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The paper contains fixed point theorems of Edelstein's type for contractive mappings in (η,ϕ)-chainable uniform spaces.

The genus of a complete graph equals the least integer greater than or equal to (E-3V+6)/6, whereE andV are the numbers of edges and vertices of the graph. This paper extends the class of graphs known to have this property, concentrating on graphs whose number of vertices is congruent to 6 modulo 12.

In this paper various functional equations which arise in the study of binary operations on the set of uniform probability distribution functions are considered and solved.

LetR be a ring. A bi-additive symmetric mappingD(.,.): R × R → R is called a symmetric bi-derivation if, for any fixedy ∈ R, a mappingx ↦ D(x, y) is a derivation. The purpose of this paper is to prove some results concerning symmetric bi-derivations on prime and semi-prime rings. We prove that the existence of a nonzero symmetric bi-derivationD(.,.): R × R → R, whereR is a prime ring of characteristic not two, with the propertyD(x, x)x = xD(x, x), x ∈ R, forcesR to be commutative. A theorem in the spirit of a classical result first proved by E. Posner, which states that, ifR is a prime ring of characteristic not two andD 1,D 2 are nonzero derivations onR, then the mappingx ↦ D 1(D 2 (x)) cannot be a derivation, is also presented.

In this paper we consider and solve a functional equation in a restricted class of operations on the space of distribution functions. As a consequence, we obtain a characterization of convolution.

For a fixed non-negative integerp, letU 2p = {U 2p (n)},n ≥ 0, denote the sequence that is defined by the initial conditions $$U_{2p} (0) = U_{2p} (1) = U_{2p} (2) = = U_{2p} (2p) = 1$$ and the restricted subadditive recursion $$U_{2p} (n + 2p + 1) = \mathop {\min }\limits_{0 \leqslant / \leqslant p} (U_{2p} (n + l) + U_{2p} (n + 2p - l)),n \geqslant 0$$ U 2p is of importance in the theory of sequential search for simple real zeros of real valued continuous 2p-th derivatives In this paper, several closed form expressions forU 2p (n), n > 2p, are determined, thereby providing insight into the structure ofU 2p Two of the properties thus illuminated are (a) the existence of exactlyp + 1 limit points (1 + 1/(p + 1 +i), 0 ≤i ≤p) of the associated sequence {U 2p (n + 1)/U 2p (n)},n ≥ 0, and (b) the relevance toU 2p of the classic number theoretic function ord

This paper adds the finishing touches to an algorithmic treatment of quadratic forms over the rational numbers. The Witt index of a rational quadratic form is explicitly computed. When combined with a recent adjustment in the Haase invariants, this gives a complete set of invariants for rational quadratic forms, a set which can be computed and which respects all of the standard natural operations (including the tensor product) for quadratic forms. The overall approach does not use (at least explicitly) anyp-adic methods, but it does give the Witt ring of thep-adics as well as the Witt ring of the rationals.

In the first part of this paper a separation theorem is proved for disjoint subsemigroups of a given abelian semigroup. Applying this result, separation theorems and characterization theorems are obtained for semiinternal functions.

Imagine that randomly oriented objects in the shape of a regularn-sided polygon are moving on a conveyor. Our aim is to specify sequences composed of two different rigid motions which, when performed on these objects, will reposition them in all possible ways. We call such sequencesfacing sequences. (Expressed in group theoretical terms, a facing sequence in a groupG is a sequence of elementsa1,a2, ...,an fromG such thatG={e,a1,a1a2, ...,a1a2 ...an}). In this paper we classify various kinds of facing sequences and determine some of their properties. The arguments are group theoretical and combinatorial in nature.

F. H. Jackson defined a generalization of the factorial function by $$1(1 + q)(1 + q + q^2 ) \cdot \cdot \cdot (1 + q + q^2 + \cdot \cdot \cdot + q^{n - 1} ) = (n!)_q $$ forq>0. He also generalized the gamma function, both for 0 1. Askey then obtained analogues of many of the classical facts about theq-gamma function for 01. It turns out that the log convexity off together with the initial condition and the functional equation no longer forcesf to be theq-gamma function. A stronger condition is needed than the log convexity, and two sufficient conditions are given in this paper. Also we will consider the behavior of theq-gamma function asq-changes forq>1.

