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Vol.25 No.1
Abstract
It is proved that 'maximum' is actually attained in the following risk measure representation <TEX>$${\rho}_m(X)={max \atop Q{\in}Q_m}E_Q[-X</TEX><TEX>]</TEX><TEX>$$</TEX>.
Abstract
The main objective of this paper is to demonstrate how one can obtain very quickly so far unknown Laplace transforms of rather general cases of the generalized hypergeometric function <TEX>$_3F_3$</TEX> by employing generalizations of classical summation theorems for the series <TEX>$_3F_2$</TEX> available in the literature. Several new as well known results obtained earlier by Kim et al. follow special cases of main findings.
Abstract
We find modular equations of degree 3 to evaluate some new values of the cubic continued fraction <TEX>$G(e^{-{\pi}\sqrt{n}})$</TEX> and <TEX>$G(-e^{-{\pi}\sqrt{n}})$</TEX> for <TEX>$n={\frac{2{\cdot}4^m}{3}}$</TEX>, <TEX>${\frac{1}{3{\cdot}4^m}}$</TEX>, and <TEX>${\frac{2}{3{\cdot}4^m}}$</TEX>, where m = 1, 2, 3, or 4.
Abstract
In [4], the authors show that if <TEX>${\mathbb{X}}$</TEX> is a <TEX>${\mathbb{k}}-configuration$</TEX> in <TEX>${\mathbb{P}}^2$</TEX> of type (<TEX>$d_1$</TEX>, <TEX>${\ldots}$</TEX>, <TEX>$d_s$</TEX>) with <TEX>$d_s$</TEX> > <TEX>$s{\geq}2$</TEX>, then <TEX>${\Delta}H_{m{\mathbb{X}}}(md_s-1)$</TEX> is the number of lines containing exactly <TEX>$d_s-points$</TEX> of <TEX>${\mathbb{X}}$</TEX> for <TEX>$m{\geq}2$</TEX>. They also show that if <TEX>${\mathbb{X}}$</TEX> is a <TEX>${\mathbb{k}}-configuration$</TEX> in <TEX>${\mathbb{P}}^2$</TEX> of type (1, 2, <TEX>${\ldots}$</TEX>, s) with <TEX>$s{\geq}2$</TEX>, then <TEX>${\Delta}H_{m{\mathbb{X}}}(m{\mathbb{X}}-1)$</TEX> is the number of lines containing exactly s-points in <TEX>${\mathbb{X}}$</TEX> for <TEX>$m{\geq}s+1$</TEX>. In this paper, we explore a standard <TEX>${\mathbb{k}}-configuration$</TEX> in <TEX>${\mathbb{P}}^2$</TEX> and find that if <TEX>${\mathbb{X}}$</TEX> is a standard <TEX>${\mathbb{k}}-configuration$</TEX> in <TEX>${\mathbb{P}}^2$</TEX> of type (1, 2, <TEX>${\ldots}$</TEX>, s) with <TEX>$s{\geq}2$</TEX>, then <TEX>${\Delta}H_{m{\mathbb{X}}}(m{\mathbb{X}}-1)=3$</TEX>, which is the number of lines containing exactly s-points in <TEX>${\mathbb{X}}$</TEX> for <TEX>$m{\geq}2$</TEX> instead of <TEX>$m{\geq}s+1$</TEX>.
Abstract
In this work, by applying the binomial expansion, some refinements of the Young and Heinz inequalities are proved. As an application, a determinant inequality for positive definite matrices is obtained. Also, some operator inequalities around the Young's inequality for semidefinite invertible matrices are proved.
Abstract
We have introduced two new notions of convexity and closedness in functional analysis. Let X be a real normed space, then <TEX>$C({\subseteq}X)$</TEX> is functionally convex (briefly, F-convex), if <TEX>$T(C){\subseteq}{\mathbb{R}}$</TEX> is convex for all bounded linear transformations <TEX>$T{\in}B$</TEX>(X, R); and <TEX>$K({\subseteq}X)$</TEX> is functionally closed (briefly, F-closed), if <TEX>$T(K){\subseteq}{\mathbb{R}}$</TEX> is closed for all bounded linear transformations <TEX>$T{\in}B$</TEX>(X, R). By using these new notions, the Alaoglu-Bourbaki-Eberlein-<TEX>${\check{S}}muljan$</TEX> theorem has been generalized. Moreover, we show that X is reflexive if and only if the closed unit ball of X is F-closed. James showed that for every closed convex subset C of a Banach space X, C is weakly compact if and only if every <TEX>$f{\in}X^{\ast}$</TEX> attains its supremum over C at some point of C. Now, we show that if A is an F-convex subset of a Banach space X, then A is bounded and F-closed if and only if every element of <TEX>$X^{\ast}$</TEX> attains its supremum over A at some point of A.