H. Sato introduced a Schwarzian derivative of a contactomorphism of R 3 and with T. Ozawa described its basic properties. In this note their construction is extended to all odd dimensions and to non-flat contact projective structures. The contact projective Schwarzian derivative of a contact projective structure is defined to be a cocycle of the contactomorphism group taking values in the space...

We study normal criterion of meromorphic functions shared values, we obtain the following. Let F be a family of meromorphic functions in a domain D, such that function f ∈ F has zeros of multiplicity at least 2, there exists nonzero complex numbers bf , cf depending on f satisfying i bf/cf is a constant; ii min{σ 0, bf , σ 0, cf , σ bf , cf ≥ m} for some m > 0; iii 1/ck−1 f f ′ k z f z / b f/c ...

Let $r>0, n\in\mathbb N, {\bf k}\in\mathbb N$. Consider the delay differential equation $$ x'(t)=g(x(t-d_1(Lx_t)),\ldots,x(t-d_{{\bf k}}(Lx_t))) for $g:( \mathbb R^n)^{{\bf k}}\supset V\to\mathbb R^n$ continuously differentiable, $L$ a continuous linear map from $C([-r,0],\mathbb R^n)$ into finite-dimensional vector-space $F$, each $d_k:F\supset W\to[0,r]$, $k=1,\ldots,{\bf k}$, and $x_t(s)=x(t...

In this paper, we introduce the notion of an action $Y_X$as a generalization of the notion of a module,and the notion of a norm $vt: Y_Xto F$, where $F$ is a field and $vartriangle(xy)vartriangle(y') =$ $ vartriangle(y)vartriangle(xy')$ as well as the notion of fuzzy norm, where $vt: Y_Xto [0, 1]subseteq {bf R}$, with $bf R$  the set of all real numbers. A great many standard mappings on algebr...

Gillespie posed two questions in [Front. Math. China 12 (2017) 97-115], one of which states that “for what rings R do we have K(AC)=K(R-Inj)?”. We give an answer to such a question. As applications, obtain new homological approach unifies some well-known conditions Krause’s recollement holds, and example show there exists Gorenstein injective module is not AC-injective. also improve Neeman’s an...

For a planar graph H $H$ , let N P ( n ) ${{\bf{N}}}_{{\mathscr{P}}}(n,H)$ denote the maximum number of copies in an $n$ -vertex graph. In this paper, we prove that 7 ~ 4 27 ${{\bf{N}}}_{{\mathscr{P}}}(n,{P}_{7})\unicode{x0007E}\frac{4}{27}{n}^{4}$ C 6 ∕ 3 ${{\bf{N}}}_{{\mathscr{P}}}(n,{C}_{6})\unicode{x0007E}{(n\unicode{x02215}3)}^{3}$ 8 ${{\bf{N}}}_{{\mathscr{P}}}(n,{C}_{8})\unicode{x0007E}{(...