Volume 69, Issue 6

A Riesz structure on a lattice ordered abelian group G is a real vector space structure where the product of a positive element of G and a positive real is positive. In this paper we show that for every cardinal k there is a totally ordered abelian group with at least k Riesz structures, all of them isomorphic. Moreover two Riesz structures on the same totally ordered group are partially isomorphic in the sense of model theory. Further, as a main result, we build two nonisomorphic Riesz structures on the same l-group with strong unit. This gives a solution to a problem posed by Conrad in 1975. Finally we apply the main result to MV-algebras and Riesz MV-algebras.

Considering the equation x 4 + y 4 = n ( u 4 + v 4 ), we first investigate a necessary condition by which the equation has solution and then, for infinitely many n ′s we present the integral solutions.

Let R ( n ) denote the number of representations of a natural number n as the sum of two squares and four cubes of primes. In this paper, it is proved that the anticipated asymptotic formula for R ( n ) fails for at most O(N14+ε )$\begin{array}{}O(N^{\frac{1}{4} + \varepsilon})\end{array}$ positive integers not exceeding N .

Let 𝓞 𝕂 be the ring of integers of the number field 𝕂 = Q(2,3)$\begin{array}{} \displaystyle \mathbb{Q}(\sqrt{2},\sqrt{3}) \end{array}$. A D ( n )- quadruple in the ring 𝓞 𝕂 is a set of four distinct non-zero elements { z 1 , z 2 , z 3 , z 4 } ⊂ 𝓞 𝕂 with the property that the product of each two distinct elements increased by n is a perfect square in 𝓞 𝕂 . We show that the set of all n ∈ 𝓞 𝕂 such that a D ( n )-quadruple in 𝓞 𝕂 exists coincides with the set of all integers in 𝕂 that can be represented as a difference of two squares of integers in 𝕂.

In this paper, we prove that blown up at seven general points admits a conic bundle structure over ℙ 1 and it can be embedded as (2, 2) divisor in ℙ 1 × ℙ 2 . Conversely, any smooth surface in the complete linear system |(2, 2)| of ℙ 1 × ℙ 2 can be obtained as an embedding of blowing up ℙ 2 at seven points. We also show that smooth surface linearly equivalent to (2, 2) in ℙ 1 × ℙ 2 has at most four (−2) curves.

Yuqun Chen and K. P. Shum in [ Rees short exact sequence of S-systems , Semigroup Forum 65 (2002), 141–148] introduced Rees short exact sequence of acts and considered conditions under which a Rees short exact sequence of acts is left and right split, respectively. To our knowledge, conditions under which the induced sequences by functors Hom( R L S , –), Hom(–, R L S ) and A S ⊗ S – (where R , S are monoids) are exact, are unknown. This article addresses these conditions. Results are different from that of modules.

In this paper the solution of the pointwise multiplier problem between weighted Copson function spaces Cop p 1 , q 1 ( u 1 , v 1 ) and weighted Cesàro function spaces Ces p 2 , q 2 ( u 2 , v 2 ) is presented, where p 1 , p 2 , q 1 , q 2 ∈ (0, ∞), p 2 ≤ q 2 and u 1 , u 2 , v 1 , v 2 are weights on (0, ∞).

