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In this paper, we present the central solutions f , g : G × G → C of the functional Eq. f (x 1 y 1 , x 2 y 2) + g (x 1 y 1 - 1 , x 2) + f (x 1 , x 2 y 2 - 1) = f (x 1 y 1 - 1 , x 2 y 2 - 1) + g (x 1 y 1 , x 2) + f (x 1 , x 2 y 2) for all x 1 , x 2 , y 1 , y 2 ∈ G , where G is a 2-divisible group. [ABSTRACT FROM AUTHOR]

In this paper, a computer program developed in the computer algebra system Maple is presented, which investigates alienness and strong alienness of linear functional equations. [ABSTRACT FROM AUTHOR]

In this paper, we determine the central solutions f : G × G → C of the functional equation f (x 1 σ y 1 , x 2 τ y 2) - f (x 1 σ y 1 , x 2) - f (x 1 , x 2 τ y 2) = f (x 1 y 1 , x 2 y 2) - f (x 1 y 1 , x 2) - f (x 1 , x 2 y 2) for all x 1 , x 2 , y 1 , y 2 ∈ G , where G is a group, σ , τ : G → G are involutions. [ABSTRACT FROM AUTHOR]

The equality problem of two-variable Bajraktarević means can be expressed as the functional equation f g - 1 ( f (x) + f (y) g (x) + g (y) ) = h k - 1 ( h (x) + h (y) k (x) + k (y) ) (x , y ∈ I) , where I is a nonempty open real interval, f , g , h , k : I → 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. [ABSTRACT FROM AUTHOR]

Let S be a monoid (= semigroup with identity), and let σ : S → S be a homomorphism such that σ ∘ σ = i d . In an earlier paper we solved the Pexiderized d'Alembert functional equation (PDFE) f (x y) + g (σ (y) x) = h (x) k (y) for unknown f , g , h , k : S → C , assuming that S is either regular or generated by its squares and that one of the unknown functions is central. The present paper has two main results. The first describes the solutions of PDFE on a general monoid in terms of multiplicative functions, solutions of a special case of the sine subtraction law, and solutions of other functional equations with just one unknown function. The second main result uses the first one to give a more detailed solution of PDFE on a larger class of monoids than has been treated previously. We also find the continuous solutions on topological monoids. Examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi's result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation F (x + y) - F (x) - F (y) = y f (x) + x f (y) considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation. [ABSTRACT FROM AUTHOR]

In the present paper we consider the problem of characterization of maps which can be expressed as an affine difference i.e. a three-place map of the form t f (x) + (1 - t) f (y) - f (t x + (1 - t) y). We give a general solution of a functional equation associated with this problem. [ABSTRACT FROM AUTHOR]

We deal with an alienation problem for an Euler–Lagrange type functional equation f (α x + β y) + f (α x - β y) = 2 α 2 f (x) + 2 β 2 f (y) assumed for fixed nonzero real numbers α , β , 1 ≠ α 2 ≠ β 2 , and the classic quadratic functional equation g (x + y) + g (x - y) = 2 g (x) + 2 g (y). We were inspired by papers of Kim et al. (Abstract and applied analysis, vol. 2013, Hindawi Publishing Corporation, 2013) and Gordji and Khodaei (Abstract and applied analysis, vol. 2009, Hindawi Publishing Corporation, 2009), where the special case g = γ f was examined. [ABSTRACT FROM AUTHOR]

For any set X we let F (X) denote the complex vector space of functions f : X → C . Let X = S be a magma, and let V be a subspace of F (S) , which is invariant under left or right translations. It is known for an abelian group S that if p 1 χ 1 , ⋯ , p n χ n ∈ F (S) are nonzero exponential polynomials with distinct exponentials χ 1 , ⋯ , χ n then p 1 χ 1 + ⋯ + p n χ n ∈ V ⇒ p 1 χ 1 , ⋯ , p n χ n ∈ V and χ 1 , ⋯ , χ n ∈ V . We extend this to magmas. Our results imply that any exponential polynomial solution f ∈ F (S) of f (x y) = f (x) χ (y) + χ (x) f (y) where χ ∈ F (S) is an exponential, has the form f = a χ where a ∈ F (S) is additive, even when χ has zeros. [ABSTRACT FROM AUTHOR]