In this paper we consider a problem posed by W Mills D Robbins and H Rumsey for a certain plane partition generating functionZn(x, m) The special caseZn(1,m) is the generating function that arose in the weak Macdonald conjecture Mills—Robbins—Rumsey conjectured thatZn(2,m) also possesses a nice finite product representation Their conjecture is proved as Theorem 1

Let (X, ℱ) be a topological space. For any functionf: D→[− ∞, ∞) (whereD ⊂ X), thelower hull mf:D →[− ∞, ∞) off is defined by $$m_f (x) = m_{f\left| T \right.} (x) = \mathop {\sup \inf }\limits_{U \in T_x \in U \cap D} f(t),x \in D,$$ where ℱx denotes the family of all open sets containing x. The main result of the paper is that, ifX is a real linear topological Baire space,D ⊂ X is convex and open, andf: D→[− ∞, ∞) isJ-convex, then the functionmf is convex and continuous. (In the case of a single real variable this result goes back to F. Bernstein and G. Doetsch, 1915.)

Let ℐ be a finite regular incidence-polytope. A realization of ℐ is given by an imageV of its vertices under a mapping into some euclidean space, which is such that every element of the automorphism group Γ(ℐ) of ℐ induces an isometry ofV. It is shown in this paper that the family of all possible realizations (up to congruence) of ℐ forms, in a natural way, a closed convex cone, which is also denoted by ℐ The dimensionr of ℐ is the number of equivalence classes under Γ(ℐ) of diagonals of ℐ, and is also the number of unions of double cosets Γ*σΓ* ∪ Γ*σ−1Γ* (σ ∉ Γ*), where Γ* is the subgroup of Γ(ℐ) which fixes some given vertex of ℐ. The fine structure of ℐ corresponds to the irreducible orthogonal representations of Γ(ℐ). IfG is such a representation, let its degree bed G , and let the subgroup ofG corresponding to Γ* have a fixed space of dimensionw G . Then the relations $$\begin{array}{l} \Sigma _G w_G d_G = \upsilon - 1, \\ \Sigma _G {\textstyle{1 \over 2}}w_G (w_G + 1) = r, \\ \Sigma _G w_G ^2 = \bar w \\ \end{array}$$ hold, where ℐ hasv vertices, and $$\bar w$$ is the number of double cosets Γ*σΓ* (σ ∉ Γ*). The second relation corresponds to the fact that the realizations associated with a given irreducible representationG form a cone of dimension 1/2w G (w G + 1), which forw G ⩾ 2 has as base the convex hull of a projective space of dimensionw G − 1 embedded in an ellipsoid of dimension 1/2w G (w G + 1) − 2. Comparison of the second and third relations leads to a curious connexion between the cone ℐ and the group Γ(ℐ), namely, that the following conditions are equivalent: (1)r = $$\bar w$$ , (2) ℐ is polyhedral, (3)w G ⩽ 1 for all irreducible orthogonal representationsG of Γ(ℐ), (4) σ-1 ∈ Γ*σΓ* for each σ ∈ Γ(ℐ). The realization cones ℐ are described for various regular polytopes.

This is an expository paper describing geometries associated to the groups of Lie type, following Jacques Tits' theory of buildings and his earlier theory of shadows. The geometry of shadows in the case of the general linear group is simply the set of subspaces of the underlying vector space with the relation of inclusion. The vertices and edges of the Tits building in this case also correspond to subspaces and inclusion respectively. That the automorphism group of this geometry is the semi-direct product of the projective general linear group and the group of field automorphisms is called thefundamental theorem of projective geometry. The theory of buildings and shadows provides analogous geometries and an analogous theorem for each group of Lie type.

