Band 71, Heft 5

We present the algebraic structures behind the approaches used to work with pairwise comparison matrices and, in general, the representation of preferences. We obtain a general definition of consistency and a universal decomposition in the space of PCMs, which allow us to define a consistency index. Also Arrow’s theorem, which is presented in a general form, is relevant. All the presented results can be seen in the main formulations of PCMs, i.e., multiplicative, additive and fuzzy approach, by the fact that each of them is a particular interpretation of the more general algebraic structure needed to deal with these theories.

The ring of dual numbers over a ring R is R [ α ] = R [ x ]/( x 2 ), where α denotes x + ( x 2 ). For any finite commutative ring R , we characterize null polynomials and permutation polynomials on R [ α ] in terms of the functions induced by their coordinate polynomials ( f 1 , f 2 ∈ R [ x ], where f = f 1 + αf 2 ) and their formal derivatives on R . We derive explicit formulas for the number of polynomial functions and the number of polynomial permutations on ℤ p n [ α ] for n ≤ p ( p prime).

In the current article, we examine some properties of analytic functions associated with cosine and exponential functions. We calculate some conditions on α so that; if 1+αz2−pf′(z)p$1+\frac{\alpha z^{2-p}f'(z)}{p}$, 1+αz2f′(z)pf(z)$1+\alpha \frac{z^{2}f'(z)}{pf(z)}$, 1+αzp+2f′(z)pf2(z)$1+\alpha \frac{z^{p+2}f'(z)}{pf^{2}(z)}$ and 1+αz2p+2f′(z)pf3(z)$1+\alpha \frac{z^{2p+2}f'(z)}{pf^{3}(z)}$ are subordinated by Janowski functions, then f(z)zp≺cos⁡(z)$\frac{f(z)}{z^{p}}\prec \cos (z)$. Further, we also discuss the same implications for f(z)zp≺ez$\frac{f(z)}{z^{p}}\prec \textrm{e}^{z}$.

The topic of analytic summability of functions was introduced and studied in 2016 by Hooshmand. He presented some inequalities and upper bounds for analytic summand functions by applying Bernoulli polynomials and numbers. In this work we apply upper bounds, represented by Hua-feng, for Bernoulli numbers to improve the inequalities and related results. Then, we observe that the inequalities are sharp and leave a conjecture about them. Also, as some applications, we use them for some special functions and obtain many particular inequalities. Moreover, we arrived at the inequality 1p+2p+3p+⋯+rp≤12rp+13rp+1(p+1)+23p!πp+1sinh⁡(πr)$1^p + 2^p + 3^p + \dots + r^p \leq \frac{1}{2}r^p + \frac{1}{3}\frac{r^{p+1}}{(p+1)} + \frac{2}{3}\frac{p!}{\pi^{p+1}}\sinh(\pi r)$, for r sums of power of natural numbers, if p ∈ ℕ e and analogously for the odd case.

We study the fourth-order boundary value problem with a sign-changing weight function: u ⁗ = λ m ( t ) u + f 1 ( t , u , u ′ , u ″ , u ‴ , λ ) + f 2 ( t , u , u ′ , u ″ , u ‴ , λ ) , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , $$\left\{\begin{array}{ll} u''''=\lambda m(t)u+f_1(t,u,u',u'',u''',\lambda)+f_2(t,u,u',u'',u''',\lambda),\qquad t\in(0,1),\\[1.3ex] u(0)=u(1)=u''(0)=u''(1)=0, \end{array}\right.$$ where λ ∈ ℝ is a parameter, f 1 , f 2 ∈ C ([0, 1] × ℝ 5 , ℝ), f 1 is not differentiable at the origin and infinity. Under some suitable conditions on nonlinear terms, we prove the existence of unbounded continua of positive and negative solutions of this problem which bifurcating from intervals of the line of trivial solutions or from infinity, respectively.

We investigate the problem with local homogeneous two-point conditions with respect to time for nonhomogeneous PDE of second order in time variable and generally infinite order in spatial variables in the case when the characteristic determinant of the problem identically equals zero. We establish the nonexistence conditions of solution of this problem in the class of entire functions.