Aichinger's equation is used to give simple proofs of several well-known characterizations of polynomial functions as solutions of certain functional equations. Concretely, we use that Aichinger's equation characterizes polynomial functions to solve, for arbitrary commutative groups, Ghurye–Olkin's functional equation, Wilson's functional equation, the Kakutani–Nagumo–Walsh functional equation, and a general version of Fréchet's unmixed functional equation. [ABSTRACT FROM AUTHOR]

Given two functions f , g : I → R and a probability measure μ on the Borel subsets of [0, 1], the two-variable mean M f , g ; μ : I 2 → I is defined by M f , g ; μ (x , y) : = (f g ) - 1 ∫ 0 1 f (t x + (1 - t) y) d μ (t) ∫ 0 1 g (t x + (1 - t) y) d μ (t) (x , y ∈ I). This class of means includes quasiarithmetic as well as Cauchy and Bajraktarević means. The aim of this paper is, for a fixed probability measure μ , to study their equality problem, i.e., to characterize those pairs of functions (f, g) and (F, G) for which M f , g ; μ (x , y) = M F , G ; μ (x , y) (x , y ∈ I) holds. Under at most sixth-order differentiability assumptions for the unknown functions f, g and F, G, we obtain several necessary conditions in terms of ordinary differential equations for the solutions of the above equation. For two particular measures, a complete description is obtained. These latter results offer eight equivalent conditions for the equality of Bajraktarević means and of Cauchy means. [ABSTRACT FROM AUTHOR]

In the present paper we are concerned with the problem of characterization of maps which can be expressed as an affine difference i.e. a map of the form t f (x) + (1 - t) f (y) - f (t x + (1 - t) y) , where t ∈ (0 , 1) is a given number. We give a general solution of the functional equation associated with this problem. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Given a probability space (Ω , A , P) , a complete separable Banach space X with the σ -algebra B (X) of all its Borel subsets, an operator Λ : Ω → L (X , X) and ξ : Ω → X we consider the B (X) ⊗ A -measurable function f : X × Ω → X given by f (x , ω) = Λ (ω) x + ξ (ω) and investigate the continuous dependence of a weak limit π f of the sequence of iterates (f n (x , ·)) n ∈ N of f, defined by f 0 (x , ω) = x , f n + 1 (x , ω) = f (f n (x , ω) , ω n + 1) for x ∈ X and ω = (ω 1 , ω 2 , ⋯) . Moreover for X taken as a Hilbert space we characterize π f via the functional equation φ f (u) = ∫ Ω φ f (Λ (ω) u) φ ξ (u) P (d ω) with the aid of its characteristic function φ f . We also indicate the continuous dependence of a solution of that equation. [ABSTRACT FROM AUTHOR]

Our main result is that we describe the solutions g , f : S → C of the functional equation g (x σ (y)) = g (x) g (y) - f (x) f (y) + α f (x σ (y)) , x , y ∈ S , where S is a semigroup, α ∈ C is a fixed constant and σ : S → S an involutive automorphism. [ABSTRACT FROM AUTHOR]

In this paper we find the solutions of the functional equation f (x y) = g (x) h (y) + ∑ j = 1 n g j (x) h j (y) , x , y ∈ M , where M is a monoid, n ≥ 2 , and g j (for j = 1 , ... , n ) are linear combinations of at least 2 distinct nonzero multiplicative functions. [ABSTRACT FROM AUTHOR]

In an earlier paper we discussed the composition of derivations of order 1 on a commutative ring R, showing that (i) the composition of n derivations of order 1 yields a derivation of order at most n, and (ii) under additional conditions on R the composition of n derivations of order exactly 1 forms a derivation of order exactly n. In the present paper we consider the composition of derivations of any orders on rings. We show that on any commutative ring R the composition of a derivation of order at most n with a derivation of order at most m results in a derivation of order at most n + m . If R is an integral domain of sufficiently large characteristic, then the composition of a derivation of order exactly n with a derivation of order exactly m results in a derivation of order exactly n + m . As in the previous paper, the results are proved using Leibniz difference operators. [ABSTRACT FROM AUTHOR]