The paper presents a functional equation approach to the construction and characterization of proportion functions on three-dimensional boxes, extending some classical considerations of plane geometry which were motivated by architectural problems. LetD : = (0, ∞) andI : = [1, ∞). A functionf: D 3 →I will be called normalized iff(x, x, x) = 1 for allx > 0 and symmetric iff(x 1,x 2,x 3) =f(x σ(1),x σ(2),x σ(3)) for allx 1,x 2,x 3 > 0 and for any permutation σ of the set {1, 2, 3}. A proportion function in three dimensions is a three-place functionf fromD 3 intoI which is normalized, symmetric and satisfies a condition of the form $$f(x,y,z) = f(\alpha (x,y,z))forallx,y,z > 0,$$ for all mappings α:D 3 →D 3 belonging to a fixed setB of bijections ofD 3. Two boxes of sidesx, y, z and ξ,ηz with the common edgez are homothetic iff{ξ, η} = {zy/x, z 2/x}. This motivates to characterize functionsf fromD 3 intoI which are normalized, symmetric and satisfy $$f(x,y,z) = f\left( {z,\frac{{zy}}{x},\frac{{z^2 }}{x}} \right)forallx,y,z > 0.$$ Also the equation $$f(x,y,\sqrt {xy} ) = f\left( {\frac{{y^2 }}{{\sqrt {xy} }},\sqrt {xy} ,y} \right)forallx,y,z > 0$$ (case of two boxes with a common face) in place of the previous one is important in this context. All the corresponding proportion functions (replace α in the definition of a proportion function by the functions in the functional equations above) are determined.

This paper is concerned with a class of iterative processes of the formu k+1 =Tu k (k = 0, 1, ⋯) for solving nonlinear operator equationsu = Tu orFu = 0. By studying the relationship between a linear functional inequalityϕ(Ah) β(h) + γ(h) ⩽ ϕ(h) and estimates for the iteration operatorT a general semilocal convergence theorem is obtained. The theorem contains as special cases theorems for various iterative methods. Numerical examples illustrate the accuracy of the error estimates for the approximationu k .

A periodic binary array is said to besequential if and only if every line of the array is occupied by a given periodic binary sequence or by some cyclic shift or reversal of this sequence. This paper extends earlier results for arrays built on the square grid, characterising further sequential arrays with two ones per period, and those with three consecutive ones per period.

It is shown how a set of canonical variables in the sense of Rund [5] can be associated with a given extremal of a multiple integral variational problem in a simple, direct manner. The definition of these variables in a previous paper [1], which is concerned with the problem of imbedding a given extremal in anr-geodesic field, is thereby clarified and abbreviated considerably. A theorem due essentially to Boerner, which is crucial to the imbedding theorem given in [1], is proved more easily and under less restrictive hypotheses than in [1]. Furthermore, it is shown how the present definition of the canonical variables allows one to eliminate from the geodesic field theory of Caratheodory the restriction that the Lagrangian be non-vanishing along the extremal.

This paper is concerned with the problem of solving linear difference equations of ordern with constant coefficients and with given initial conditions in which the variable runs not only through the integers but over ℝ. The main idea is the introduction of a suitable commutative ring of functions with discrete convolution as multiplication rule which works, although it is not a field. The existence of inverses is studied and, after the introduction of suitable functions, the problem is reduced by means of a Laplace-like relation to an algebraic equation. Examples of application are given. Finally some remarks make the connection with the Operational Calculus of Mikusinski and with the Operational Calculus of Fenyo. The advantages of this method lie in the fact that it is applicable to functions others than the step functions, in its simplicity from the theoretical point of view, in its usefulness even when computation is required and in its formal similarity to the classical treatment of linear differential equations with constant coefficients.

In this paper we consider the dimer problem forM×N rectangular arrays, whereM andN are positive integers,M being small. A unified method for solving such problems is given, and is applied to the casesM=2 (the solution of which is already known (see [1, 4]) andM=3 which, it seems, has not previously been solved. The method is also applicable to a wider class of problems, and some examples of such applications are given. In theory it is always possible to obtain a closed solution to these problems in the form of rational generating functions. In practice this is feasible only for very small values ofM, but the methods described will enable numerical results for larger values ofM to be found by means of a computer program.