Let 𝓐 denote the family of analytic functions f with f (0) = f ′(0) – 1 = 0, in the open unit disk Δ. We consider a class Scs∗ (α ):=f∈ A:zf′(z)f(z)− 1≺ z1+α − 1z− α z2,z∈ Δ ,$$\begin{array}{} \displaystyle \mathcal{S}^{\ast}_{cs}(\alpha):=\left\{f\in\mathcal{A} : \left(\frac{zf'(z)}{f(z)}-1\right)\prec \frac{z}{1+\left(\alpha-1\right) z-\alpha z^2}, \,\, z\in \Delta\right\}, \end{array}$$ where 0 ≤ α ≤ 1/2, and ≺ is the subordination relation. The methods and techniques of geometric function theory are used to get characteristics of the functions in this class. Further, the sharp inequality for the logarithmic coefficients γ n of f ∈ Scs∗ $\begin{array}{} \mathcal{S}^{\ast}_{cs} \end{array}$( α ): ∑ n=1∞ γ n2≤ 141+α 2π 26− 2Li2− α +Li2α 2,$$\begin{array}{} \displaystyle \sum_{n=1}^{\infty}\left|\gamma_n\right|^2 \leq \frac{1}{4\left(1+\alpha\right)^2}\left(\frac{\pi^2}{6}-2 \mathrm{Li}_2\left(-\alpha\right)+ \mathrm{Li}_2\left(\alpha^2\right)\right), \end{array}$$ where Li 2 denotes the dilogarithm function are investigated.

The purpose of the paper is to show that noncanonical operator Ly=r2(t)r1(t)y′(t)′′$$\begin{array}{} \displaystyle \mathcal {L}\,y=\left(r_2(t)\left(r_1(t)y'(t)\right)'\right)' \end{array}$$ can be easily written in essentially unique canonical form Ly=q3(t)q2(t)q1(t)q0(t)y(t)′′′$$\begin{array}{} \displaystyle \mathcal {L}\,y = q_3(t)\left(q_2(t)\left(q_1(t)\left(q_0(t)y(t)\right)'\right)'\right)' \end{array}$$ such that ∫∞1qi(s)ds=∞,i=1,2.$$\begin{array}{} \displaystyle \int\limits^\infty \frac{1}{q_i(s)}\,\text{d}{s}=\infty, \quad i=1,2. \end{array}$$ The canonical representation is applied for examination of the third order noncanonical equations r2(t)r1(t)y′(t)′′+p(t)y(τ(t))=0.$$\begin{array}{} \displaystyle \left(r_2(t)\left(r_1(t)y'(t)\right)'\right)'+p(t)y(\tau(t))=0. \end{array}$$

In this paper we prove some existence and regularity results for nonlinear unilateral problems with degenerate coercivity via the penalty method.

In the Orlicz type spaces 𝓢 M , we prove direct and inverse approximation theorems in terms of the best approximations of functions and moduli of smoothness of fractional order. We also show the equivalence between moduli of smoothness and Peetre K -functionals in the spaces 𝓢 M .

In this paper, a kind of new analogue of Phillips operators based on ( p , q )-integers is introduced. The moments of the operators are established. Then some local approximation for the above operators is discussed. Also, the rate of convergence and weighted approximation by these operators by means of modulus of continuity are studied. Furthermore, the Voronovskaja type asymptotic formula is investigated.

In this paper, we introduce a new type of set-valued mappings satisfying a contractive condition involving SR -functions. Then we prove a best proximity point theorem concerning the aforementioned mappings. Some new best proximity and fixed point results are obtained as consequences. An example is provided to support the theory.

In this paper, we study the relationship between the class of weakly demicompact linear operators, introduced in [KRICHEN, B.—O’REGAN, D.: On the class of relatively weakly demicompact nonlinear operators , Fixed Point Theory 19 (2018), 625–630], and measures of weak noncompactness of linear operators with respect to an axiomatic one. Moreover, some Fredholm and perturbation results involving the class of weakly demicompact linear operators are investigated. Our results are then used to investigate the relationship between the relative essential spectrum of the sum of two linear operators and the relative essential spectrum of each of these operators.

In this paper, we consider F 𝓡 -generalized contractivity condition and utilized the same to establish some fixed point results for a self-mapping in partial metric spaces endowed with an amorphous binary relation. Our results generalize several core results of the existing literature. We also furnish some examples to exhibit the utility of our results. Finally, we further deduce fixed point result for cyclic contractions in partial metric spaces.