We study the functional equation for unknown functions f , g , h , k , ℓ : S → C , where S is a semigroup and m 1 , m 2 : S → C are multiplicative functions. The study is divided into two main parts: m 1 = m 2 and m 1 ≠ m 2 . In some cases we assume that one or more of the unknown functions is central and/or that S is a monoid. The solutions are found in all cases under the umbrella assumption that S is a commutative monoid. The continuous solutions on topological semigroups are also found. [ABSTRACT FROM AUTHOR]

Let G be a locally compact Hausdorff group and ψ : G → G is a continuous anti-endomorphism that need not be involutive. Our main goal is to extend previous results about Kannappan's functional equation g (x y z 0) + g (x ψ (y) z 0) = g (x) g (y) , where z 0 is an arbitrarily fixed element in G. We introduce and study the continuous solutions h , f , g : G → C of the integral-functional equation ∫ G { h (x y t) + μ (y) h (x ψ (y) t) } d ν (t) = f (x) g (y) , x , y ∈ G , where μ : G → C is a continuous multiplicative function satisfying μ (x ψ (x)) = 1 for all x ∈ G , and ν is a regular, compactly supported, complex-valued Borel measure on G. [ABSTRACT FROM AUTHOR]

We consider a functional equation which is a generalization of Whitehead's functional equation on groups. We present its general solution on free groups. Solutions on other selected groups are also given. [ABSTRACT FROM AUTHOR]

Several results concerning the homomorphism functional equation f x ∨ y = f x ∨ f y , in the class of semilattices, as well as other similar equations, are presented in the paper. [ABSTRACT FROM AUTHOR]

A derivation (of order 1) satisfies the reduction formula f (x k) = k x k - 1 f (x) for any integer k. In this article we find corresponding reduction formulas for derivations of higher order on commutative rings. More precisely, for every derivation f of order n and every positive integer k we find an explicit formula for f (x k) as a linear combination of x k - 1 f (x) , x k - 2 f (x 2) , ... , x k - n f (x n) . The proof hinges on the hypergeometric identity ∑ k ≥ 0 (- 1) k n k n + d + 1 - k n - j d + j - k j = n j for any positive integer n, nonnegative integer d, and integer j satisfying 0 ≤ j ≤ n . We prove this identity via the WZ-method. [ABSTRACT FROM AUTHOR]

We resume a problem of Gy. Szabó concerning the existence of non-trivial even orthogonally additive mappings. Using methods from projective geometry, we derive several equivalent conditions for the existence of such mappings. As a by-product, we get Szabó's main result in the general context of orthogonality spaces in Rätz's sense over arbitrary ordered fields. [ABSTRACT FROM AUTHOR]

In this paper, we determine the complex-valued solutions of the functional equation f (x α σ (y)) + h (τ (y) x) = 2 f (x) k (y) for all x , y ∈ G , where G is a group, α ∈ G is a fixed element and σ , τ : G → G are involutions. [ABSTRACT FROM AUTHOR]

The aim of this work is to study the conditional distributivity given by the following functional equation: F (x , U (y , z)) = U (F (x , y) , F (x , z)) , x , y , z ∈ [ 0 , 1 ] , 0 < U (y , z) < 1 where F is a nullnorm and U is a uninorm. The paper discusses one case: F is continuous and U ∈ U max ( U min ) with continuous underlying operators. [ABSTRACT FROM AUTHOR]

The general solution of the multiplicative symmetric functional equation f (f (x) y) = f (x f (y)) , in abelian groups, was given by Dhombres. But it is still an open problem in non-abelian groups and abelian quasi-groups. In this paper, by using a new approach introduced by the author, we give a vast class of its solutions in arbitrary semigroups. It also works for non-abelian groups, and gives its general solution in abelian groups with a different representation from Dhombres's solution. Also, we introduce some related equations and closed relationships between decomposer type and multiplicative symmetric equations. It is worth noting that the new approach and methods can be used for studying the equation on magmas (groupoids) as the most general algebraic structures for the topic. At last we pose a conjecture that guesses the general solution in arbitrary groups (and possibly semigroups). [ABSTRACT FROM AUTHOR]