In this paper, we show that the following three-dimensional system of difference equations x n + 1 = y n x n − 2 a x n − 2 + b z n − 1 , y n + 1 = z n y n − 2 c y n − 2 + d x n − 1 , z n + 1 = x n z n − 2 e z n − 2 + f y n − 1 , n ∈ N 0 , $$\begin{equation*}x_{n+1}=\frac{y_{n}x_{n-2}}{ax_{n-2}+bz_{n-1}}, \quad y_{n+1}=\frac{z_{n}y_{n-2}}{cy_{n-2}+dx_{n-1}}, \quad z_{n+1}=\frac{x_{n}z_{n-2}}{ez_{n-2}+fy_{n-1}}, \quad n\in \mathbb{N}_{0},\end{equation*}$$ where the parameters a , b , c , d , e , f and the initial values x − i , y − i , z − i , i ∈ {0, 1, 2}, are complex numbers, can be solved, extending further some results in the literature. Also, we determine the forbidden set of the initial values by using the obtained formulas. Finally, an application concerning a three-dimensional system of difference equations are given.

We discuss critical and subcritical linear second-order difference equations, and we observe several identities and inequalities which such equations satisfy depending on their coefficients. Later, we investigate the limit behaviour depending on the coefficients of solutions and of the sequences which appear when finding said solutions. We will see that certain identity is preserved in limit under weaker assumptions. Finally, we investigate a class of fourth-order linear difference equations and show that they are always 1-critical.

In the present work, we prove a Korovkin theorem for statistical e -convergence on two dimensional weighted spaces. We show that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. We also study the rate of statistical e -convergence by using the weighted modulus of continuity and afterwards we present an application in support of our result.

The main objective of this paper is to construct a new sequence of operators involving Apostol-Genocchi polynomials based on certain parameters. We investigate the rate of convergence of the operators given in this paper using second-order modulus of continuity and Voronovskaja type approximation theorem. Moreover, we find weighted approximation result of the given operators. Finally, we derive the Kantorovich variant of the given operators and discussed the approximation results.

The incomplete gamma function Γ( a , u ) is defined by Γ ( a , u ) = ∫ u ∞ t a − 1 e − t d t , $$\Gamma(a,u)=\int\limits_{u}^{\infty}t^{a-1}\textrm{e}^{-t}\textrm{d} t,$$ where u > 0. Using the incomplete gamma function, we define a new Poisson like regular matrix P(μ)=(pnkμ)$\mathfrak{P}(\mu)=(p^{\mu}_{nk})$ given by p n k μ = n ! Γ ( n + 1 , μ ) e − μ μ k k ! ( 0 ≤ k ≤ n ) , 0 ( k > n ) , $$p^{\mu}_{nk}= \begin{cases} \dfrac{n!}{\Gamma(n+1,\mu)}\dfrac{\textrm{e}^{-\mu}\mu^k}{k!} \quad &(0\leq k\leq n), \\[1ex] 0\quad & (k>n), \end{cases}$$ where μ > 0 is fixed. We introduce the sequence space ℓp(P(μ))$\ell_p(\mathfrak{P}(\mu))$ for 1 ≤ p ≤ ∞ and some topological properties, inclusion relations and generalized duals of the newly defined space are discussed. Also we characterize certain matrix classes and compact operators related to the space ℓp(P(μ))$\ell_p(\mathfrak{P}(\mu))$. We obtain Gurarii’s modulus of convexity and investigate some geometric properties of the new space. Finally, spectrum of the operator P(μ)$\mathfrak{P}(\mu)$ on sequence space c 0 has been investigated.

Let G be a locally compact group (not necessarily abelian) and B be a homogeneous Banach space on G , which is in a good situation with respect to a homogeneous function algebra on G . Feichtinger showed that there exists a minimal Banach space B min in the family of all homogenous Banach spaces C on G , containing all elements of B with compact support. In this paper, the amenability and super amenability of B min with respect to the convolution product or with respect to the pointwise product are showed to correspond to amenability, discreteness or finiteness of the group G and conversely. We also prove among other things that B min is a symmetric Segal subalgebra of L 1 ( G ) on an IN-group G , under certain conditions, and we apply our results to study pseudo-amenability and some other homological properties of B min on IN-groups. Furthermore, we determine necessary and sufficient conditions on A under which Amin$\mathcal{A}_{\min}$ with the pointwise product is an abstract Segal algebra or Segal algebra in A , whenever A is a homogeneous function algebra with an approximate identity. We apply these results to study amenability of some Feichtinger algebras with respect to the pointwise product.

In this note, we give a sufficient and necessary condition for weighted translations, generated by group actions, to be disjoint topologically transitive in terms of the weights, the group element and the measure. The characterization of disjoint topological mixing is obtained as well. Moreover, we apply the results to the quotient spaces of locally compact groups and hypergroups.

