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 Aequationes Mathematicae   [SJR: 0.882]   [H-I: 23]   [2 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 1420-8903 - ISSN (Online) 0001-9054    Published by Springer-Verlag  [2355 journals]
• Lower estimation of the difference between quasi-arithmetic means
• Authors: Paweł Pasteczka
Pages: 7 - 24
Abstract: Abstract In the 1960s Cargo and Shisha introduced a metric in a family of quasi-arithmetic means defined on a common interval as the maximal possible difference between these means taken over all admissible vectors with corresponding weights. During the years 2013–2016 we proved that, having two quasi-arithmetic means, we can majorize the distance between them in terms of the Arrow–Pratt index. In this paper we are going to prove that this operator can also be used to establish certain lower bounds of this distance.
PubDate: 2018-02-01
DOI: 10.1007/s00010-017-0513-8
Issue No: Vol. 92, No. 1 (2018)

• Around Jensen’s inequality for strongly convex functions
Pages: 25 - 37
Abstract: Abstract In this paper we use basic properties of strongly convex functions to obtain new inequalities including Jensen type and Jensen–Mercer type inequalities. Applications for special means are pointed out as well. We also give a Jensen’s operator inequality for strongly convex functions. As a corollary, we improve the Hölder-McCarthy inequality under suitable conditions. More precisely we show that if $$Sp\left( A \right) \subset \left( 1,\infty \right)$$ , then \begin{aligned} {{\left\langle Ax,x \right\rangle }^{r}}\le \left\langle {{A}^{r}}x,x \right\rangle -\frac{{{r}^{2}}-r}{2}\left( \left\langle {{A}^{2}}x,x \right\rangle -{{\left\langle Ax,x \right\rangle }^{2}} \right) ,\quad r\ge 2 \end{aligned} and if $$Sp\left( A \right) \subset \left( 0,1 \right)$$ , then \begin{aligned} \left\langle {{A}^{r}}x,x \right\rangle \le {{\left\langle Ax,x \right\rangle }^{r}}+\frac{r-{{r}^{2}}}{2}\left( {{\left\langle Ax,x \right\rangle }^{2}}-\left\langle {{A}^{2}}x,x \right\rangle \right) ,\quad 0<r<1 \end{aligned} for each positive operator A and $$x\in \mathcal {H}$$ with $$\left\ x \right\ =1$$ .
PubDate: 2018-02-01
DOI: 10.1007/s00010-017-0496-5
Issue No: Vol. 92, No. 1 (2018)

• Relation-induced connectedness in the digital plane
• Authors: Josef Šlapal
Pages: 75 - 90
Abstract: Abstract We introduce and discuss a connectedness induced by n-ary relations ( $$n>1$$ an integer) on their underlying sets. In particular, we focus on certain n-ary relations with the induced connectedness allowing for a definition of digital Jordan curves. For every integer $$n>1$$ , we introduce one such n-ary relation on the digital plane $${\mathbb {Z}}^2$$ and prove a digital analogue of the Jordan curve theorem for the induced connectedness. It follows that these n-ary relations may be used as convenient structures on the digital plane for the study of geometric properties of digital images. For $$n=2$$ , such a structure coincides with the (specialization order of the) Khalimsky topology and, for $$n>2$$ , it allows for a variety of Jordan curves richer than that provided by the Khalimsky topology.
PubDate: 2018-02-01
DOI: 10.1007/s00010-017-0508-5
Issue No: Vol. 92, No. 1 (2018)

• Super-biderivations of classical simple Lie superalgebras
• Authors: Jixia Yuan; Xiaomin Tang
Pages: 91 - 109
Abstract: Abstract In this paper, we characterize super-biderivations of classical simple Lie superalgebras over the complex field $$\mathbb {C}$$ . Furthermore, we prove that all super-biderivations of classical simple Lie superalgebras are inner super-biderivations. As an application, the super-biderivations of a general linear Lie superalgebra are studied. We find that there exist non-inner and non-super-skewsymmetric super-biderivations. Finally, using the results on biderivations we characterize linear super commuting maps on the classical simple Lie superalgebras and general linear Lie superalgebras.
PubDate: 2018-02-01
DOI: 10.1007/s00010-017-0503-x
Issue No: Vol. 92, No. 1 (2018)