We consider the differentiability of the solution F : I × R → I of the translation equation, where I is an interval. The stability of this equation in classes of differentiable functions is considered, too. [ABSTRACT FROM AUTHOR]

The connected door space is an enigmatic topological space in which every proper nonempty subset is either open or closed, but not both. This paper provides an elementary proof of the classification theorem of connected door spaces. More importantly, we show that connected door topologies can be viewed as solutions of the valuation f(A)+f(B)=f(A∪B)+f(A∩B) and the equation f(A)+f(B)=f(A∪B). In addition, some special solutions, which can be regarded as a union of connected door spaces, are provided. [ABSTRACT FROM AUTHOR]

The general solutions of the functional equations F (p q) = q α G (p) + p α H (q) + K (p) L (q) , F (p q) = q β G (p) + p α H (q) + c K (p) K (q) and f (p q) = q α f (p) + p α f (q) + c g (p) g (q) with g (1) = 0 and c, a given nonzero real constant, are obtained. Here F, G, H, K, L, f and g are real-valued functions each with domain I, the unit closed interval and 1 ≠ α > 0 , α ∈ R ; 1 ≠ β > 0 , β ∈ R . [ABSTRACT FROM AUTHOR]

This paper examines various aspects related to the Cauchy functional equation $$f(x+y)=f(x)+f(y)$$ , a fundamental equation in the theory of functional equations. In particular, it considers its solvability and its stability relative to subsets of multi-dimensional Euclidean spaces and tori. Several new types of regularity conditions are introduced, such as one in which a complex exponent of the unknown function is locally measurable. An initial value approach to analyzing this equation is considered too and it yields a few by-products, such as the existence of a non-constant real function having an uncountable set of periods which are linearly independent over the rationals. The analysis is extended to related equations such as the Jensen equation, the multiplicative Cauchy equation, and the Pexider equation. The paper also includes a rather comprehensive survey of the history of the Cauchy equation. [ABSTRACT FROM AUTHOR]

Let S be a semigroup. We describe the solutions f , g : S → C of the functional equation f (x y) = f (x) g (y) + g (x) f (y) - g (x) g (y) , x , y ∈ S , in terms of multiplicative functions on S and solutions of the particular case φ (x y) = φ (x) χ (y) + φ (y) χ (x) , x , y ∈ S , of the sine addition law in which χ : S → C is a non-zero multiplicative function and φ : S → C . [ABSTRACT FROM AUTHOR]

Let X, Y be linear spaces over a field K . Assume that f : X 2 → Y satisfies the general linear equation with respect to the first and with respect to the second variables, that is, for all x , x i , y , y i ∈ X and with a i , b i ∈ K \ { 0 } , A i , B i ∈ K ( i ∈ { 1 , 2 } ). It is easy to see that such a function satisfies the functional equation for all x i , y i ∈ X ( i ∈ { 1 , 2 } ), where C 1 : = A 1 B 1 , C 2 : = A 1 B 2 , C 3 : = A 2 B 1 , C 4 : = A 2 B 2 . We describe the form of solutions and study relations between (∗) and (∗ ∗) . [ABSTRACT FROM AUTHOR]

We investigate the orthogonality preserving property for pairs of operators on inner product C ∗ -modules. Employing the fact that the C ∗ -valued inner product structure of a Hilbert C ∗ -module is determined essentially by the module structure and by the orthogonality structure, pairs of linear and local orthogonality-preserving operators are investigated, not a priori bounded. We obtain that if A is a C ∗ -algebra and T , S : E → F are two bounded A -linear operators between full Hilbert A -modules, then ⟨ x , y ⟩ = 0 implies ⟨ T (x) , S (y) ⟩ = 0 for all x , y ∈ E if and only if there exists an element γ of the center Z (M (A)) of the multiplier algebra M (A) of A such that ⟨ T (x) , S (y) ⟩ = γ ⟨ x , y ⟩ for all x , y ∈ E . Varying the conditions on the operators T and S we obtain further affirmative results for local operators and for pairs of a bounded and an unbounded A -linear operator with bounded inverse. [ABSTRACT FROM AUTHOR]

