Sentence examples for real-valued functional from inspiring English sources

This paper provides a fixed-width confidence interval of a real-valued functional, which involves the comparison between the univariate distribution functions, under a response-adaptive design in which a Stein-type two-stage procedure is carried out.

A real-valued functional (mathcal{J}:mathcal{H}rightarrow mathbb{R} ) is called convex on a convex class (mathcal{F}subset mathcal{H}) if (mathcal{J} ( mu f+ ( 1-mu ) g ) leqmu mathcal{J} ( f ) + ( 1-mu ) mathcal{J} ( g ) ) for all f and g in (mathcal{F}) and (0leqmuleq1).

Moreover, for (mu>0), (0< r<1), the real-valued functional mathcal{J} ( f ) = biggl( frac{1}{2pi} int_{0}^{2pi }biglvert f bigl( re^{itheta} bigr) bigrvert ^{mu}dtheta biggr) ^{1/mu} quad( fin{ mathcal{H}} ) is continuous on (mathcal{H}).

For all fixed values of (m,n,lambdainmathbb{N}), (zin{mathbb{D}}), the following real-valued functionals are continuous and convex on (mathcal{H}): mathcal{J} ( f ) =vert a_{n}vert, qquad mathcal{J} ( f ) =vert b_{n}vert, qquadmathcal {J} ( f ) =biglvert f ( z ) bigrvert, qquadmathcal{J} (f ) =biglvert J_{mathcal{ H}^{lambda}f ( z ) bigrvert quad ( fin{ mathcal{H}} ).

We denote by ℓ ∞ ( S ) the Banach space of all bounded real-valued functionals on S with supremum norm.

Let (N_{i}, eta_{i}:H_{i}times H_{i}rightarrow H_{i}), (F_{i}:H_{1}times H_{2}rightarrow CB(H_{i})), (A_{i}, B_{i}:H_{i}rightarrow CB(H_{i})) be mappings, let mappings (m_{i}:H_{i}rightarrow H_{i}) satisfy (1.2), and let (b_{i}:H_{i}times H_{i}rightarrow R) be real-valued functionals satisfying the properties in Theorem  2.1 and properties (i - iv).

The idea is that if we treat φ x,t) as an infinite-dimensional vector, indexed by x and t, functionals (with square brackets) are the analogues of real-valued functions; functional differentiation is the analogue partial differentiation; and functional integration the analogue of volume integration over the vector space.

Another interesting topic in vector optimization is to characterize the generalized cone-convexity of the vector-valued (or set-valued) objective functions in terms of usual generalized convexity of certain real-valued functions, by means of some appropriate scalarization functionals.

If, in particular, H is a Schrödinger-type operator and U a real-valued function, then the gradient of the trace functional is the quantum mechanical expression of the particle density with respect to an equilibrium distribution function f="−F′.

if for each (pin E^{prime}), the support functional (xmapsto sigma_{Phi(x)}(p)=sup_{yinPhi(x)}langle p,yrangle) is upper semicontinuous as an extended real-valued function on X. Always u.s.c.s.c

A real-valued function ║.,...,...