Grüss-type and Ostrowski-type inequalities in approximation theory

We discuss the Grüss inequalities on spaces of continuous functions defined on a compact metric space. Using the least concave majorant of the modulus of continuity, we obtain the Grüss inequality for the functional L(f) = H(f; x), where H:C[a, b] → C[a, b] is a positive linear operator and x ∈ [a, b] is fixed. We apply this inequality in the case of known operators, e.g., the Bernstein operator, the Hermite–Fejér interpolation operator, and convolution-type operators. Moreover, we deduce Grüss-type inequalities using the Cauchy mean-value theorem, thus generalizing results of Chebyshev and Ostrowski. The Grüss inequality on a compact metric space for more than two functions is given, and an analogous Ostrowski-type inequality is obtained. The latter, in turn, leads to one further version of the Grüss inequality. In the appendix, we prove a new result concerning the absolute first-order moments of the classic Hermite–Fejér operator.     

Some bounds for integrals with refinements of the Grüss inequality

Some integral bounds are obtained which provide refinements and extensions of the Grüss type inequalities. We also get an alternative formulation of the Grüss inequality.     

Inequalities for the expectation of Δ-monotone functions

Summary For some subsets of the set of all -monotone functions on [0,1] ⁿ we characterize distribution functions F, G such that E _F fE_G f for all f within these subsets. Furthermore, we determine sharp upper and lower bounds of integrals of functions in these subsets w.r.t. all distributions with fixed marginals and give some applications of these results.     

Verified bounds for singular values,in particular for the spectral norm of a matrix and its inverse

The singular value decomposition and spectral norm of a matrix are ubiquitous in numerical analysis. They are extensively used in proofs, but usually it is not necessary to compute them. However, there are some important applications in the realm of verified error bounds for the solution of ordinary and partial differential equations where reasonably tight error bounds for the spectral norm of a matrix are mandatory. We present various approaches to this together with some auxiliary useful estimates.     

Some bounds and limits in the theory of Riemann’s zeta function

For any real $a>0$ we determine the supremum of the real $\sigma$ such that $\zeta (\sigma +it)=a$ for some real $t$. For $0<a<1$, $a=1$, and $a>1$ the results turn out to be quite different.We also determine the supremum $E$ of the real parts of the ‘turning points’, that is points $\sigma +it$ where a curve $\mathrm{Im}\zeta (\sigma +it)=0$ has a vertical tangent. This supremum $E$ (also considered by Titchmarsh) coincides with the supremum of the real $\sigma$ such that ${\zeta }^{\prime }(\sigma +it)=0$ for some real $t$.We find a surprising connection between the three indicated problems: $\zeta \left(s\right)=1$, ${\zeta }^{\prime }\left(s\right)=0$ and turning points of $\zeta \left(s\right)$. The almost extremal values for these three problems appear to be located at approximately the same height.     

Upper and lower bounds for the Hilbert–Schmidt norm of a potential operator

Archiv der Mathematik - We give upper and lower bounds for the Hilbert–Schmidt norm of logarithmic potential on a planar domain in terms of its area and inradius.     

New Grüss’ type inequalities for functions of bounded variation and applications

Some new sharp Grüss’ type inequalities for functions of bounded variation and applications for selfadjoint operators in Hilbert spaces are given.     

A solution to a problem of Grünbaum and Motzkin and of Erdős and Purdy about bichromatic configurations of points in the plane

Let P be a set of n blue points in the plane, not all on a line. Let R be a set of m red points such that P ∩ R = ? and every line determined by P contains a point from R. We provide an answer to an old problem by Grünbaum and Motzkin [9] and independently by Erd?s and Purdy [6] who asked how large must m be in terms of n in such a case? More specifically, both [9] and [6] were looking for the best absolute constant c such that m ≥ cn. We provide an answer to this problem and show that m ≥ (n?1)/3.     

Gröbner–Shirshov Basis for HNN Extensions of Groups and for the Alternating Group

In this article, we generalize the Shirshov's Composition Lemma by replacing the monomial order for others. By using Gröbner–Shirshov bases, the normal forms of HNN extension of a group and the alternating group are obtained.     

Convergence and continuous dependence for the Brinkman–Forchheimer equations

This paper investigates the flow of fluid in a porous medium which is described in the Brinkman–Forchheimer equations and obtains the structural stability results for the coefficients.     

Sharp frequency bounds for eigenfunctions of the Ornstein–Uhlenbeck operator

We prove sharp bounds for the growth rate of eigenfunctions of the Ornstein–Uhlenbeck operator and its natural generalizations. The bounds are sharp even up to lower order terms and have important applications to geometric flows.     

On the coincidence of Grünberg-Kegel graphs of a finite simple group and its proper subgroup

Let G be a finite group. The spectrum of G is the set ω(G) of orders of all its elements. The subset of prime elements of ω(G) is denoted by π(G). The spectrum ω(G) of a group G defines its prime graph (or Grünberg-Kegel graph) Γ(G) with vertex set π(G), in which any two different vertices r and s are adjacent if and only if the number rs belongs to the set ω(G). We describe all the cases when the prime graphs of a finite simple group and of its proper subgroup coincide.     

Reflected Brownian motion in the quadrant: tail behavior of the stationary distribution

Let Z be a two-dimensional Brownian motion confined to the non-negative quadrant by oblique reflection at the boundary. Such processes arise in applied probability as diffusion approximations for two-station queueing networks. The parameters of Z are a drift vector, a covariance matrix, and a “direction of reflection” for each of the quadrant’s two boundary rays. Necessary and sufficient conditions are known for Z to be a positive recurrent semimartingale, and they are the only restrictions imposed on the process data in our study. Under those assumptions, a large deviations principle (LDP) is conjectured for the stationary distribution of Z, and we recapitulate the cases for which it has been rigorously justified. For sufficiently regular sets B, the LDP says that the stationary probability of xB decays exponentially as x→∞, and the asymptotic decay rate is the minimum value achieved by a certain function I(?) over the set B. Avram, Dai and Hasenbein  (Queueing Syst.: Theory Appl. 37, 259–289, 2001) provided a complete and explicit solution for the large deviations rate function I(?). In this paper we re-express their solution in a simplified form, showing along the way that the computation of I(?) reduces to a relatively simple problem of least-cost travel between a point and a line.     

Grüss type and related integral inequalities in probability spaces

Aequationes mathematicae - In this paper we study Grüss type inequalities for real and complex valued functions in probability spaces. Some earlier Grüss type inequalities are extended...