It is, of course, well known to students of calculus that the extremal nature of a function f at a critical point is decided if the so-called discriminant (or Hessian) given by the expression A =f2-fxxfyy evaluated at the critical point is nonzero. It is also known through simple examples that, in the so-called degenerate case when A = 0 at the point, f may have either an extremum or a saddle point. Consequently, the extremal nature of fin this case is indeterminate from a knowledge of the second derivatives at the point in question alone and higher order partial derivatives at the point must be considered. (It is interesting that during the last century a certain confusion existed concerning the degenerate case, even apparently in the minds of some renowned mathematicians. For a short account of this history of the degenerate case see [1].) Systematic, yet straightforward and simple methods by which to take into account the higher order derivatives seem, however, difficult to come by. In fact, the only method known to the author which offers an essentially complete account of this case is due to Freedman [2]. (His techniques are concerned with the solution of the equationf (x, y)= 0 for x = x (y) but implicitly yield information about extrema as well. He also considers cases other than the degenerate case. Also in a recent paper I-3] Butler and Freedman consider the case when the lowest order terms off are cubic or higher; as stated below, we do not consider this case here.) The purpose of this note is to present a complete method for determining the extremal nature of f on the basis of its derivatives at the point in question under the two assumptions that (i)f possesses the necessary number of partial derivatives and (ii) the lowest order terms in its Taylor expansion with remainder at the point are quadratic. Under these conditions we will show how the extremal nature off may be decided in the degenerate case through a sequence of tests each involving a discriminant and each having a degenerate case, whose occurence, however, can be followed by the next test of the sequence. Each test has the same format as the standard discriminant test using A, which itself may be considered as simply the first test of the sequence. Although they accomplish more or less the same ends, the details

in the field of measurable functions? Hosszfi solved equation (1) under the assumption of differentiability. The aim of this paper is to answer the question as originally posed. We will give the most general solution of (l) in the domain of distributions. Let us introduce the following notations. Ak: the linear space of L. Schwartz test functions of k variables ; A~: the linear space of distributions on Ak; D r =dr/dtr; D~=Or/c~xr; D~z=c~r/c3y r ( r = l , 2,.. .); D o: the identity operator; AI(~ ) = {h:heA I and Drh=O for x = ~ , r = 0 , 1, 2,.. .}; ~(~) = {h:heAl(c 0 and c~¢supp h} (i.e. there exists a neighborhood of ~ in which h vanishes identically); A~(~): A~(~): c~ : g6 A ~(a) i f fge A 2 and D ~D"~g = 0 along the straight line x = a (n, m = 0, 1,2,.. .); ge A ~ (c¢) iffg~ A 2 and D'~D~g = 0 along the straight line y = a (n, m = 0, 1,2,...) ; the linear space of all infinitely differentiable functions of k variables (with arbitrary support), all derivatives of which are everywhere continuous

The parameters S and M were introduced in (1), (3) and (4) in connection with the stochastic rank and the term rank of a matrix. The results of this paper involve these same parameters. Two of these results are the following. Let A be a matrix of non-negative integers with sum S and maximum row or column sum M and let a = [S/M] (the greatest integer in S/M). It is shown in theorem 1 that A is expressible as a sum of subpermutat ion matrices of rank a and a sub-permutation matrix of rank q, 0 ~< q < a. Further, let K be a bipartite graph with finite vertex sets X and Y. Let S be the total number of edges and let M be the maximum number of edges which have the same vertex. It is shown as a consequence of theorem 3 that K has a transversal G of order a = [S/M] with vertex sets U and V such that every vertex of X which has M edges is an element of U and every vertex of Y which has M edges is an element of V. Theorem 1 may be proved as a corollary of theorem 3 but the connection of theorem 1 with the doubly stochastic extension of a matrix is interesting. It is on this connection that the proof of theorem 1 in section 3 is based.

