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In Szirmai (Ars Math Contemp 16:349–358, 2019) we proved that to each saturated congruent hyperball packing there exists a decomposition of the 3-dimensional hyperbolic space $$\mathbb {H}^3$$ H 3 into truncated tetrahedra. Therefore, in order to get a density upper bound for hyperball packings, it is sufficient to determine the density upper bound of hyperball packings in truncated simplices. In this paper we prove, using the above results and results of the papers Miyamoto (Topology 33(4): 613–629, 1994) and Szirmai (Mat Vesn 70(3): 211–221, 2018), that the density upper bound of the saturated congruent hyperball (hypersphere) packings related to the corresponding truncated tetrahedron cells is realized in regular truncated tetrahedra with density $$\approx 0.86338$$ ≈ 0.86338 . Furthermore, we prove that the density of locally optimal congruent hyperball arrangement in a regular truncated tetrahedron is not a monotonically increasing function of the height (radius) of the corresponding optimal hyperball, unlike the ball (sphere) and horoball (horosphere) packings.

Investigating the future value F(K, s, t) of a capital K invested between dates s and t, the “natural” condition $$F(K,s,t)\ge K$$ F ( K , s , t ) ≥ K has lost its naturality because of the strange fact of negative interest rates. This leads to the task of describing the possible solutions of the multiplicative Sincov equation $$f(s,u)=f(s,t)f(t,u)$$ f ( s , u ) = f ( s , t ) f ( t , u ) for $$s\le t\le u$$ s ≤ t ≤ u where $$f(s,t)=0$$ f ( s , t ) = 0 may happen. In this paper we solve this task and discuss connections to the theory of investments.

In this paper we consider a generalized monomial or polynomial $$ f : \mathbb {R}\rightarrow \mathbb {R}$$ f : R → R that satisfies the additional equation $$ f(x) f(y) = 0 $$ f ( x ) f ( y ) = 0 for the pairs $$ (x,y) \in D \,$$ ( x , y ) ∈ D , where $$ D \subseteq {\mathbb {R}}^{2} $$ D ⊆ R 2 is given by some algebraic condition. In the particular cases when f is a generalized polynomial and there exist non-constant regular polynomials p and q that fulfill $$\begin{aligned} D = \{\, (p(t),q(t)) \,\vert \, t \in \mathbb {R}\,\} \end{aligned}$$ D = { ( p ( t ) , q ( t ) ) | t ∈ R } or f is a generalized monomial and there exists a positive rational m fulfilling $$\begin{aligned} D = \{\, (x,y) \in {\mathbb {R}}^{2} \,\vert \, x^2 - m y^2 = 1 \,\}, \end{aligned}$$ D = { ( x , y ) ∈ R 2 | x 2 - m y 2 = 1 } , we prove that $$ f(x) = 0 $$ f ( x ) = 0 for all $$ x \in \mathbb {R}\,$$ x ∈ R .

The equality problem of two-variable Bajraktarević means can be expressed as the functional equation $$\begin{aligned} \left( \frac{f}{g}\right) ^{-1}\bigg (\frac{f(x)+f(y)}{g(x)+g(y)}\bigg ) =\left( \frac{h}{k}\right) ^{-1}\bigg (\frac{h(x)+h(y)}{k(x)+k(y)}\bigg )\qquad (x,y\in I), \end{aligned}$$ 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, $$f,g,h,k:I\rightarrow {\mathbb {R}}$$ 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.

Spectral analysis and synthesis studies translation invariant function spaces, so-called varieties over topological groups. The basic building blocks are the finite dimensional varieties. In the commutative case finite dimensional varieties are spanned by exponential polynomials. In non-commutative situations no relevant results exist. In this paper we consider finite dimensional left translation invariant linear spaces of continuous complex valued functions over the Heisenberg group. Using basic knowledge about Lie algebra we describe all left varieties of this type. In particular, it turns out that those function spaces are spanned by exponential polynomials as well.

We study the analogue of Wendel’s equality in random polytope models in which the hull of the random points is formed by intersections of congruent balls, called the spindle (or hyper-) convex hull. According to the classical identity of Wendel the probability that the origin is contained in the (linear) convex hull of n i.i.d. random points distributed according to an origin symmetric probability distribution in the d-dimensional Euclidean space $$\mathbb {R}^{d}$$ R d that assigns measure zero to hyperplanes is a constant depending only on n and d. While in the classical convex case one gets nonzero probabilities only for $$n\ge d+1$$ n ≥ d + 1 points in $$\mathbb {R}^{d}$$ R d , for the spindle convex hull this happens for all $$n\ge 2$$ n ≥ 2 . We study this question for the uniform and normally distributed random models.

This article introduces the novel notion of dimension preserving approximation for continuous functions defined on [0, 1] and initiates the study of it. Restrictions and extensions of continuous functions in regards to fractal dimensions are also investigated.

