A family of the Cauchy type mean-value theorems

The Cauchy type mean-value theorems for the Riemann-Liouville fractional derivative are deduced here from known mean-value theorems of the Lagrange type. A general method for deducing these Cauchy type formulas is extracted. Two Cauchy type formulas are then deduced without a priori knowledge about the Lagrange type mean-value theorems.     

A functional equation and its application to the characterization of gamma distributions

The functional equation $$ f\left(x\right)g\left(y\right)=p\left(x+y\right)q\left(\frac{x}{y} \right) $$ is investigated for almost all ${\left(x,\,y\right)\in\mathbb{R}^{2}_{+}}$ and for the measurable functions ${f,\,g,\,p,\,q:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}}$ . This equation is related to the Lukács characterization of gamma distribution.     

A certain family of summation-integral type operators

In the present paper, the authors introduce and investigate a new sequence of linear positive operators G_n,c, which includes some well-known operators as its special cases. The results obtained here include an estimate on the rate of convergence of G_n,c (f, x) by means of the decomposition technique for functions of bounded variation.     

A family of Bochner-Riesz type multipliers on the two-dimensional disk

Let s=(ν,μ)∈R² and define on R². Given p∈[1,+∞[, we prove some necessary and sufficient conditions on s such that ms be a Fourier multiplier for Lp. We employ two different techniques, according to ν=0 or ν≠0. In the latter case, the results obtained are optimal.     

A Hamada type characterization of the classical geometric designs

The dimension of a combinatorial design ${{\mathcal D}}$ over a finite field F = GF(q) was defined in (Tonchev, Des Codes Cryptogr 17:121–128, 1999) as the minimum dimension of a linear code over F that contains the blocks of ${{\mathcal D}}$ as supports of nonzero codewords. There it was proved that, for any prime power q and any integer n ≥ 2, the dimension over F of a design ${{\mathcal D}}$ that has the same parameters as the complement of a classical point-hyperplane design PG _n-1(n, q) or AG _n-1(n, q) is greater than or equal to n + 1, with equality if and only if ${{\mathcal D}}$ is isomorphic to the complement of the classical design. It is the aim of the present paper to generalize this Hamada type characterization of the classical point-hyperplane designs in terms of associated codes over F = GF(q) to a characterization of all classical geometric designs PG _d (n, q), where 1 ≤ d ≤ n ? 1, in terms of associated codes defined over some extension field E?=?GF(q ^t ) of F. In the affine case, we conjecture an analogous result and reduce this to a purely geometric conjecture concerning the embedding of simple designs with the parameters of AG _d (n, q) into PG(n, q). We settle this problem in the affirmative and thus obtain a Hamada type characterization of AG _d (n, q) for d = 1 and for d > (n ? 2)/2.     

A new characterization of euler's gamma function by a functional equation

This paper gives a new characterization of Euler's gamma function from the aspect of complex analysis. To this end the Gauss multiplication formula is used.     

A differential equations approach to the modal location for a family of bivariate gamma distributions

Analytical and numerical results are given for determining the location of the mode of a class of bivariate gamma densities as a function of the parameters. The model location for a class of bivariate gammas as considered by Kibble (1941, Sankhya A 5 137–150) is shown to satisfy a nonlinear differential equation in ϱ, the correlation coefficient for fixed shape parameter. Qualitative and asymptotic properties of the modal location are also given. Whenever the shape parameters are unequal, analytical and numerical results are used to provide a conjecture for the modal location in the general case.     

A family of Steffensen type methods with seventh-order convergence

In this paper, based on some known fourth-order Steffensen type methods, we present a family of three-step seventh-order Steffensen type iterative methods for solving nonlinear equations and nonlinear systems. For nonlinear systems, a development of the inverse first-order divided difference operator for multivariable function is applied to prove the order of convergence of the new methods. Numerical experiments with comparison to some existing methods are provided to support the underlying theory.     

A characterization of some geometries of Lie type

For geometries associated with permutation representations of the groups of Lie type E ₆, E ₇, E ₈ on certain maximal parabolic subgroups (e.g. the stabilizers of root subgroups), axiom systems are given that characterize them in terms of points and lines.     

A Helmholtz-Lie type characterization of ellipsoids,II

A closed convex surfaceS in is an ellipsoid if and only if for anyx, y εS there is an affinity mappingx ontoy and a neighborhood ofx inS onto a neighborhood ofy inS.     

A helmholtz-lie type characterization of ellipsoids,I

A closed convex surfaceS in withd odd, is an ellipsoid if and only if it has the following property: for any pair of pointsx, y inS there is an affine transformation which mapsx ontoy and a suitable neighborhood ofx inS onto a neighborhood ofy inS.     

A characterization of bounded symmetric domains of type IV

Let ?? be a bounded symmetric domain of type IV and dimension bigger than four. We show that a Stein manifold of the same dimension as ?? and with the same automorphism group is biholomorphic to ??.     

A characterization of selfinjective algebras of strictly canonical type

We prove that the class of selfinjective algebras of strictly canonical type, investigated in Kwiecień and Skowroński (2009) [27], Kwiecień and Skowroński (2009) [28], coincides with the class of selfinjective algebras having triangular Galois coverings with infinite cyclic group and the Auslander–Reiten quiver with quasi-tubes maximally saturated by simple and projective modules, satisfying natural conditions.