The lower bound was subsequently improved by Wilker [8] to 1.059, and by the author [2] to 1.28467. An improved upper bound of 1.5403 . . . . (9+x/41)/10 was proved in [3], but the arguments there, although they apply to sphere packings in higher dimensions, are too general in nature to yield a significant improvement of this bound. In this paper, we present a method of attack which gives both upper and lower bounds. In fact, for any integer x, we obtain bounds 2 (K)

wheref~k and h~ are given functions and cp~ are unknown. Theorem 1 below yields the uniqueness of solutions cp~ (i= 1 .... , n) in certain classes B~. In the case n = 1 this result is related to the investigations of B. Choczewski [2] and M. Kuczma [4] regarding the asymptotic behavior at the fixed points of the func t ionfof solutions (the so-called regular solutions) of the equation ~o (x) = h (x, q~ I f (x) ] ) (2) or of its particular cases. As a corollary to Theorem 1 we obtain a uniqueness theorem for the differentiable solutions of the system (1). The main results concerning the differentiable solutions of equation (2) are due to B. Choczewski [1] (cf. also [3], Chapter IV). Some recent contributions are contained in the author's papers [7] and [8]. In general the solution of equation (2) as well as that of system (1) depends on an arbitrary function (cf. [3], Theorems 3.1, 4.1, 12.5), therefore conditions ensuring the uniqueness of solutions are of considerable importance in the theory of functional equations in a single variable (cf. [5]).

A well known theorem of H. Zassenhaus [6], which also appears in M. Hall [1], p. 382, states that a finite sharply 2-transitive permutation group 1) is isomorphic to the group of linear transformations x--', a + m . x on a finite near-field. M ore generally, one can show that the group of linear transformations on an algebraic structure called a near-domain (see Definition A) is sharply 2-transitive and that, up to isomorphism as permutation groups, each sharply 2-transitive group is isomorphic to the group of linear transformations on a uniquely determined near-domain, [2], [3], [4]. Hence the class of sharply 2-transitive groups is completely characterized by the class of neardomains. To the authors' knowledge, the question as to the existence of near-domains which are not near-fields is open. Some results on this question are given in [4]. In this paper, a theorem is proved which states that a near-domain is a near-field if and only if a certain subset is finite. Thus the theorem of Zassenhaus which states, in the terminology used here, that every finite near-domain is a near-field, can be generalized to more relaxed conditions of finiteness (Theorem A, Coroliaries 1 and 2). The latter two results are then interpreted in terms of sharply 2-transitive groups (Corollary 3).

The intent of this paper is to show that first integrals of the discrete Euler equation can be determined explicitly by investigating the invariance properties of the discrete Lagrangian. The result obtained is a discrete analog of the classical theorem of E. Noether in the Calculus of Variations.

The problem of tiling space by translates of certain star bodies, called ‘crosses’ and ‘semicrosses’, is intimately connected with finding a subsetA of a finite abelian groupG such that for a particular subset of the integersS each non-zero element ofG is uniquely expressible in the forms·g withs inS andg inA. This paper examines some of the algebraic questions raised; in particular it obtains bounds on the number of elements inS, constructs factorizations ofZ p n , and presents an example of a setS that factors no group.

A function B:G x G ~ A is called an additive bimorphism of a group G into a commutative group A if B (xy, z) = B (x, z) + B (y, z) B (z, xy) = B (z, x) + B (z, y) holds for all x, y, zEG. A function f : G ~ A is a quadratic form on G if f ( x y ) + f ( x y -1) = 2 f ( x ) + 2 f ( y ) (1) holds for all x, yeG. We assume that aeA and 2a = 0 implies a =0. I f B is an additive bimorphism then x ~ g (x, x) is a quadratic form on G. The main results of this paper are the following two theorems.

Many theorems of integration theory are true for a wide range of definitions of integrals. One such theorem is that giving integration by parts, and we discuss it in this paper.

