A Note on Finite Quadrature Rules with a Kind of Freud Weight Function

We introduce a finite class of weighted quadrature rules with the weight function |x|−2exp(−1/2) on (−∞,∞) as ∫∞−∞||−2exp(−1/2∑)()==1()+[], where are the zeros of polynomials orthogonal with respect to the introduced weight function, are the corresponding coefficients, and [] is the error value. We show that the above formula is valid only for the finite values of . In other words, the condition ≥{max}+1/2 must always be satisfied in order that one can apply the above quadrature rule. In this sense, some numerical and analytic examples are also given and compared

On the main invariant of elements algebraic over a Henselian valued field

Let $v$ be a henselian valuation of a field $K$ with value group $G$, let $\bar{v}$ be the (unique) extension of $v$ to a fixed algebraic closure $\bar{K}$ of $K$ and let $(\tilde{K},\tilde{v})$ be a completion of $(K,v)$. For $\alpha\in\bar{K}\setminus K$, let $M(\alpha,K)$ denote the set $\{\bar{v}(\alpha-\beta):\beta\in\bar{K},\ [K(\beta):K] \lt [K(\alpha):K]\}$. It is known that $M(\alpha,K)$ has an upper bound in $\bar{G}$ if and only if $[K(\alpha):K]=[\tilde{K}(\alpha):\tilde{K}]$, and that the supremum of $M(\alpha,K)$, which is denoted by $\delta_{K}(\alpha)$ (usually referred to as the main invariant of $\alpha$), satisfies a principle similar to the Krasner principle. Moreover, each complete discrete rank 1 valued field $(K,v)$ has the property that $\delta_{K}(\alpha)\in M(\alpha,K)$ for every $\alpha\in\bar{K}\setminus K$. In this paper the authors give a characterization of all those henselian valued fields $(K,v)$ which have the property mentioned above

On the Main Invariant . . . Henselian Valued Field

Let v be a henselian valuation of a eld K with value group G, v be the (unique) extension of v to a xed algebraic closure K of K and ( K; ~ v) be completion of (K; v). For 2 K\K, let M(;K) denote the set fv( ) : 2 K; [K( ) : K] < [K() : K]g. It is known that M(;K) has an upper bound in G, if and only if [K() : K] = [ K] and that the supremum of M(;K); which is denoted by K () (usually referred to as the main invariant of ), satis es a principle similar to the Krasner's principle [S. K. Khanduja and J. Saha. Mathematika 46 (1999) 83-92]. Moreover each complete discrete rank 1 valued eld (K; v) has the property that K () 2 M(;K) for every 2 K\K: In this paper, the authors give a characterization of all those henselian valued elds (K; v) which have the property mentioned above

On chains associated with elements algebraic over a henselian valued field

Let v be a henselian valuation of a field K,and v̅ be the (unique) extension of v to a fixed algebraic closure of K̅. For an element θ∈K̅\K, a chain θ= θ0, θ1,...θmof elements of K̅ such that θ̅(θi−1−θi)=SUP{ v̅(θi−1−β)|[K(β):K]<[K(θi−1):K]} and θm ∈ K, is called a complete distinguished chain for θ with respect to (K, v). In 1995, Popescu and Zaharescu proved the existence of a complete distinguished chain for each θ∈K̅\K when (K, v) is a complete discrete rank one valued field (cf. [10]). In this paper, for a henselian valued field (K, v) of arbitrary rank, we characterize those elements θ∈K̅\K for which there exists a complete distinguished chain. It is shown that a complete distinguished chain for θ gives rise to several invariants associated to θ which are same for all the K-conjugates of θ