Chebyshev-type quadrature formulas for new weight classes

We give Chebyshev-type quadrature formulas for certain new weight classes. These formulas are of highest possible degree when the number of nodes is a power of 2. We also describe the nodes in a constructive way, which is important for applications. One of our motivations to consider these type of problems is the Faraday cage phenomenon for discrete charges as discussed by J. Korevaar and his colleagues.

A CHEBYSHEV-TYPE QUADRATURE RULE WITH SOME INTERESTING PROPERTIES

We give here an n-point Chebyshev-type rule of algebraic degree of precision n - 1, but having nodes that can be given explicitly. This quadrature rule also turns out to be one with an ''almost'' highest algebraic degree of precision.

New Hardy-Type Inequalities with Singular Weights

2010 Mathematics Subject Classification: 26D10.

Ostrowski Type Inequalities over Spherical Shells

2000 Mathematics Subject Classification: 26D10, 26D15.

Landau and Kolmogoroff type polynomial inequalities

Let 0type inequalitiesparallel to f((f))parallel to(2) less than or equal to A parallel to f(m)parallel to + B parallel to f parallel to/ A theta(k) + B mu(k),with certain constants theta(k)e mu(k), which hold for every f is an element of pi(n) (pi(n) denotes the space of real algebraic polynomials of degree not exceeding n).For the particular case j=1 and m=2, we provide a complete characterisation of the positive constants A and B, for which the corresponding Landau type polynomial inequalities parallel to f'parallel to less than or equal toA parallel to f parallel to + B parallel to f parallel to/ A theta(k) + B mu(k)hold. In each case we determine the corresponding extremal polynomials for which equalities are attained.

Landau and Kolmogoroff type polynomial inequalities II

Let 0 < j < m ≤ n. Kolmogoroff type inequalities of the form ∥f(j)∥2 ≤ A∥f(m)∥ 2 + B∥f∥2 which hold for algebraic polynomials of degree n are established. Here the norm is defined by ∫ f2(x)dμ(x), where dμ(x) is any distribution associated with the Jacobi, Laguerre or Bessel orthogonal polynomials. In particular we characterize completely the positive constants A and B, for which the Landau weighted polynomial inequalities ∥f′∥ 2 ≤ A∥f″∥2 + B∥f∥ 2 hold. © Dynamic Publishers, Inc.

Norm convolution inequalities in Lebesgue spaces

We obtain upper and lower estimates of the (p; q) norm of the con-volution operator. The upper estimate sharpens the Young-type inequalities due to O'Neil and Stepanov.

Horospheres in hyperbolic geometry

In this paper we investigate the role of horospheres in Integral Geometry and Differential Geometry. In particular we study envelopes of families of horocycles by means of “support maps”. We define invariant “linear combinations” of support maps or curves. Finally we obtain Gauss-Bonnet type formulas and Chern-Lashof type inequalities.

Ruin problems under IBNR Dynamics

In this paper, we consider a discrete-time risk process allowing for delay in claim settlement, which introduces a certain type of dependence in the process. From martingale theory, an expression for the ultimate ruin probability is obtained, and Lundberg-type inequalities are derived. The impact of delay in claim settlement is then investigated. To this end, a convex order comparison of the aggregate claim amounts is performed with the corresponding non-delayed risk model, and numerical simulations are carried out with Belgian market data.

The Brownian traveller on manifolds

We study the inuence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the at case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the at case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation.

On Majorization for Matrices

In this paper, we give several results for majorized matrices by using continuous convex function and Green function. We obtain mean value theorems for majorized matrices and also give corresponding Cauchy means, as well as prove that these means are monotonic. We prove positive semi-definiteness of matrices generated by differences deduced from majorized matrices which implies exponential convexity and log-convexity of these differences and also obtain Lypunov's and Dresher's type inequalities for these differences.

Contribution of modifiable risk factors to social inequalities in type 2 diabetes: prospective Whitehall II cohort study.

OBJECTIVE: To assess the contribution of modifiable risk factors to social inequalities in the incidence of type 2 diabetes when these factors are measured at study baseline or repeatedly over follow-up and when long term exposure is accounted for. DESIGN: Prospective cohort study with risk factors (health behaviours (smoking, alcohol consumption, diet, and physical activity), body mass index, and biological risk markers (systolic blood pressure, triglycerides and high density lipoprotein cholesterol)) measured four times and diabetes status assessed seven times between 1991-93 and 2007-09. SETTING: Civil service departments in London (Whitehall II study). PARTICIPANTS: 7237 adults without diabetes (mean age 49.4 years; 2196 women). MAIN OUTCOME MEASURES: Incidence of type 2 diabetes and contribution of risk factors to its association with socioeconomic status. RESULTS: Over a mean follow-up of 14.2 years, 818 incident cases of diabetes were identified. Participants in the lowest occupational category had a 1.86-fold (hazard ratio 1.86, 95% confidence interval 1.48 to 2.32) greater risk of developing diabetes relative to those in the highest occupational category. Health behaviours and body mass index explained 33% (-1% to 78%) of this socioeconomic differential when risk factors were assessed at study baseline (attenuation of hazard ratio from 1.86 to 1.51), 36% (22% to 66%) when they were assessed repeatedly over the follow-up (attenuated hazard ratio 1.48), and 45% (28% to 75%) when long term exposure over the follow-up was accounted for (attenuated hazard ratio 1.41). With additional adjustment for biological risk markers, a total of 53% (29% to 88%) of the socioeconomic differential was explained (attenuated hazard ratio 1.35, 1.05 to 1.72). CONCLUSIONS: Modifiable risk factors such as health behaviours and obesity, when measured repeatedly over time, explain almost half of the social inequalities in incidence of type 2 diabetes. This is more than was seen in previous studies based on single measurement of risk factors.