This present paper is inspired by one of the questions posed by Okeke (Results Math 78(96):1-30, 2023, see Remark 2.10b). In particular, we aim to develop a robust computer code based on the theoretical results obtained in Okeke (2023), which determines the polynomial solutions of the following functional equation, 0.1∑i=1nγiF(aix+biy)=∑j=1m(αjx+βjy)f(cjx+djy),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \ extstyle \\sum \\limits _{i=1}^n \\gamma _i F(a_i x + b_i y)=\ extstyle \\sum \\limits _{j=1}^m(\\alpha _j x + \\beta _j y) f(c_j x + d_j y), \\end{aligned}$$\\end{document}for all x,y∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x,y\\in \\mathbb {R}$$\\end{document}, γi,αj,βj∈R,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma _i,\\alpha _j,\\beta _j \\in \\mathbb {R},$$\\end{document} and ai,bi,cj,dj∈Q,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a_i,b_i,c_j,d_j \\in \\mathbb {Q},$$\\end{document} and their special forms. The primary motivation for writing such a computer code is that solving even simple equations belonging to class (0.1) needs long and tiresome calculations. Therefore, one of the advantages of such a computer code is that it allows us to solve complicated problems quickly, easily, and efficiently. Additionally, the computer code will significantly improve the level of accuracy in calculations. Along with that, there is also the factor of speed. We point out that the computer code will operate with symbolic calculations provided by the programming language Python, which means that it does not contain any numerical or approximate methods, and it yields the exact solutions of the equations considered.

In this paper, we consider the affine variety codes obtained evaluating the polynomials by=akxk+…+a1x+a0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$by=a_kx^k+\\ldots +a_1x+a_0$$\\end{document}, b,ai∈Fqr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b,a_i\\in {\\mathbb {F}}_{q^r}$$\\end{document}, at the affine Fqr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {F}}}_{q^r}$$\\end{document}-rational points of the Norm-Trace curve. In particular, we investigate the weight distribution and the set of minimal codewords. Our approach, which uses tools of algebraic geometry, is based on the study of the absolute irreducibility of certain algebraic varieties.

The purpose of the present paper is to show that in certain classes of real (or complex) functions, the Bernoulli polynomials are essentially the only ones satisfying the Raabe functional equation. For the class of the real 1-periodic functions which are expandable as Fourier series, we point out new solutions of the Raabe functional equation, not relating to the Bernoulli polynomials. Furthermore, we will give for the considered classes various proofs, making the mathematical content of the paper quite rich.

We provide three large classes of control functions that ensure the hyperstability of the Cauchy equation on restricted domains included in various types of commutative semigroups. Among other consequences, we obtain significant improvements on similar results known from the literature for several Aoki-Rassias-type control functions.

For a set $$\mathcal {H}$$ of connected graphs, a spanning subgraph H of G is called an $$\mathcal {H}$$ -factor of G if each component of H is isomorphic to an element of $$\mathcal {H}$$ . A graph G is called an $$\mathcal {H}$$ -factor uniform graph if for any two edges $$e_1$$ and $$e_2$$ of G, G has an $$\mathcal {H}$$ -factor covering $$e_1$$ and excluding $$e_2$$ . Let each component in $$\mathcal {H}$$ be a path with at least d vertices, where $$d\ge 2$$ is an integer. Then an $$\mathcal {H}$$ -factor and an $$\mathcal {H}$$ -factor uniform graph are called a $$P_{\ge d}$$ -factor and a $$P_{\ge d}$$ -factor uniform graph, respectively. In this article, we verify that (i) a 2-edge-connected graph G is a $$P_{\ge 3}$$ -factor uniform graph if $$\delta (G)>\frac{\alpha (G)+4}{2}$$ ; (ii) a $$(k+2)$$ -connected graph G of order n with $$n\ge 5k+3-\frac{3}{5\gamma -1}$$ is a $$P_{\ge 3}$$ -factor uniform graph if $$|N_G(A)|>\gamma (n-3k-2)+k+2$$ for any independent set A of G with $$|A|=\lfloor \gamma (2k+1)\rfloor $$ , where k is a positive integer and $$\gamma $$ is a real number with $$\frac{1}{3}\le \gamma \le 1$$ .

In Shulman (J Funct Anal 263:1468–1484, Theorem 2.2) the author estimated the cardinality of the global image of a subadditive map from a group to an arbitrary power set and applied the result to functional equations of Levi-Civita type (the relations of this result to covering a group by subgroups will be shortly discussed in the present paper). Here we will establish the metric stability of this theorem. Namely it will be shown that if a set-valued map F from a group G to a metric space (mathcal {X}, d) satisfies the condition dF(gh),F(g)∪F(h)<δ,for anyg,h∈Gand someδ>0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} d \\left( F(gh), F(g)\\cup F(h)\\right) < \\delta , \\;\\; \ ext {for any }\\ g,h\\in G \\; \ ext { and some } \\delta >0, \\end{aligned}$$\\end{document}and the cardinality of each F(g) does not exceed nin mathbb {N}, then the set F(G):= cup _{gin G}F(g) can be covered by n(n+3)/2 balls of radius (3cdot 6^{n-1}-1)delta .