Aequationes Mathematicae最新文献

Recently, through the study of q-generalized higher-order Stirling numbers, q-generalized finite multiple zeta functions have been naturally introduced, and their values at roots of unity have been explicitly obtained. When (qrightarrow 1), they are finite multiple zeta functions. In this paper, we obtain some explicit expressions for certain q-multiple t-values at roots of unity. In other words, such expressions are multiple zeta values when the indices of the sum are limited to odd numbers. These finite functions are closely related to Stirling numbers of type B, which have strong relations to the Coxeter group and Artin basis.

For a digraph ({mathcal {D}}) with n vertices, a arcs and the outdegree sequence (d_1^{+}, d_2^{+},dots , d_n^{+}) of vertices of ({mathcal {D}}). The first outdegree Zagreb index of ({mathcal {D}}) is (Zg^{+}({mathcal {D}})), which is defined as (Zg^{+}({mathcal {D}})=sum limits _{i=1}^{n}(d_i^{+})^2). This work establishes new upper and lower bounds for the first outdegree Zagreb index (Zg^{+}({mathcal {D}})) of a digraph ({mathcal {D}}), expressed in terms of various structural invariants. The digraphs that achieve these extremal bounds are fully characterized. In particular, we investigate the problem of determining the orientations that maximize or minimize the first outdegree Zagreb index for the wheel graph (W_n).

We say that a map (f:S_X rightarrow S_Y) between the unit spheres of two Banach spaces X and Y is a phase-isometry if it satisfies

Given two arbitrary index sets (Gamma ) and (Delta ), and real Hilbert spaces H and K with (pin [1, infty ]), we show that every surjective phase-isometry between (S_{ell ^p(Gamma ,H)}) and (S_{ell ^p(Delta , K)}) can be extended to a surjective phase-isometry from (ell ^p(Gamma ,H)) onto (ell ^p(Delta , K)), which is phase equivalent to a linear isometry.

We celebrate Janos Azcel’s 100th anniversary. His earliest paper on mean values and his famous bisymmetry equation are revisited for our enjoyment and inspiration.

We introduce the functional equation (g(xyz) - g(x)g(yz) - g(y)g(xz) - g(z)g(xy) + 2g(x)g(y)g(z) = 0) for an unknown function g mapping a semigroup S into a field K. It seems reasonable to call this a cosine functional equation because when (S = ({mathbb R},+)) and (K = {mathbb R}) the function (g = cos ) is a solution. It is not very surprising to find that this equation has a strong connection with the sine addition formula. We show that for any solution g there exists a function (f:S rightarrow K) such that (f(xy) = f(x)g(y) + g(x)f(y)) for all (x,y in S). The converse is true if (f ne 0). For the case (K = {mathbb C}) we show that all solutions of the cosine equation are arithmetic means of two multiplicative functions. Some more general equations are also solved.

We consider a class of functional equations in one variable that, in some settings, describe polynomials on groups. Namely, a binomial equation of order n, for functions from a group G to an Abelian group K, is the equation of the form

We call solutions of this equation binomial functions of order n. Our aim is to consider diverse connections between binomial functions and (semi)polynomials on G. We prove that all sufficiently smooth (mathbb {R})-valued binomial functions on (mathbb {R}^d) are polynomials. Furthermore, we show that continuous binomial functions on groups with dense union of compact subgroups (e.g. on the groups of all triangular matrices whose diagonal elements are of module 1) are constant. On the other hand, we construct examples showing that, in distinction to the case of semipolynomials, there are non-constant binomial functions on some groups topologically generated by compact subgroups, e.g. on the groups (SL(n,mathbb {R})) of unimodular matrices.

For (s in {mathbb {C}}) and (0< a <1), let (zeta (s,a)) and (mathrm{{Li}}_s (e^{2pi ia})) be the Hurwitz and periodic zeta functions, respectively. For (0 < a le 1/2), put (Z(s,a):= zeta (s,a) + zeta (s,1-a)), (P(s,a):= mathrm{{Li}}_s (e^{2pi ia}) + mathrm{{Li}}_s (e^{2pi i(1-a)})), (Y(s,a):= zeta (s,a) - zeta (s,1-a)) and (O(s,a):= -i bigl ( mathrm{{Li}}_s (e^{2pi ia}) - mathrm{{Li}}_s (e^{2pi i(1-a)}) bigr )). Let (n ge 0) be an integer and (b:= r/q), where (q>r>0) are coprime integers. In this paper, we prove that the values (Z(-n,b)), (pi ^{-2n-2} P(2n+2,b)), (Y(-n,b)) and (pi ^{-2n-1} O(2n+1,b)) are rational numbers, in addition, (pi ^{-2n-2} Z(2n+2,b)), (P(-n,b)), (pi ^{-2n-1} Y(2n+1,b)) and (O(-n,b)) are polynomials of (cos (2pi /q)) and (sin (2pi /q)) with rational coefficients. Furthermore, we show that (Z(-n,a)), (pi ^{-2n-2} P(2n+2,a)), (Y(-n,a)) and (pi ^{-2n-1} O(2n+1,a)) are polynomials of (0<a<1) with rational coefficients, in addition, (pi ^{-2n-2} Z(2n+2,a)), (P(-n,a)), (pi ^{-2n-1} Y(2n+1,a)) and (O(-n,a)) are rational functions of (exp (2 pi ia)) with rational coefficients. Note that the rational numbers, polynomials and rational functions mentioned above are given explicitly. Moreover, we show that (P(s,a) equiv 0) for all ( 0< a < 1/2) if and only if s is a negative even integer. We also prove similar assertions for Z(s, a), Y(s, a), O(s, a) and so on. Furthermore, we prove that the function Z(s, |a|) appears as the spectral density of some stationary self-similar Gaussian distributions.

In [4], it was observed that each tw-variable weighted quasiarithmetic mean is weakly associative, i.e. it satisfies the equality (Mleft( Mleft( x,yright) ,xright) =Mleft( x,Mleft( y,xright) right) ) for all x, y. In the present paper a broader class of non-symmetric weakly associative means is presented. A conjecture that a two-variable formal power series (Mleft( x,yright) =sum _{k=1}^{infty }sum _{j=0}^{k}a_{k-j,j}x^{k-j}y^{j}) with (a_{1,0}ne a_{0,1},) is weakly associative if and only if (Mleft( x,yright) =a_{1,0}x+left( 1-a_{1,0}right) y) is formulated. This conjecture allows to characterize the class of weighted quasiarithmetic means, as well as a new, broader class of means. Looking for translative weakly associative functions we arrive to an open questions concerning the composite functional equation

Let (Z^n) be a set of n-person income profiles over two time periods. The notion that a profile (zin Z^n) exhibits higher mobility than (z') is expressed as (zsuccsim z'). Cowell and Flachaire give a set of principles, stated as formal axioms, we wish (succsim ) to fulfill. Numeric measures, m, are sought to represent (succsim ) so that (zsuccsim z') corresponds with (m(z)ge m(z')). We report that the Jensen differences of strictly convex functions are useful in constructing examples of measures that meet their first four axioms. Their fifth axiom is found incompatible with the first four.

The notion of integral means of set-valued maps with values in a Banach space is introduced and investigated. A set-valued counterpart of the classical mean value theorem for integrals is presented. It is also proved that if a set-valued map is convex on an interval I then its integral mean is Schur-convex on (I^2).