• Separation by Jensen and affine stochastic processes
• Authors: Dawid Kotrys; Kazimierz Nikodem
Pages: 111 - 122
Abstract: Abstract Characterizations of pairs of stochastic processes that can be separated by Jensen and by affine stochastic processes are presented. As a consequence, some stability results of the Hyers–Ulam-type are obtained.
PubDate: 2018-02-01
DOI: 10.1007/s00010-017-0528-1
Issue No: Vol. 92, No. 1 (2018)

• Weighted entropy and optimal portfolios for risk-averse Kelly investments
• Authors: M. Kelbert; I. Stuhl; Y. Suhov
Pages: 165 - 200
Abstract: Abstract Following a series of works on capital growth investment, we analyse log-optimal portfolios where the return evaluation includes ‘weights’ of different outcomes. The results are twofold: (A) under certain conditions, the logarithmic growth rate leads to a supermartingale, and (B) the optimal (martingale) investment strategy is a proportional betting. We focus on properties of the optimal portfolios and discuss a number of simple examples extending the well-known Kelly betting scheme. An important restriction is that the investment does not exceed the current capital value and allows the trader to cover the worst possible losses. The paper deals with a class of discrete-time models. A continuous-time extension is a topic of an ongoing study.
PubDate: 2018-02-01
DOI: 10.1007/s00010-017-0515-6
Issue No: Vol. 92, No. 1 (2018)

• Some comments on floating and centroid bodies in the plane
• Authors: Z. Guerrero-Zarazua; J. Jerónimo-Castro
Abstract: Abstract Consider a long, convex, homogenous cylinder with horizontal axis and with a planar convex body K as transversal section. Suppose the cylinder is immersed in water and let $$K_w$$ be the wet part of K. In this paper we study some properties of the locus of the centroid of $$K_w$$ and prove an analogous result to Klamkin–Flanders’ theorem when the locus is a circle. We also study properties of bodies floating at equilibrium when either the origin or the centroid of the body is pinned at the water line. In some sense this is the floating body problem for a density varying continuously. Finally, in the last section we give an isoperimetric type inequality for the perimeter of the centroid body (defined by C. M. Petty in Pacific J Math 11:1535–1547, 1961) of convex bodies in the plane.
PubDate: 2018-01-12
DOI: 10.1007/s00010-017-0525-4

• Jensen–Steffensen inequality for diamond integrals, its converse and
improvements via Green function and Taylor’s formula
• Authors: Ammara Nosheen; Rabia Bibi; Josip Pečarić
Abstract: Abstract In this paper we define the Jensen–Steffensen inequality and its converse for diamond integrals. Then we give some improvements of these inequalities using Taylor’s formula and the Green function. We investigate bounds for the identities related to improvements of the Jensen–Steffensen inequality and its converse.
PubDate: 2018-01-08
DOI: 10.1007/s00010-017-0527-2

• New characterizations of the Takagi function via functional equations
Pages: 1001 - 1007
Abstract: Abstract We provide two new characterizations of the Takagi function as the unique bounded solution of some systems of two functional equations. The results are independent of those obtained by Kairies (Wyż Szkoł Ped Krakow Rocznik Nauk Dydakt Prace Mat 196:73–82, 1998), Kairies (Aequ Math 53:207–241, 1997), Kairies (Aequ Math 58:183–191, 1999) and Kairies et al. (Rad Mat 4:361–374, 1989; Errata, Rad Mat 5:179–180, 1989).
PubDate: 2017-12-01
DOI: 10.1007/s00010-017-0501-z
Issue No: Vol. 91, No. 6 (2017)

