We study the kernels of the remainder term Rn,s(f) of GaussTurán quadrature formulas ∫ 1 −1 f(t)w(t) dt = n ∑

This paper considers the birth of algebraic proof by looking at the works of Cardano, Viète, Harriot and Pell. The transition from geometric to algebraic proof was mediated by appeals to the Eudoxan theory of proportions in book V of Euclid. The crucial notational innovation was the development of brackets. By the middle of the seventeenth century, geometric proof was unsustainable as the sole ...

We show that, for a finite set A of real numbers, the size of the set A + A A + A = { a + b c + d : a, b, c, d ∈ A, c + d 6= 0 } is bounded from below by ∣∣∣∣A + A A + A ∣∣∣∣ |A|2+1/4 |A/A|1/8 log |A| . This improves a result of Roche-Newton (2016).

The exact order of deterministic, stochastic and average error bounds of multidimensional quadrature formulas for anisotropic Sobolev class W r pðIÞ; Nikolskii class H pðIÞ and their periodic analogue is obtained. It is proved that if p41; then the stochastic and average error bounds are essentially smaller than the deterministic error bounds. Nonlinear methods, adaptive methods, or even method...

We prove that continuous sentences preserved by the ultramean construction (a generalization of the ultraproduct construction) are exactly those sentences which are approximated by linear sentences. Continuous sentences preserved by linear elementary equivalence are exactly those sentences which are approximated in the Riesz space generated by linear sentences. Also, characterizations for linea...

We consider a Gaussian type quadrature rule for some classes of integrands involving highly oscillatory functions of the form f (x) = f 1 (x) sin ζ x + f 2 (x) cos ζ x, where f 1 (x) and f 2 (x) are smooth, ζ ∈ R. We find weights σ ν and nodes x ν , ν = 1, 2,. .. , n, in a quadrature formula of the form 1 −1 f (x) dx ≈ n ν=1 σ ν f (x ν) such that it is exact for all polynomials f 1 (x) and f 2 ...

Density-functional chemical shielding calculations are reported for the alanine dipeptide with a variety of backbone torsion angles and for methane and N-methylacetamide complexes with rare gases, monatomic ions, water, and other amides. These fragment systems model electrostatic, nonbonded, and hydrogen bonding interactions in proteins and have been investigated at a variety of geometries. The...

We deene a notion of polarization in linear logic (LL) coming from the polarities of Jean-Yves Girard's classical sequent calculus LC 4]. This allows us to deene a translation between the two systems. Then we study the application of this polarization constraint to proof-nets for full linear logic described in 7]. This yields an important simpli-cation of the correctness criterion for polarized...

We deene a notion of polarization in linear logic (LL) coming from the polarities of Jean-Yves Girard's classical sequent calculus LC 4]. This allows us to deene a translation between the two systems. Then we study the application of this polarization constraint to proof-nets for full linear logic described in 7]. This yields an important simpli-cation of the correctness criterion for polarized...