The equality problem of two-variable Bajraktarević means can be expressed as the functional equation fg-1(f(x)+f(y)g(x)+g(y))=hk-1(h(x)+h(y)k(x)+k(y))(x,y∈I),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left( \\frac{f}{g}\\right) ^{-1}\\bigg (\\frac{f(x)+f(y)}{g(x)+g(y)}\\bigg ) =\\left( \\frac{h}{k}\\right) ^{-1}\\bigg (\\frac{h(x)+h(y)}{k(x)+k(y)}\\bigg )\\qquad (x,y\\in I), \\end{aligned}$$\\end{document}where I is a nonempty open real interval, f,g,h,k:Irightarrow {mathbb {R}} are continuous functions, g, k are positive and f/g, h/k are strictly monotone. This functional equation, for the first time, was solved by Losonczi in 1999 under 6th-order continuous differentiability assumptions. Additional and new characterizations of this equality problem have been found recently by Losonczi, Páles and Zakaria under the same regularity assumptions in 2021. In this paper it is shown that the same conclusion can be obtained under substantially weaker regularity conditions, namely, assuming only first-order differentiability.

We find all polyhedral graphs such that their complements are still polyhedral. These turn out to be all self-complementary.

In this paper we introduce the concept of translation invariant functions: considering an arbitrary set emptyset ne S subset {mathbb {R}}^n, the function F : S longrightarrow {mathbb {R}} is translation invariant if F(x) = F(y) implies F(x+t)=F(y+t) for any vectors x,y,t in {mathbb {R}}^n such that x ,y , x+t ,y+t in S. In our main results we shall consider an open, connected set emptyset ne D subset {mathbb {R}}^n. We prove that if F : D longrightarrow {mathbb {R}} is a translation invariant, continuous function, then there exists a vector a = (a_1, dots , a_n) in {mathbb {R}}^n and a strictly monotone, continuous function f such that F(x1,⋯,xn)=f(a1x1+⋯+anxn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} F(x_1, \\dots , x_n) = f (a_1 x_1 + \\dots + a_n x_n) \\end{aligned}$$\\end{document}holds for all (x_1, dots , x_n) in D ,. Using this result we also show that continuous solutions F : D longrightarrow {mathbb {R}} of the system of functional equations F(x1,⋯,xj+tj,⋯,xn)=Ψj(F(x1,⋯,xj,⋯,xn),tj)(j=1,⋯,n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} F(x_1 , \\dots , x_j + t_j , \\dots , x_n) = \\Psi _j (F (x_1 , \\dots , x_j , \\dots , x_n), t_j) \\ \\ (j=1,\\dots ,n) \\end{aligned}$$\\end{document}can be represented as the composition of a strictly monotone, continuous function and a linear functional as well. Applying the latter theorem, we give a characterization of Cobb–Douglas type utility functions.

We answer some questions on the asymptotics of ballot walks raised in [S. B. Ekhad and D. Zeilberger, April 2021] and prove that these models are not D-finite. This short note demonstrates how the powerful tools developed in the last decades on lattice paths in convex cones help us to answer some challenging problems that were out of reach for a long time. On the way we generalize tandem walks to the family of large tandem walks whose steps are of arbitrary length and map them bijectively to a generalization of ballot walks in three dimensions.

We produce a series of results extending information-theoretical inequalities (discussed by Dembo–Cover–Thomas in 1988–1991) to a weighted version of entropy. Most of the resulting inequalities involve the Gaussian weighted entropy; they imply a number of new relations for determinants of positive-definite matrices. Unlike the Shannon entropy where the contribution of an outcome depends only upon its probability, the weighted (or context-dependent) entropy takes into account a ‘value’ of an outcome determined by a given weight function $${\varphi }$$ . An example of a new result is a weighted version of the strong Hadamard inequality (SHI) between the determinants of a positive-definite $$d\times d$$ matrix and its square blocks (sub-matrices) of different sizes. When $${\varphi }\equiv 1$$ , the weighted inequality becomes a ‘standard’ SHI; in general, the weighted version requires some assumptions upon $${\varphi }$$ . The SHI and its weighted version generalize a widely known ‘usual’ Hadamard inequality $${\mathrm{det}}\,{{\mathbf {C}}}\le \prod \nolimits _{j=1}^dC_{jj}$$ .

We examine the family of all (at most) three-point symmetric quadratures on [-1,1] which are exact on polynomials of order 3 to find all possible inequalities between them in the class of 3-convex functions. Next we optimise them by using convex combinations of the quadratures considered. We find the optimal quadrature and use it to construct the adaptive method of approximate integration. An effective method to estimate the error of this method is also given. It needs a considerably fewer number of subdivisions of the interval of integration than the classical adaptive methods as well as the method developed by the second-named author in his recent paper (Wa̧sowicz in Aequ Math 94(5):887–898, 2020).