A new canonical formalism for unconstrained m-fold integral problems in the calculus of variations has recently been introduced by Rund [6]. This theory possesses several advantages over the earlier canonical formalisms developed by Carath6odory [2] and de Donder-Weyl [3], [8], [9]. In particular, it is based on the geodesic field theory of Carath6odory (op. cir.) and as such is applicable to problems with variable boundaries (unlike the field theory of Weyl, cf. [5] pp. 248-250). It is, moreover, far less abstruse than the formalism of Carath6odory, and has the advantage that it reduces to the usual canonical formalism (with but a slight modification) for the case m = 1. The purpose of the present paper is twofold. In the first place, it is shown in Sections 2 and 3 how the formalism of Rund can be extended to cover also the case of multiple-integral problems subject to constraints (problem of Lagrange). This formalism also reduces, with only some slight changes, to the usual formalism for singleintegral problems of Lagrange for the case m = 1. The applications of this formalism to the field theory of Carath6odory are briefly discussed in Section 4. Secondly, two theorems are proved in Section 5 which are generalizations of a basic result of Carath6odory ([1] § 30) according to which

Since the monograph by Amir that appeared in 1986, a lot of attention has been given to the problem of characterizing, by means of properties of the norms, when a Banach space is indeed a Hilbert space, i.e., when the norm derives from an inner product. In this paper, similar investigations will be carried out for smooth spaces instead of inner product spaces. We consider the approximate orthogonalities in real normed spaces. We show that the relations approximate semi-orthogonality and approximate ρ+-orthogonality are generally incomparable (unless the normed space is smooth). As a result, we give a characterization of smooth spaces in terms of those approximate orthogonalities.

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 2 + y 2 = 1, then f is identically equal to 0.

Let X be a real linear space. We characterize solutions \({f, g : X \rightarrow \mathbb{R}}\) of the equation f(x + g(x)y) = f(x)f(y), where f is continuous on rays. Our result refers to papers by Brzdȩk (Acta Math Hungar 101:281–291, 2003), Chudziak (Aequat Math, doi:10.1007/s00010-013-0228-4, 2013) and Jablonska (J Math Anal Appl 375:223–229, 2011).

The aim of this paper is to solve the orthogonality equation with two unknown functions. This problem was posed by J. Chmieli´nski during two international conferences.

In this paper we consider stability in the Ulam–Hyers sense, and in other similar senses, for the five equivalent definitions of one-dimensional dynamical systems.

A graceful labeling of a graph G with q edges is an injective assignment of labels from {0, 1, . . . , q} to the vertices of G so that when each edge is assigned the absolute value of the difference of the vertex labels it connects, the resulting edge labels are distinct. A labeling of the first kind for coronas \({C_n \odot K_1}\) occurs when vertex labels 0 and q = 2n are assigned to adjacent vertices of the n-gon. A labeling of the second kind occurs when q = 2n is assigned to a pendant vertex. Previous research has shown that all coronas \({C_n \odot K_1}\) have a graceful labeling of the second kind. In this paper we show that all coronas \({C_n \odot K_1}\) with \({n \equiv 3, 4\, {\rm (mod\, 8)}}\) also have a graceful labeling of the first kind.

This is an expository paper in which we present some simple observations on the stability of some inhomogeneous functional equations. In particular, we state several stability results for the inhomogeneous Cauchy equation $$f(x+y)=f(x)+f(y)+d(x,y)$$ and for the inhomogeneous forms of the Jensen and linear functional equations.

Although it is customary to define polygons as certain families of edges, when considering polyhedra it is usual to view polygons as 2-dimensional pieces of the plane. If this rather illogical point of view is replaced by consistently understanding polygons as 1-dimensional complexes, the theory of polyhedra becomes richer and more satisfactory. Even with the strictest definition of regularity this approach leads to 17 individual regular polyhedra in the Euclidean 3-space and 12 infinite families of such polyhedra, besides the traditional ones (which consist of 5 Platonic polyhedra, 4 Kepler—Poinsot polyhedra, 3 planar tessellations and 3 Petrie—Coxeter polyhedra). Among the many still open problems that naturally arise from the new point of view, the most obvious one is the question whether the regular polyhedra found in the paper are the only ones possible in the Euclidean 3-space.

A loop which satisfies the identitiesx2 =e, xe = ex = x, andx(yx) = (xy) x = y is called a generalized Steiner loop. In this paper it is shown that a generalized Steiner loop is a groupoid with a single lawx(((yy) z) x) = z.

