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We consider semilinear parabolic equations of the form

This paper describes a problem of finding the contributions of multiple variables to a change in their function. Such a problem is well known in economics, for example, in the decomposition of a change in the mean price via the varying in time prices and volumes of multiple products. Commonly, it is considered by the tools of index analysis, the formulae of which present rather heuristic constructs. As shown in this work, the multivariate version of the Lagrange mean value theorem can be seen as an equation of the function’s finite change and solved with respect to an interior point whose value is used in the estimation of the contribution of the independent variables. Consideration is performed on the example of the weighted mean value function, which is the main characteristic of statistical estimation in various fields. The solution for this function can be obtained in the closed form, which helps in the analysis of results. Numerical examples include the cases of Simpson’s paradox, and practical applications are discussed.

If a differentiable function f : [ a , b ] → R and a point η ∈ [ a , b ] satisfy f ( η ) − f ( a ) = f ′ ( η ) ( η − a ) , then the point η is called a Flett’s mean value point of f in [ a , b ] . The concept of Flett’s mean value points can be generalized to the 2-dimensional Flett’s mean value points as follows: For the different points r ^ and s ^ of R × R , let L be the line segment joining r ^ and s ^ . If a partially differentiable function f : R × R → R and an intermediate point ω ^ ∈ L satisfy f ( ω ^ ) − f ( r ^ ) = ω ^ − r ^ , f ′ ( ω ^ ) , then the point ω ^ is called a 2-dimensional Flett’s mean value point of f in L. In this paper, we will prove the Hyers–Ulam stability of 2-dimensional Flett’s mean value points.

Using a theorem of Ulam and Hyers, we will prove the Hyers-Ulam stability of two-dimensional Lagrange’s mean value points ( η , ξ ) which satisfy the equation, f ( u , v ) − f ( p , q ) = ( u − p ) f x ( η , ξ ) + ( v − q ) f y ( η , ξ ) , where ( p , q ) and ( u , v ) are distinct points in the plane. Moreover, we introduce an efficient algorithm for applying our main result in practical use.

Denote by Δ1(x;φ) the error term in the classical Rankin–Selberg problem. Denote by ζ(s) the Riemann zeta-function. We establish an upper bound for this integral ∫0TΔ1(t;φ)ζ12+it2dt. In addition, when 2≤k≤4 is a fixed integer, we will derive an asymptotic formula for the integral ∫1TΔ1k(t;φ)ζ12+it2dt. The results rely on the power moments of Δ1(t;φ) and E(t), the classical error term in the asymptotic formula for the mean square of ζ12+it.

In this paper, we study fuzzy calculus in two main branches differential and integral. Some rules for finding limit and $gH$-derivative of $gH$-difference, constant multiple of two fuzzy-valued functions are obtained and we also present fuzzy chain rule for calculating $gH$-derivative of a composite function. Two techniques namely, Leibniz's rule and integration by parts are introduced for fuzzy integrals. Furthermore, we prove three essential theorems such as a fuzzy intermediate value theorem, fuzzy mean value theorem for integral and mean value theorem for $gH$-derivative. We derive a Bolzano's theorem, Rolle's theorem and some properties for $gH$-differentiable functions. To illustrate and explain these rules and theorems, we have provided several examples in details.

Mean value interpolation is a method for fitting a smooth function to piecewise-linear data prescribed on the boundary of a polygon of arbitrary shape, and has applications in computer graphics and curve and surface modelling. The method generalizes to transfinite interpolation, i.e., to any continuous data on the boundary but a mathematical proof that interpolation always holds has so far been missing. The purpose of this note is to complete this gap in the theory.

The article is a natural continuation of the systematic research of the properties of the generalized concept of differentiability for functions with a domain X⊂Rn that is not necessarily open, at points that allow a neighbourhood ray in the domain. In the new context, the well-known Lagrange’s mean value theorem for scalar functions is stated and proved, even for the case when the differential is not unique at all points of the observed segment in the domain. Likewise, it has been proven that its variant is valid for vector functions as well. Additionally, the paper provides a proof of the generalization of the mean value theorem for continuous scalar functions continuously differentiable in the interior of a compact domain.

We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds and in doubling metric measure spaces. We show that the strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajłasz–Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions.

In this paper we will show a new way to represent functions as infinite series, finding some conditions under which a function is expandable with this method, and showing how it allows us to find the values of many interesting series. At the end, we will prove one of the main results of the paper, a Representation Theorem.