‎for any integer $kge 1$‎, ‎a minus $k$-dominating function is a‎ ‎function $f‎ : ‎v (g)rightarrow {-1,0‎, ‎1}$ satisfying $sum_{win‎‎n[v]} f(w)ge k$ for every $vin v(g)$‎, ‎where $n(v) ={u in‎‎v(g)mid uvin e(g)}$ and $n[v] =n(v)cup {v}$‎. ‎the minimum of‎‎the values of $sum_{vin v(g)}f(v)$‎, ‎taken over all minus‎‎$k$-dominating functions $f$‎, ‎is called the minus $k$-domination‎‎number and i...

The aim of this study was to test whether both phonetic and phonological representations of intonation can be rapidly modified when imitating utterances belonging to a different regional variety of the same language. Our main hypothesis was that tonal alignment, just as other phonetic features of speech, would be rapidly modified by Italian speakers when imitating pitch accents of a different (...

Abstract: Let G=(V,E) be a graph and let f:V(G)→{0,1,2} be a function‎. ‎A vertex v is protected with respect to f‎, ‎if f(v)>0 or f(v)=0 and v is adjacent to a vertex of positive weight‎. ‎The function f is a co-Roman dominating function‎, ‎abbreviated CRDF if‎: ‎(i) every vertex in V is protected‎, ‎and (ii) each u∈V with positive weight has a neighbor v∈V with f(v)=0 such that the func...

For any integer  ‎, ‎a minus  k-dominating function is a‎function  f‎ : ‎V (G)  {-1,0‎, ‎1} satisfying w) for every  vertex v, ‎where N(v) ={u V(G) | uv  E(G)}  and N[v] =N(v)cup {v}. ‎The minimum of ‎the values of  v)‎, ‎taken over all minus‎k-dominating functions f,‎ is called the minus k-domination‎number and is denoted by $gamma_k^-(G)$ ‎. ‎In this paper‎, ‎we ‎introduce the study of minu...

A total Roman dominating function on a graph $G$ is a function $f: V(G) rightarrow {0,1,2}$ such that for every vertex $vin V(G)$ with $f(v)=0$ there exists a vertex $uin V(G)$ adjacent to $v$ with $f(u)=2$, and the subgraph induced by the set ${xin V(G): f(x)geq 1}$ has no isolated vertices. The total Roman domination number of $G$, denoted $gamma_{tR}(G)$, is the minimum weight $omega(f)=sum_...

Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simplegraph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph$G$ is the set consisting of $e$ and all edges having a commonend-vertex with $e$. A signed Roman edge $k$-dominating function(SREkDF) on a graph $G$ is a function $f:E rightarrow{-1,1,2}$ satisfying the conditions that (i) for every edge $e$of $G$, $sum _{xin N[e]} f...

A Roman dominating function on a graphG is a function f : V (G) → {0, 1, 2} satisfying the condition that every vertex u ∈ V (G) for which f(u) = 0 is adjacent to at least one vertex v ∈ V (G) for which f(v) = 2. The weight of a Roman dominating function is the value f(V (G)) = ∑ u∈V (G) f(u). The Roman domination number γR(G) of G is the minimum weight of a Roman dominating function on G. A Ro...

The author analyzes a historical, long, and tortuous way to discover the cerebrospinal fluid. At least 35 physicians and anatomists described in the text have laid the fundamentals of recognition of this biological fluid's presence. On the basis of crucial anatomical, experimental, and clinical works there are four greatest physicians who should be considered as equal cerebrospinal fluid's disc...