We study the complexity of approximating monotone Boolean functions with disjunctive normal form (DNF) formulas, exploring two main directions. First, we construct DNF approximators for arbitrary monotone functions achieving one-sided error: we show that every monotone f can be ε-approximated by a DNF g of size 2n−Ωε( √ n) satisfying g(x) ≤ f(x) for all x ∈ {0, 1}. This is the first non-trivial...

This paper proposes new notions of polynomial depth (called monotone poly depth), based on a polynomial version of monotone Kolmogorov complexity. We show that monotone poly depth satisfies all desirable properties of depth notions i.e., both trivial and random sequences are not monotone poly deep, monotone poly depth satisfies the slow growth law i.e., no simple process can transform a non dee...

Abstract In this paper we consider semilinear PDEs, with a convex nonlinearity, in sector-like domain. Using cylindrical coordinates $$(r, \theta , z)$$ ( r , θ z ) investigate the shape of possibly sig...

The classes of monotone or convex (and necessarily monotone) densities on ℝ(+) can be viewed as special cases of the classes of k-monotone densities on ℝ(+). These classes bridge the gap between the classes of monotone (1-monotone) and convex decreasing (2-monotone) densities for which asymptotic results are known, and the class of completely monotone (∞-monotone) densities on ℝ(+). In this pap...

This paper deals with numerical solving semilinear parabolic problems based on the method of upper and lower solutions. A monotone iterative method with quadratic convergence rate is constructed. The monotone iterative method combines an explicit construction of initial upper and lower solutions and the modified accelerated monotone iterative method. The monotone iterative method leads to the e...

This study is concerned with the visualization of monotone data using a piecewise C rational trigonometric interpolating scheme. Four positive shape parameters are incorporated in the structure of rational trigonometric spline. Conditions on two of these parameters are derived to attain the monotonicity of monotone data and other two are left free. Figures are used widely to exhibit that the pr...

and Applied Analysis 3 4 strictly η-monotone if 〈 u − v, ηx, y > 0, ∀x, y ∈ H, x / y, u ∈ M x , v ∈ M ( y ) ; 2.5 5 strongly monotone if there exists a constant r > 0 satisfying 〈 u − v, x − y ≥ r∥x − y∥2, ∀x, y ∈ H, u ∈ M x , v ∈ My; 2.6 6 strongly η-monotone if there exists a constant r > 0 satisfying 〈 u − v, ηx, y ≥ r∥x − y∥2, ∀x, y ∈ H, u ∈ M x , v ∈ My; 2.7 7 maximal monotone resp., maxim...

Shape constrained densities are encountered in many nonparametric estimation problems. The classes of monotone or convex (and monotone) densities can be viewed as special cases of the classes of k−monotone densities. A density g is said to be k−monotone if (−1)g is nonnegative, nonincreasing and convex for l = 0, . . . , k−2 if k ≥ 2, and g is simply nonincreasing if k = 1. These classes of sha...

a theory is presented for the dispersion relations of the nonlinear phonon-polaritons arising when phonons are coupled to the electromagnetic waves in multilayered structures of nonlinear materials. the calculations are applied to a multilayered structure consisting of a thin film surrounded by semi-infinite bounding media where each layer may have a frequency dependent dielectric function and ...