• Report of Meeting
• Pages: 1157 - 1204
PubDate: 2017-12-01
DOI: 10.1007/s00010-017-0530-7
Issue No: Vol. 91, No. 6 (2017)

• Invariance under outer inverses
• Authors: R. E. Hartwig; P. Patrício
Abstract: Abstract We shall use the minus partial order combined with Pierce’s decomposition to derive the class of outer inverses for idempotents, units and group invertible elements. Subsequently we show, for matrices over a field $${\mathbb {F}}$$ , that the triplet $$B\hat{A}C$$ is invariant under all choices of outer inverses of A if and only if $$B = 0$$ or $$C = 0$$ .
PubDate: 2017-12-27
DOI: 10.1007/s00010-017-0524-5

• A perimeter-based angle measure in Minkowski planes
• Authors: Martin Obst
Abstract: Abstract Measuring angles in the Euclidean plane is a well-known topic, but for general normed planes there exists a variety of different concepts. These can be of a special kind, e.g. also preserving special orthogonality types. But these concepts are no angle measures in the sense of measure theory since they are not additive. This motivates us to define a new angle measure for normed planes that is in fact a measure in the sense of measure theory. Furthermore, we look at related types of rotation and reflection.
PubDate: 2017-12-27
DOI: 10.1007/s00010-017-0526-3

• On functional equations stemming from actuarial mathematics
• Authors: Jacek Chudziak
Abstract: Abstract We determine extensions of some implicitly defined functionals stemming from actuarial mathematics.
PubDate: 2017-12-20
DOI: 10.1007/s00010-017-0519-2

• One-sided invertibility of discrete operators and their applications
• Authors: Irina Asekritova; Yuri Karlovich; Natan Kruglyak
Abstract: Abstract For $$p\in [1,\infty ]$$ , we establish criteria for the one-sided invertibility of binomial discrete difference operators $${{\mathcal {A}}}=aI-bV$$ on the space $$l^p=l^p(\mathbb {Z})$$ , where $$a,b\in l^\infty$$ , I is the identity operator and the isometric shift operator V is given on functions $$f\in l^p$$ by $$(Vf)(n)=f(n+1)$$ for all $$n\in \mathbb {Z}$$ . Applying these criteria, we obtain criteria for the one-sided invertibility of binomial functional operators $$A=aI-bU_\alpha$$ on the Lebesgue space $$L^p(\mathbb {R}_+)$$ for every $$p\in [1,\infty ]$$ , where $$a,b\in L^\infty (\mathbb {R}_+)$$ , $$\alpha$$ is an orientation-preserving bi-Lipschitz homeomorphism of $$[0,+\infty ]$$ onto itself with only two fixed points 0 and $$\infty$$ , and $$U_\alpha$$ is the isometric weighted shift operator on $$L^p(\mathbb {R}_+)$$ given by $$U_\alpha f= (\alpha ^\prime )^{1/p}(f\circ \alpha )$$ . Applications of binomial discrete operators to interpolation theory are given.
PubDate: 2017-12-19
DOI: 10.1007/s00010-017-0522-7

• Geometry on the lines of spine spaces
• Authors: Krzysztof Petelczyc; Mariusz Żynel
Abstract: Abstract Spine spaces can be considered as fragments of a projective Grassmann space. We prove that the structure of lines together with a binary coplanarity relation, as well as with the binary relation of being in one pencil of lines, is a sufficient system of primitive notions for these geometries. It is also shown that, over a spine space, the geometry of pencils of lines can be reconstructed in terms of the two binary relations.
PubDate: 2017-12-19
DOI: 10.1007/s00010-017-0523-6

