Enhancement of the algebraic precision of a linear operator and consequences under positivity

Let Ω be a compact convex domain in and let L be a bounded linear operator that maps a subspace of C(Ω) into C(Ω). Suppose that L reproduces polynomials up to degree m. We show that for appropriately defined coefficients a_mrj the operator

Necessary and sufficient conditions for the validity of Jensen’s inequality

We consider the d-dimensional Jensen inequality $$ T[\varphi(f_1, \dots, f_d)]\, \ge \, \varphi(T[f_1], \dots, T[f_d])\quad\quad(\ast)$$ T [ φ ( f 1 , … , f d ) ] ≥ φ ( T [ f 1 ] , … , T [ f d ] ) ( * ) as it was established by McShane in 1937r. Here T is a functional, φ is a convex function defined on a closed convex set ${K\subset \mathbb{R}^d}$ K ? R d , and f ₁, . . . , f _d are from some linear space of functions. Our aim is to find necessary and sufficient conditions for the validity of (*). In particular, we show that if we exclude three types of convex sets K, then Jensen’s inequality holds for a sublinear functional T if and only if T is linear, positive, and satisfies T[1] = 1. Furthermore, for each of the excluded types of convex sets, we present nonlinear, sublinear functionals T for which Jensen’s inequality holds. Thus the conditions on K are optimal. Our contributions generalize or complete several known results.     

Sharp Integral Inequalities of the Hermite–Hadamard Type

We consider a family of two-point quadrature formulae and establish sharp estimates for the remainders under various regularity conditions. Improved forms of certain integral inequalities due to Hermite and Hadamard, Iyengar, Milovanovi and Pecari , and others are obtained as special cases. Our results can also be interpreted as analogues to a theorem of Ostrowski on the deviation of a function from its averages. Furthermore, we establish a generalization of a result of Fink concerning L^p estimates for the remainder of the trapezoidal rule and present the best constants in the error bounds.     

Lower bound of effective yield strength domain for random heterogeneous materials

A random heterogeneous material is represented by a finite family of microstructures, the environment of each microstructure being subjected to a perfect mix condition whose validity is justified both theoretically and numerically. Necessary conditions, satisfied by any permissible strain rate field on a representative and unlimited domain of the material, are highlighted. They enable one to obtain, by solving a discrete infmax problem, a lower bound of the effective yield strength domain of the material, that is rigorous and more predictive than the classical bound of Reuss. An analytical application on a porous medium illustrates the methodology.     

A class of nonlinear four-point subdivision schemes

We consider four-point subdivision schemes of the form $$ (Sf)_{2i} = f_i,\qquad (Sf)_{2i+1} = \frac{f_i+f_{i+1}}{2} - \frac{1}{{8}}M\!\left(\strut \Delta^2f_{i-1}, \Delta^2f_i\right) $$ with any M that is originally defined as a positive-valued function for positive arguments and is extended to the whole of ?² by setting $M(x,y):=- M(\left|x\right|,\left|y\right|)$ if x?<?0, y?<?0 and M(x, y)?:?=?0 if xy????0. For these schemes, we study analytic properties, such as convexity preservation, convergence, smoothness of the limit function, stability and approximation order, in terms of simple and easily verifiable conditions on M. Fourth-order approximation on intervals of strict convexity is also investigated. All the results known for the most frequently used schemes, the PPH scheme and the power-p schemes, are included as special cases or improved, and extended to more general situations. The various statements are illustrated by two examples and tested by numerial experiments.