Let G be an abelian group, $${\mathbb{C}}$$ be the field of complex numbers, $${\alpha \in G}$$ be any fixed element and $${\sigma : G \to G}$$ be an involution. In this paper, we determine the general solution $${f, g : G \to \mathbb{C}}$$ of the functional equation $${f(x + \sigma y + \alpha) + g(x + y + \alpha) = 2f(x)f(y)}$$ for all $${x, y \in G}$$ . [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and n a positive integer. We show that the composition of n derivations of order 1 results in a derivation of order n on R. If in addition R is an integral domain of characteristic 0, then the composition of n nontrivial derivations of order 1 forms a nontrivial derivation of order n. This is also true for integral domains of characteristic larger than n!, but not for integral domains of characteristic n!, nor for commutative rings (even of characteristic 0) which are not integral domains. We prove our results by the use of Leibniz differences. One corollary is that nontrivial derivations of all orders exist on R . [ABSTRACT FROM AUTHOR]

This paper studies the solutions of the integral Van Vleck's functional equation for the sine and of the integral Kannappan's functional equation. Topics discussed include complex-valued solutions of the functional equation, d'Alembert's functional equation, and generalization of the lemmas of H. Stetkær.

In the present paper, we determine the complex-valued solutions ( f, g) of the functional equation in the setting of groups and monoids that need not be abelian, where $${\sigma,\tau}$$ are involutive automorphisms. We prove that their solutions can be expressed in terms of multiplicative and additive functions. [ABSTRACT FROM AUTHOR]

For a family (A x) x ∈ (0 , 1) of integral quasi-arithmetic means satisfying certain measurability-type assumptions we search for an integral mean K such that K ( (A x (P)) x ∈ (0 , 1) ) = K (P) for every compactly supported probability Borel measure P . Also some results concerning the uniqueness of invariant means will be given. [ABSTRACT FROM AUTHOR]

In this paper we prove that if a generalized polynomial function f satisfies the condition f( x) f( y) = 0 for all solutions of the equation x + y = 1, then f is identically equal to 0. [ABSTRACT FROM AUTHOR]

Let M be a topological monoid, let ψ : M → M be a continuous anti-homomorphism of M, and let μ : M → C be a continuous multiplicative function such that μ (x ψ (x)) = 1 for all x ∈ M . We describe, in terms of multiplicative functions, additive functions and characters of 2-dimensional representations of M, the solutions (w, g), where w , g : M → C , such that w is central and g is continuous, of the new functional equation w (x y) + μ (y) w (x ψ (y)) = 2 w (x) g (y) , x , y ∈ M. We also treat the equation on compact groups and the special equation (when μ = g = 1 ): Jensen's functional equation. [ABSTRACT FROM AUTHOR]

Let X be a real Polish vector space, D ⊂ X be a universally Baire non-Haar-meager set, and let f : X → R satisfy f (x + y) - f (x) - f (y) ∈ Z for all x , y ∈ X and f (D) ⊂ (- 1 3 , 1 3) + Z . We prove that there exists a unique linear functional L : X → R with f (x) - L (x) ∈ Z for x ∈ X and next we apply it to functionals which are orthogonally additive modulo Z . These new results are topological counterparts to some results of Brzdȩk (Results Math 30:25–38, 1996) and also generalize (to some extent) his results. [ABSTRACT FROM AUTHOR]

Our main result is the solution of the functional equation f (x σ (y)) + h (τ (y) x) = 2 f (x) k (y) for complex-valued functions f, h, k on monoids generated by their squares. Here σ and τ are involutive automorphisms of the monoid. [ABSTRACT FROM AUTHOR]

Let C be an affine plane curve. We consider additive functions f:K→K for which f(x)f(y)=0, whenever (x,y)∈C. We show that if K=R and C is the hyperbola with defining equation xy=1, then there exist nonzero additive functions with this property. Moreover, we show that such a nonzero f exists for a field K if and only if K is transcendental over Q or over Fp, the finite field with p elements. We also consider the general question when K is a finite field. We show that if the degree of the curve C is large enough compared to the characteristic of K, then f must be identically zero. [ABSTRACT FROM AUTHOR]