• Packing chromatic number versus chromatic and clique number
• Authors: Boštjan Brešar; Sandi Klavžar; Douglas F. Rall; Kirsti Wash
Abstract: Abstract The packing chromatic number $$\chi _{\rho }(G)$$ of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets $$V_i$$ , $$i\in [k]$$ , where each $$V_i$$ is an i-packing. In this paper, we investigate for a given triple (a, b, c) of positive integers whether there exists a graph G such that $$\omega (G) = a$$ , $$\chi (G) = b$$ , and $$\chi _{\rho }(G) = c$$ . If so, we say that (a, b, c) is realizable. It is proved that $$b=c\ge 3$$ implies $$a=b$$ , and that triples $$(2,k,k+1)$$ and $$(2,k,k+2)$$ are not realizable as soon as $$k\ge 4$$ . Some of the obtained results are deduced from the bounds proved on the packing chromatic number of the Mycielskian. Moreover, a formula for the independence number of the Mycielskian is given. A lower bound on $$\chi _{\rho }(G)$$ in terms of $$\Delta (G)$$ and $$\alpha (G)$$ is also proved.
PubDate: 2017-12-13
DOI: 10.1007/s00010-017-0520-9

• From r -dual sets to uniform contractions
• Authors: Károly Bezdek
Abstract: Abstract Let $$\mathbb {M}^d$$ denote the d-dimensional Euclidean, hyperbolic, or spherical space. The r-dual set of a given set in $$\mathbb {M}^d$$ is the intersection of balls of radii r centered at the points of the a given set. In this paper we prove that for any set of given volume in $$\mathbb {M}^d$$ the volume of the r-dual set becomes maximal if the set is a ball. As an application we prove the following. The Kneser–Poulsen Conjecture states that if the centers of a family of N congruent balls in Euclidean d-space is contracted, then the volume of the intersection does not decrease. A uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers, that is, when the pairwise distances of the two sets are separated by some positive real number. We prove a special case of the Kneser–Poulsen conjecture namely, we prove the conjecture for uniform contractions (with sufficiently large N) in $$\mathbb {M}^d$$ .
PubDate: 2017-11-23
DOI: 10.1007/s00010-017-0516-5

• Erdős–Gyárfás conjecture for some families of Cayley
graphs
• Authors: Mohammad Hossein Ghaffari; Zohreh Mostaghim
Abstract: Abstract The Paul Erdős and András Gyárfás conjecture states that every graph of minimum degree at least 3 contains a simple cycle whose length is a power of two. In this paper, we prove that the conjecture holds for Cayley graphs on generalized quaternion groups, dihedral groups, semidihedral groups and groups of order $$p^3$$ .
PubDate: 2017-11-21
DOI: 10.1007/s00010-017-0518-3

• Authors: Bruce Ebanks
Abstract: Abstract In 1990, Benz asked whether a real additive mapping satisfying $$xf(y)=yf(x)$$ for all points (x, y) on the unit circle must be linear. In 2005, Boros and Erdei showed that it must be so. Here we generalize the problem to a pair of additive functions f, g related by the functional equation $$xf(y)=yg(x)$$ for all points (x, y) on a specified curve. We find that for many (but not all) types of curves this forces f and g to be equal and linear.
PubDate: 2017-10-23
DOI: 10.1007/s00010-017-0514-7

• Inequalities between remainders of quadratures
• Authors: Andrzej Komisarski; Szymon Wąsowicz
Abstract: Abstract It is well-known that in the class of convex functions the (nonnegative) remainder of the Midpoint Rule of approximate integration is majorized by the remainder of the Trapezoid Rule. Hence the approximation of the integral of a convex function by the Midpoint Rule is better than the analogous approximation by the Trapezoid Rule. Following this fact we examine remainders of certain quadratures in classes of convex functions of higher orders. Our main results state that for 3-convex (5-convex, respectively) functions the remainder of the 2-point (3-point, respectively) Gauss–Legendre quadrature is non-negative and it is not greater than the remainder of Simpson’s Rule (4-point Lobatto quadrature, respectively). We also check 2-point Radau quadratures for 2-convex functions to demonstrate that similar results fail to hold for convex functions of even orders. We apply the Peano Kernel Theorem as a main tool of our considerations.
PubDate: 2017-10-05
DOI: 10.1007/s00010-017-0505-8

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