A graceful labeling of a graph $G=(V,E)$ with $m$ edges is aninjection $f: V(G) rightarrow {0,1,ldots,m}$ such that the resulting edge labelsobtained by $|f(u)-f(v)|$ on every edge $uv$ are pairwise distinct. For natural numbers $n$ and $k$, where $n > 2k$, a generalized Petersengraph $P(n, k)$ is the graph whose vertex set is ${u_1, u_2, cdots, u_n} cup {v_1, v_2, cdots, v_n}$ and its edge set...

chemokines are classified in four distinct groups as cxc, cc, cx3c and c, depending on the presence or absence of a motif called elr (arg-leu-glu) before the first cysteine residue in their structure. cxc chemokines are also subdivided into elr + and elr - . increasing evidence has indicated the existence of a chemokine network in the liver which is involved in both physiological responses and,...

 Let $P(lambda)$ be an $n$-square complex matrix polynomial, and $1 leq k leq n$ be a positive integer. In this paper, some algebraic and geometrical properties of the $k$-numerical range of $P(lambda)$ are investigated. In particular, the relationship between the $k$-numerical range of $P(lambda)$ and the $k$-numerical range of its companion linearization is stated. Moreover, the $k$-numerical...

Let $(K, v)$ be a Henselian discrete valued field with residue $\widehat K$ of characteristic $p$, and Brd$_{p}(K)$ the Brauer $p$-dimension $K$. This paper shows that Brd$_{p}(K) \ge n$, if $[\widehat K\colon \widehat K ^{p}] = p ^{n}$, for some $n \in \mathbb{N}$. It proves \infty $ only $.

The Zygmund sums of a function $f\in L_{1}$ are trigonometric polynomials the form $$Z^{s}_{n-1}(f;t):=\frac{a_{0}}{2}+\sum_{k=1}^{n-1}\Big(1-\big(\frac{k}{n}\big)^{s}\Big) \big(a_{k}(f)\cos kt+b_{k}(f)\sin kt\big), s>0,$$ where $a_{k}(f)$ and $b_{k}(f)$ Fourier coefficients $f$. We establish exact-order estimates uniform approximations by $Z^{s}_{n-1}$ $2\pi$-periodic continuous functions f...

in this paper, we introduce new classes $sum_{k,p,n}(alpha ,m,lambda ,l,rho )$ and $mathcal{t}_{k,p,n}(alpha ,m,lambda ,l,rho )$ of p-valent meromorphic functions defined by using the extended multiplier transformation operator. we use a strong convolution technique and derive inclusion results. a radius problem and some other interesting properties of these classes are discussed.