Given two functions $$f,g:I\rightarrow \mathbb {R}$$ f , g : I → R and a probability measure $$\mu $$ μ on the Borel subsets of [0, 1], the two-variable mean $$M_{f,g;\mu }:I^2\rightarrow I$$ M f , g ; μ : I 2 → I is defined by $$\begin{aligned} M_{f,g;\mu }(x,y) :=\bigg (\frac{f}{g}\bigg )^{-1}\left( \frac{\int _0^1 f\big (tx+(1-t)y\big )d\mu (t)}{\int _0^1 g\big (tx+(1-t)y\big )d\mu (t)}\right) \quad (x,y\in I). \end{aligned}$$ 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 $$\mu $$ μ , to study their equality problem, i.e., to characterize those pairs of functions (f, g) and (F, G) for which $$\begin{aligned} M_{f,g;\mu }(x,y)=M_{F,G;\mu }(x,y) \quad (x,y\in I) \end{aligned}$$ 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.

The purpose of this paper is to investigate the equality problem of generalized Bajraktarevic means, i.e., to solve the functional equation * $$\begin{aligned} f^{(-1)}\bigg (\frac{p_1(x_1)f(x_1)+\dots +p_n(x_n)f(x_n)}{p_1(x_1)+\dots +p_n(x_n)}\bigg )=g^{(-1)}\bigg (\frac{q_1(x_1)g(x_1)+\dots +q_n(x_n)g(x_n)}{q_1(x_1)+\dots +q_n(x_n)}\bigg ), \end{aligned}$$ which holds for all $$(x_1,\dots ,x_n)\in I^n$$ , where $$n\ge 2$$ , I is a nonempty open real interval, the unknown functions $$f,g:I\rightarrow {\mathbb {R}}$$ are strictly monotone, $$f^{(-1)}$$ and $$g^{(-1)}$$ denote their generalized left inverses, respectively, and $$p=(p_1,\dots ,p_n):I\rightarrow {\mathbb {R}}_{+}^n$$ and $$q=(q_1,\dots ,q_n):I\rightarrow {\mathbb {R}}_{+}^n$$ are also unknown functions. This equality problem in the symmetric two-variable (i.e., when $$n=2$$ ) case was already investigated and solved under sixth-order regularity assumptions by Losonczi (Aequationes Math 58(3):223–241, 1999). In the nonsymmetric two-variable case, assuming the three times differentiability of f, g and the existence of $$i\in \{1,2\}$$ such that either $$p_i$$ is twice continuously differentiable and $$p_{3-i}$$ is continuous on I, or $$p_i$$ is twice differentiable and $$p_{3-i}$$ is once differentiable on I, we prove that (*) holds if and only if there exist four constants $$a,b,c,d\in {\mathbb {R}}$$ with $$ad\ne bc$$ such that $$\begin{aligned} cf+d>0,\qquad g=\frac{af+b}{cf+d},\qquad \text{ and }\qquad q_\ell =(cf+d)p_\ell \qquad (\ell \in \{1,\dots ,n\}). \end{aligned}$$ In the case $$n\ge 3$$ , we obtain the same conclusion with weaker regularity assumptions. Namely, we suppose that f and g are three times differentiable, p is continuous and there exist $$i,j,k\in \{1,\dots ,n\}$$ with $$i\ne j\ne k\ne i$$ such that $$p_i,p_j,p_k$$ are differentiable.

In this paper, we investigate maps on sets of positive operators which are induced by the continuous functional calculus and transform a Kubo–Ando mean $$\sigma $$ σ into another $$\tau $$ τ . We establish that under quite mild conditions, a mapping $$\phi $$ ϕ can have this property only in the trivial case, i.e. when $$\sigma $$ σ and $$\tau $$ τ are nontrivial weighted harmonic means and $$\phi $$ ϕ stems from a function which is a constant multiple of the generating function of such a mean. In the setting where exactly one of $$\sigma $$ σ and $$\tau $$ τ is a weighted arithmetic mean, we show that under fairly weak assumptions, the mentioned transformer property never holds. Finally, when both of $$\sigma $$ σ and $$\tau $$ τ are such a mean, it turns out that the latter property is only satisfied in the trivial case, i.e. for maps induced by affine functions.