In this paper, we obtain hyperstability results for the $${\sigma}$$ -Drygas functional equation and the inhomogeneous $${\sigma}$$ -Drygas functional equation on semigroups. Namely, we show that a function satisfying the $${\sigma}$$ -Drygas equation approximately must be exactly the solution of it. [ABSTRACT FROM AUTHOR]

Let G be a group, $${\mathbb{C}}$$ be the field of complex numbers, z be any fixed, nonzero element in the center Z( G) of the group G, and $${\sigma : G \to G}$$ be an involution. The main goals of this paper are to study the functional equations $${f(x{\sigma}yz_{0}) - f(xyz_{0}) = 2f(x)f(y)}$$ and $${f(x{\sigma}yz_{0}) + f(xyz_{0}) = 2f(x)f(y)}$$ for all $${x, y \in G}$$ and some fixed element z in the center Z( G) of the group G. [ABSTRACT FROM AUTHOR]

This paper treats a functional equation of d'Alembert, namely f( x + y) − f( x − y) = g( x) h( y), in the setting of groups and monoids that need not be abelian. Of the several possibilities for generalizing x − y to the non-abelian case, here we replace it by σ( y) x where σ is an involutive automorphism, and find the complex-valued solutions ( f, g, h) of f( xy) − f( σ( y) x) = g( x) h( y) in terms of multiplicative and additive functions. [ABSTRACT FROM AUTHOR]

Cain, Clark and Rose defined a function $${{f\colon\mathbb{R}^n \to \mathbb{R}}}$$ to be vertically rigid if graph( cf) is isometric to graph( f) for every c ≠ 0. It is horizontally rigid if graph $${(f(c \vec{x}))}$$ is isometric to graph( f) for every c ≠ 0 (see Cain et al., Real Anal. Exch. 31:515-518, ). In Balka and Elekes (J. Math. Anal. Appl. 345:880-888, ) the authors of the present paper settled Janković's conjecture by showing that a continuous function of one variable is vertically rigid if and only if it is of the form a + bx or $${{a+be^{kx} (a,b,k \in \mathbb{R})}}$$. Later they proved in Balka and Elekes (Real. Anal. Exch. 35:139-156, ) that a continuous function of two variables is vertically rigid if and only if after a suitable rotation around the z-axis it is of the form $${a + bx + dy, a +s(y)e^{kx}}$$ or $${{a + be^{kx} + dy (a,b,d,k \in \mathbb{R}, k \neq 0, s\colon \mathbb{R}\to \mathbb{R}\,{\rm is\, continuous})}}$$. The problem remained open in higher dimensions. The characterization in the case of horizontal rigidity is surprisingly simpler. Richter (Real Anal. Exch. 35:343-354, ) proved that a continuous function of one variable is horizontally rigid if and only if it is of the form $${{a+bx (a,b\in \mathbb{R})}}$$. The goal of the present paper is to prove that a continuous function of two variables is horizontally rigid if and only if it is of the form $${{a+ bx + dy (a,b,d \in \mathbb{R})}}$$. This problem also remains open in higher dimensions. The main new ingredient of the present paper is the use of functional equations. [ABSTRACT FROM AUTHOR]

We observe that bisymmetry is in fact the assertion of the Fubini theorem and we describe the form of general bisymmetric operations on some function spaces. [ABSTRACT FROM AUTHOR]

We solve two Pexiderized extensions of the quadratic functional equation on monoids. The equation is extended by replacing the group inversion in the classical quadratic equation with an involution, which may be either homomorphic or anti-homomorphic. Also two extensions of the Drygas functional equation are solved in the same setting. In all results it is assumed that at least one unknown function is central. [ABSTRACT FROM AUTHOR]

We solve in some cases the equation in the definition of the concomitant of a geometric object for each of the four functions in this equation. Remarks and examples are given, too. [ABSTRACT FROM AUTHOR]