. H. Zhou], Disk-cyclicity and Codisk-cyclicity of certain shift operators, Operators and Matrices, \textbf{9}(2015), 831--846] is not correct.

In this paper, we study a uniqueness question of meromorphic functions concerning certain linear differential polynomials that share a nonzero finite value with the same of L-functions. The results in this paper extend the corresponding results from Li[6] and Li & Li[7].

Let $\Omega$ be a Jordan domain in the complex plane whose boundary is piecewise analytic, and let $A(\Omega )$ be the algebra of all holomorphic functions on $\Omega$ that are continuous up to the boundary. We prove the existence of dense linear subspaces and of infinitely generated subalgebras in $A(\Omega )$ all of whose nonzero members are, in a strong sense, not differentiable at almost any point of the boundary. We also obtain infinite-dimensional closed subspaces consisting of functions that are not differentiable at any point of a dense subset of the boundary. In the case of the unit disc, those dense linear subspaces can be found with their functions being nowhere differentiable in the unit circle.

We record various properties of twisted Becker-Gottlieb transfer maps and study their multiplicative properties analogous to Becker-Gottlieb transfer. We show these twisted transfer maps factor through Becker-Schultz-Mann-Miller-Miller transfer; some of these might be well known. We apply this to show that $BSO(2n+1)_+$ splits off $MTO(2n)$, which after localisation away from $2$, refines to a homotopy equivalence $MTO(2n)\simeq BO(2n)_+$ as well as $MTO(2n+1)\simeq *$ for all $n\geqslant0$. This reduces the study of $MTO(n)$ to the $2$-localized case. At the prime $2$ our splitting allows us to identify some algebraically independent classes in mod $2$ cohomology of $\Omega^\infty MTO(2n)$. We also show that $BG_+$ splits off $MTK$ for some pairs $(G,K)$ at appropriate set of primes $p$, and investigate the consequences for characteristic classes, including algebraic independence and non-divisibility of some universally defined characteristic classes, generalizing results of Ebert and Randal-Williams.

In this article we study an almost $f$-cosymplectic manifold admitting a Ricci soliton. We first prove that there do not exist Ricci solitons on an almost cosymplectic $(\kappa,\mu)$-manifold. Further, we consider an almost $f$-cosymplectic manifold admitting a Ricci soliton whose potential vector field is the Reeb vector field and show that a three dimensional almost $f$-cosymplectic is a cosymplectic manifold. Finally we classify a three dimensional $\eta$-Einstein almost $f$-cosymplectic manifold admitting a Ricci soliton..