In this article we study Einstein-Weyl structures on a 3-dimensional trans-Sasakian manifold M of type ( α , β ). First, we prove that a 3-dimensional trans-Sasakian manifold admitting both Einstein-Weyl structures W ± = ( g , ± θ ) is Einstein, or is homothetic to a Sasakian manifold if α ≠ 0. Next for β ≠ 0 it is proved that M is Einstein, or is homothetic to an f -Kenmotsu manifold if it admits an Einstein-Weyl structure W = ( g , κη ) for some nonzero constant κ . Finally, a classification is obtained when a trans-Sasakian manifold admits a closed Einstein-Weyl structure. Further, if M is compact we also obtain two corollaries.

Our purpose in this paper is to study complete linear Weingarten space-like hypersurface immersed in locally symmetric Lorentz space obeying some curvature conditions. Our approach is based on the use of a Simons type formula related to an appropriated Cheng-Yau modified operator jointly with some generalized maximum principles, we obtain that such a space-like hypersurface must be either totally umbilical or isometric to an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. This result corresponds to a natural improvement of previous ones due to de Lima, dos Santos, Velásquez [ On the umbilicity of complete linear Weingarten spacelike hypersurfaces immersed in a locally symmetric Lorentz space , São Paulo J. Math. Sci. 11 (2017), 456–470] and Alías, de Lima, dos Santos [ New characterizations of linear Weingarten spacelike hypersurfaces in de Sitter space , Pacific J. Math. 292 (2018), 1–19].

In this paper, we consider *-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if ( M , g ) is a Kenmotsu manifold and g is a *-Ricci soliton, then soliton constant λ is zero. For 3-dimensional case, if M admits a *-Ricci soliton, then we show that M is of constant sectional curvature –1. Next, we show that if M admits a *-Ricci soliton whose potential vector field is collinear with the characteristic vector field ξ , then M is Einstein and soliton vector field is equal to ξ . Finally, we prove that if g is a gradient almost *-Ricci soliton, then either M is Einstein or the potential vector field is collinear with the characteristic vector field on an open set of M . We verify our result by constructing examples for both *-Ricci soliton and gradient almost *-Ricci soliton.

Given random variables X and Y having finite moments of all orders, their uncorrelatedness set is defined as the set of all pairs ( j , k ) ∈ ℕ 2 , for which X j and Y k are uncorrelated. It is known that, broadly put, any subset of ℕ 2 can serve as an uncorrelatedness set. This claim is no longer valid for random variables with prescribed distributions, in which case the need arises so as to identify the possible uncorrelatedness sets. This paper studies the uncorrelatedness sets for positive random variables uniformly distributed on three points. Some general features of these sets are derived. Two related Vandermonde-type determinants are examined and applied to describe uncorrelatedness sets in some special cases.

In this paper, we consider the wavelet estimators of a nonparametric regression model based on widely orthant dependent random errors. The moment consistency and the completely consistency for wavelet estimators under some more mild moment conditions are investigated. The results obtained in the paper improve and extend the corresponding ones for dependent random variables. Finally, we provide a numerical simulation to verify the validity of our results.

This paper considers the nonparametric regression model with negatively super-additive dependent (NSD) noise and investigates the convergence rates of thresholding estimators. It is shown that the term-by-term thresholding estimator achieves nearly optimal and the block thresholding estimator attains optimal (or nearly optimal) convergence rates over Besov spaces. Additionally, some numerical simulations are implemented to substantiate the validity and adaptivity of the thresholding estimators with the presence of NSD noise.

This paper is a continuation of [Bisht, R. K.—Rakočević, V.: Generalized Meir-Keeler type contractions and discontinuity at fixed point , Fixed Point Theory 19 (1) (2018), 57–64]. We obtain some fixed point theorems under contractive conditions which admit discontinuity at the fixed point. Our results subsume all the known results of similar type, provide new answers to the question of continuity of contractive mappings at their fixed points, and also rectify some recent results by Bisht and Rakočević. Many examples are given to illustrate our results.