In the present paper, a general class of linear functional equations is considered and a computer program is described, which determines the exact solutions of systems of equations belonging to this class.

In this paper, we consider the condition$$\sum _{i=0}^{n+1}\varphi _i(r_ix+q_iy)\in {\mathbb {Z}}$$∑i=0n+1φi(rix+qiy)∈Zfor real valued functions defined on a linear spaceV. We derive necessary and sufficient conditions for functions satisfying this condition to be decent in the following sense: there exist functions$$f_i:V\rightarrow {\mathbb {R}}$$fi:V→R,$$g_i:V\rightarrow {\mathbb {Z}}$$gi:V→Zsuch that$$\varphi _i=f_i+g_i$$φi=fi+gi,$$(i=0,\dots ,n+1)$$(i=0,⋯,n+1)and$$\sum _{i=0}^{n+1}f_i(r_ix+q_iy)=0$$∑i=0n+1fi(rix+qiy)=0for all$$x, y\in V$$x,y∈V.

According to the famous and pioneering result of Laurent Schwartz, any closed translation invariant linear space of continuous functions on the reals is synthesizable from its exponential monomials. Due to a result of D. I. Gurevič there is no straightforward extension of this result to higher dimensions. Following Székelyhidi (Acta Math Hungar 153(1):120–142, 2017), with the aid of Gelfand pairs and K-spherical functions, K-synthesizability of K-varieties can be described. In this paper we contribute to this direction in the special case when K is the symmetric group of order d.

Let$$I\subseteq \mathbb {R}$$I⊆Rbe a nonempty open subinterval. We say that a two-variable mean$$M:I\times I\rightarrow \mathbb {R}$$M:I×I→Renjoys thebalancing propertyif, for all$$x,y\in I$$x,y∈I, the equality$$\begin{aligned} {M\big (M(x,M(x,y)),M(M(x,y),y)\big )=M(x,y)} \end{aligned}$$M(M(x,M(x,y)),M(M(x,y),y))=M(x,y)holds. The above equation has been investigated by several authors. The first remarkable step was made by Georg Aumann in 1935. Assuming, among other things, thatMisanalytic, he solved (1) and obtained quasi-arithmetic means as solutions. Then, two years later, he proved that (1) characterizesregularquasi-arithmetic means among Cauchy means, where, the differentiability assumption appears naturally. In 2015, Lucio R. Berrone, investigating a more general equation, having symmetry and strict monotonicity, proved that the general solutions are quasi-arithmetic means, provided that the means in question arecontinuously differentiable. The aim of this paper is to solve (1), without differentiability assumptions in a class of two-variable means, which contains the class ofMatkowski means.

A problem in the calculus of variations is of the second order if the Lagrangian contains second order derivatives of the dependent variables. This type of problem appears to have been neglected somewhat in the past and in the present paper some aspects of the theory presented in I-9] and [1] for the first order case are discussed in terms of that of the second order. Previous analysis (cf e.g. [7] for the first order problem and [2] for the second order one) assumed that the determinant of the Hamiltonian complex be nonvanishing, that is,*

Using the decomposition of an automorphism of the ring of formal power series in several variables, in a semisimple and a unipotent automorphism, I prove in this paper that an automorphism allows a continuous iteration if and only if it is the exponential of a derivation. This result implies a number of results recently obtained by Reich, Schwaiger, and Bucher.

In this paper, we discuss the pairs (f, h) of arithmetical functions satisfying the functional equation in the title, whereF is the product off andh under the Dirichlet convolution; that is,F(n) = Σd|n ƒ(d)h(n/d) andS(m n) = Σd|(m, n) ƒ(d)h(n/d). The well-known Holder's identity is a special case of this functional equation (ƒ(n) =n, h(n) = μ(n)). We also generalize the functional equation in the title to any arbitrary regular arithmetical convolution and discuss the pairs of solutions (f, h) of the generalized functional equation and pose some problems relating to the characterization of all pairs of solutions.