Translativity of the beta-type function $$B_{f}:I^{2}\rightarrow \left( 0,\infty \right) $$ B f : I 2 → 0 , ∞ , $$\begin{aligned} B_{f}\left( x,y\right) :=\frac{f\left( x\right) f\left( y\right) }{f\left( x+y\right) }, \end{aligned}$$ B f x , y : = f x f y f x + y , where f is a single variable function defined either on $$I={\mathbb {R}}$$ I = R or $$I=[ 0,\infty )$$ I = [ 0 , ∞ ) , or $$I=\left( 0,\infty \right) $$ I = 0 , ∞ , is considered. In each of these three cases a complete solution is given.

In this paper, we present an application of the viscosity approximation type iterative method introduced by Nandal et al. (Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators, Symmetry, 2019) to visualize and analyse the Julia and Mandelbrot sets for a complex polynomial of the type $$T(z) = z^{n} + p z + r$$ T ( z ) = z n + p z + r , where $$p, r\in {\mathbb {C}}$$ p , r ∈ C , and $$n \ge 2$$ n ≥ 2 . This iterative method has many applications in solving various fixed point problems. We derive an escape criterion to visualize Julia and Mandelbrot sets via the proposed viscosity approximation type method. Moreover, we present several graphical examples of the fractals generated with the proposed iteration method.

In the paper there are two kinds of functional symmetry considered, i.e., (1)-power symmetry and $$(-1)$$ ( - 1 ) -power symmetry, in a rich subfamily $$\mathcal {B}$$ B of the family of all complex valued functions on a symmetric set $$\mathcal {G}\subset \mathbb {C}$$ G ⊂ C . The first one defines the values $$f(-z)$$ f ( - z ) as identical with f(z) and the other as the inverse of the f(z) in $$\mathcal {G}$$ G . The announced notions allow a unique decomposition of functions $$f\in \mathcal {B}$$ f ∈ B into a product of two factors $$f_{1},f_{-1}$$ f 1 , f - 1 having the (1)- and the $$(-1)$$ ( - 1 ) -power symmetry property, respectively. In the paper there are also given examples of such partitions $$f=f_{1}\cdot f_{-1}$$ f = f 1 · f - 1 , for various $$f\in \mathcal {B},$$ f ∈ B , a solution of the problem of invariance of the above functional symmetries with respect to some one- and two-argument function operations and two applications of the mentioned function decomposition.

We investigate the functional equation $$f(p(x)) = q(f(x))$$ f ( p ( x ) ) = q ( f ( x ) ) where p and q are given real functions. In the paper “On solvability of$$f(p(x)) = q(f(x))$$ f ( p ( x ) ) = q ( f ( x ) ) for given real functionsp, q, Aequat. Math. 90 (2016), 471 - 494”, we solved the problem of the solvability of $$f(p(x)) = q(f(x))$$ f ( p ( x ) ) = q ( f ( x ) ) under the assumption that p, q are strictly increasing continuous real functions. Now, we extend the solutions of this problem for any strictly monotonous continuous real functions p, q. Thereby, we use the methods of the just mentioned paper. Further, we present computations of the so called characteristics of the given functions p, q using the results of this paper and, finally, present a quite short algorithm with input p, q and output ’solvable/not solvable’.

In this paper we introduce the concept of translation invariant functions: considering an arbitrary set $$\emptyset \ne S \subset {\mathbb {R}}^n$$ ∅ ≠ S ⊂ R n , the function $$F : S \longrightarrow {\mathbb {R}}$$ F : S ⟶ R is translation invariant if $$F(x) = F(y)$$ F ( x ) = F ( y ) implies $$F(x+t)=F(y+t)$$ F ( x + t ) = F ( y + t ) for any vectors $$x,y,t \in {\mathbb {R}}^n$$ x , y , t ∈ R n such that $$x ,y , x+t ,y+t \in S$$ x , y , x + t , y + t ∈ S . In our main results we shall consider an open, connected set $$\emptyset \ne D \subset {\mathbb {R}}^n$$ ∅ ≠ D ⊂ R n . We prove that if $$F : D \longrightarrow {\mathbb {R}}$$ F : D ⟶ R is a translation invariant, continuous function, then there exists a vector $$a = (a_1, \dots , a_n) \in {\mathbb {R}}^n$$ a = ( a 1 , ⋯ , a n ) ∈ R n and a strictly monotone, continuous function f such that $$\begin{aligned} F(x_1, \dots , x_n) = f (a_1 x_1 + \dots + a_n x_n) \end{aligned}$$ F ( x 1 , ⋯ , x n ) = f ( a 1 x 1 + ⋯ + a n x n ) holds for all $$(x_1, \dots , x_n) \in D \,$$ ( x 1 , ⋯ , x n ) ∈ D . Using this result we also show that continuous solutions $$ F : D \longrightarrow {\mathbb {R}}$$ F : D ⟶ R of the system of functional equations $$\begin{aligned} F(x_1 , \dots , x_j + t_j , \dots , x_n) = \Psi _j (F (x_1 , \dots , x_j , \dots , x_n), t_j) \ \ (j=1,\dots ,n) \end{aligned}$$ F ( x 1 , ⋯ , x j + t j , ⋯ , x n ) = Ψ j ( F ( x 1 , ⋯ , x j , ⋯ , x n ) , t j ) ( j = 1 , ⋯ , n ) 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.