We investigate joint temporal and contemporaneous aggregation of N independent copies of strictly stationary INteger-valued AutoRegressive processes of order 1 (INAR(1)) with random coefficient α ∈ (0, 1) and with idiosyncratic Poisson innovations. Assuming that α has a density function of the form ψ ( x ) (1 − x ) β , x ∈ (0, 1), with β ∈ (−1, ∞) and limx↑1ψ(x)=ψ1∈(0,∞)$\lim\limits_{x\uparrow 1} \psi(x) = \psi_1 \in (0, \infty)$, different limits of appropriately centered and scaled aggregated partial sums are shown to exist for β ∈ (−1, 0] in the so-called simultaneous case, i.e., when both N and the time scale n increase to infinity at a given rate. The case β ∈ (0, ∞) remains open. We also give a new explicit formula for the joint characteristic functions of finite dimensional distributions of the appropriately centered aggregated process in question.

The authors introduce a new generalized distribution called the Marshall-Olkin Lindley-Log-logistic (MOLLLoG) distribution and discuss its distributional properties. The properties include hazard function, quantile function, moments, conditional moments, mean and median deviations, Bonferroni and Lorenz curves, distribution of the order statistics and Rényi entropy. A Monte Carlo simulation study was used to examine the bias, relative bias and mean square error of the maximum likelihood estimators. The betterness of the new distribution compared to other distributions is illustrated by means of two real life datasets.

This article proposes a class of generator for classical statistical distribution called the shifted Gompertz-G (SHIGO-G) distribution for generating new continuous distributions. Special models of the proposed model were examined together with some of its statistical properties in closed form which makes it tractable for censored data. Its major properties include heavy tail, approximately symmetric, left and right skewed with a combination of exponential and a reverted Gumbel distributions called the Gompertz. The bivariate SHIGO-G is introduced. The parameters estimate of the proposed model was obtained by maximum likelihood method. A Monte Carlo simulation study was employed to investigate the performance of the estimators of the proposed model mean, variance, bias and mean square error. A two real life illustration was used to examine the empirical goodness-of-fit of the test statistic of the proposed model. The results of the real life applications show that the SHIGO-G model provides a better fit for the data set used.

The problem of finding optimal tests in the family of uniform distributions is investigated. The general forms of the uniformly most powerful and generalized likelihood ratio tests are derived. Moreover, the problem of finding the uniformly most powerful unbiased test for testing two-sided hypothesis in the presence of nuisance parameter is investigated, and it is shown that such a test is equivalent to the generalized likelihood ratio test for the same problem. The simulation study is performed to evaluate the performance of power function of the tests.

In this paper, we obtain some norm inequalities involving convex and concave functions, which are the generalizations of the classical Clarkson inequalities. Let A 1 , …, A n be bounded linear operators on a complex separable Hilbert space H$\mathcal{H}$ and let α 1 , …, α n be positive real numbers such that ∑j=1nαj=1$\sum\limits^{n}_{j=1}\alpha_{j}=1$. We show that for every unitarily invariant norm, If f is a non-negative function on [0, ∞) such that f (0) = 0 and g(t)=f(t)$g(t)=f(\sqrt{t})$ is convex, then |||∑j=1nαjf(|Aj|)|||≥|||∑j,k∈Sℓ(f(αjαk4αℓ(1−αℓ)|Aj+Ak−2∑j=1nαjAj|)+f(αjαk(2αℓ−1)4αℓ(1−αℓ)|Aj−Ak|))+f(|∑j=1nαjAj|)|||$$\begin{align*}\bigg|\bigg|\bigg|\sum\limits^{n}_{j=1}\alpha_{j}f(|A_{j}|)\bigg|\bigg|\bigg|&\geq\bigg|\bigg|\bigg|\sum\limits_{j,k\in S_{\ell}}\bigg(f\bigg(\sqrt{\frac{\alpha_{j}\alpha_{k}}{4\alpha_{\ell}(1-\alpha_{\ell})}}\;\bigg|A_{j}+A_{k}-2\sum\limits^{n}_{j=1}\alpha_{j}A_{j}\bigg|\bigg)\\&\qquad+f\bigg(\sqrt{\frac{\alpha_{j}\alpha_{k}(2\alpha_{\ell}-1)}{4\alpha_{\ell}(1-\alpha_{\ell})}}|A_{j}-A_{k}|\bigg)\bigg)+f\bigg(\bigg|\sum\limits^{n}_{j=1}\alpha_{j}A_{j}\bigg|\bigg)\bigg|\bigg|\bigg|\end{align*}$$ for ℓ = 1, …, n . If f is a non-negative function on [0, ∞) such that g(t)=f(t)$g(t)=f(\sqrt{t})$ is concave, then the inverse inequality holds. Here, the symbol S ℓ = {1, …, n } ∖ {ℓ} for ℓ ∈ {1, …, n }. In addition, we provide some applications of the above inequalities.