Abstract:
By introducing a parameter, we give a unified generalization of some quadrature rules, which not only unify the recent results about error bounds for generalized mid-point, trapezoid and Simpson's rules, but also give some new error bounds for other quadrature rules as special cases. Especially, two sharp error inequalities are derived when n is an odd and an even integer, respectively.

Abstract:
We present Montgomery identity for Riemann-Liouville fractional integral as well as for fractional integral of a function with respect to another function . We further use them to obtain Ostrowski type inequalities involving functions whose first derivatives belong to spaces. These inequalities are generally sharp in case and the best possible in case . Application for Hadamard fractional integrals is given. 1. Introduction The following Ostrowski inequality is well known [1]: It holds for every whenever is continuous on and differentiable on with derivative bounded on ; that is, Ostrowski proved this inequality in 1938, and since then it has been generalized in a number of ways. Over the last few decades, some new inequalities of this type have been intensively considered together with their applications in numerical analysis, probability, information theory, and so forth. This inequality can easily be proved by using the following Montgomery identity (see, for instance, [2]): where is the Peano kernel, defined by The Riemann-Liouville fractional integral of order is defined by where is a gamma function When (5) is the Riemann definition of fractional integral. In case , fractional integral reduces to classical integral. In [3], the following Montgomery identity for fractional integrals is obtained. Theorem 1. Let , differentiable on , integrable on , and . Then, the following identity holds: where In [3], the authors used this identity to obtain the following Ostrowski type inequality for fractional integrals. Theorem 2. Let , differentiable on , and for every ; then, the following Ostrowski inequality holds: These results were further generalized in [4], while in [5] generalizations are obtained for fractional integral of a function with respect to another function (defined in Section 3). In the present paper, we give another, simpler new generalization of Montgomery identity for Riemann-Liouville fractional integral of order , which holds for a larger set of ; that is, . We also obtain Montgomery identity for fractional integral of a function with respect to another function . We further use these identities to obtain generalizations of Ostrowski inequality for fractional integrals of a function with respect to another function for functions whose first derivatives belong to spaces. These inequalities are generally sharp in case and the best possible in case . As a special case, application for Hadamard fractional integrals is given. 2. Montgomery Identity for Fractional Integrals In this section we give another, simpler new generalization of

Abstract:
Sharp quadrature formulas for integrals of complex rational functions on circles, real axis and its segments are obtained. We also find sharp quadrature formulas for calculation of $L_2$-norms of rational functions on such sets. Basing on quadrature formulas for rational functions, in particular, for simple partial fractions and polynomials, we derive sharp inequalities for different metrics (Nikol'skii type inequalities).

Abstract:
The new extension of the weighted Montgomery identity is given by using Fink identity and is used to obtain some Ostrowski-type inequalities and estimations of the difference of two integral means.

Abstract:
In this paper, we want to construct a one-to-one correspondence from the set of diffeomorphism classes of spin $d$-twisted homology $\mc P^3$ to the set of isotopy classes of the embedding from $S^3$ to $S^6$, which is a generalization of the Montgomery-Yang correspondence. Furthermore, we will apply this generalized correspondence to prove the existence of free involution on these $d$-twisted homology $\mc P^3$.

Abstract:
A new extension of the weighted Montgomery identity is given, by using Taylor's formula, and used to obtain some Ostrowski type inequalities and the estimations of the difference of two integral means.

Abstract:
A new extension of the weighted Montgomery identity is given, by using Taylor's formula and used to obtain some Ostrowski type inequalities and estimations of the difference of two integral means.

Abstract:
We prove a generalization of the Capelli identity. As an application we obtain an isomorphism of the Bethe subalgebras actions under the (gl(N),gl(M)) duality.

Abstract:
In this paper, new sharp weighted generalizations of Ostrowski and generalized trapezoid type inequalities for the Riemann--Stieltjes integrals are proved. Several related inequalities are deduced and investigated. New Simpson's type inequalities for $\mathcal{RS}$--integral are pointed out. Finally, as application; an error estimation of a general quadrature rule for $\mathcal{RS}$--integral via Ostrowski--generalized trapezoid quadrature formula is given.

Abstract:
In this note we give a generalization of the well-known Menon's identity. This is based on applying the Burnside's lemma to a certain group action.