A partition into distinct parts is refinable if one of its parts a can be replaced by two different integers which do not belong to the partition and whose sum is a, and it is unrefinable otherwise. Clearly, the condition of being unrefinable imposes on the partition a non-trivial limitation on the size of the largest part and on the possible distributions of the parts. We prove a $$O(n^{1/2})$$ O ( n 1 / 2 ) -upper bound for the largest part in an unrefinable partition of n, and we call maximal those which reach the bound. We show a complete classification of maximal unrefinable partitions for triangular numbers, proving that if n is even there exists only one maximal unrefinable partition of $$n(n+1)/2$$ n ( n + 1 ) / 2 , and that if n is odd the number of such partitions equals the number of partitions of $$\lceil n/2\rceil $$ ⌈ n / 2 ⌉ into distinct parts. In the second case, an explicit bijection is provided.

We observe that every quadrature of numerical integration together with a strictly increasing and continuous function generates a mean and then we study this family of means. We characterize the functions which generate weighted arithmetic means in this way and we show how to obtain comparison type theorems for means generated by different quadratures.

For a finite sequence $$(x_i)_{i=1}^N$$ ( x i ) i = 1 N in the unit interval, we introduce the gap ratio function which measures the size of the maximal gap length relative to all other gap lengths. This function (asymptotically) captures a lot of information about the degree of uniformity of the sequence. We discuss connections to the theories of dispersion, discrepancy, pair correlation statistics and covering numbers. Furthermore, we explicitly calculate the gap ratio function for some important classes of uniformly distributed sequences.

In this paper we will study a connection between some generalization of AC-subspaces, vector valued invariant $$\lambda $$ λ -means and $$\lambda $$ λ -complementability.

We determine the 6-dimensional nilpotent metric Lie algebras such that the Lie algebra $${\mathfrak {n}}$$ n has a descending series of ideals invariant under all automorphisms of $${\mathfrak {n}}$$ n and the dimension of the consecutive members of the series decreases by one. We call them metric Lie algebras having a framing determined by ideals. We classify the isometry equivalence classes and determine the isometry groups of connected and simply connected Riemannian nilmanifolds on 6-dimensional nilpotent Lie groups having a Lie algebra $${\mathfrak {n}}$$ n as their Lie algebra.

In this paper we will study Hyers-Ulam stability for a general linear partial differential equation of first order in a Banach space.

Given a probability space $$(\Omega ,\mathcal {A},\mathbb {P})$$ ( Ω , A , P ) , a complete separable Banach space X with the $$\sigma $$ σ -algebra $$\mathcal B(X)$$ B ( X ) of all its Borel subsets, an operator $$\Lambda :\Omega \rightarrow L(X,X)$$ Λ : Ω → L ( X , X ) and $$\xi :\Omega \rightarrow X$$ ξ : Ω → X we consider the $$\mathcal {B}(X)\otimes \mathcal A$$ B ( X ) ⊗ A -measurable function $$f:X\times \Omega \rightarrow X$$ f : X × Ω → X given by $$f(x,\omega )=\Lambda (\omega )x+\xi (\omega )$$ f ( x , ω ) = Λ ( ω ) x + ξ ( ω ) and investigate the continuous dependence of a weak limit $$\pi ^f$$ π f of the sequence of iterates $$(f^n(x,\cdot ))_{n\in \mathbb {N}}$$ ( f n ( x , · ) ) n ∈ N of f, defined by $$f^0(x,\omega )=x, f^{n+1}(x,\omega )=f(f^n(x,\omega ),\omega _{n+1})$$ f 0 ( x , ω ) = x , f n + 1 ( x , ω ) = f ( f n ( x , ω ) , ω n + 1 ) for $$x\in X$$ x ∈ X and $$\omega =(\omega _1,\omega _2,\dots )$$ ω = ( ω 1 , ω 2 , ⋯ ) . Moreover for X taken as a Hilbert space we characterize $$\pi ^f$$ π f via the functional equation $$\begin{aligned} \varphi ^f(u)=\int _{\Omega }\varphi ^f(\Lambda (\omega )u)\varphi ^{\xi }(u)\mathbb {P}(d\omega ) \end{aligned}$$ φ f ( u ) = ∫ Ω φ f ( Λ ( ω ) u ) φ ξ ( u ) P ( d ω ) with the aid of its characteristic function $$\varphi ^f$$ φ f . We also indicate the continuous dependence of a solution of that equation.