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The aim of the paper is to investigate the relationship among NMV-algebras, commutative basic algebras and naBL-algebras (i.e., non-associative BL-algebras). First, we introduce the notion of strong NMV-algebra and prove that (1)a strong NMV-algebra is a residuated l-groupoid (i.e., a bounded integral commutative residuated lattice-ordered groupoid)(2)a residuated l-groupoid is commutative basic algebra if and only if it is a strong NMV-algebra. Secondly, we introduce the notion of NMV-filter and prove that a residuated l-groupoid is a strong NMV-algebra (commutative basic algebra) if and only if its every filter is an NMV-filter. Finally, we introduce the notion of weak naBL-algebra, and show that any strong NMV-algebra (commutative basic algebra) is weak naBL-algebra and give some counterexamples.

Given a topological space X, Jenkins and McKnight have shown how ideals of the ring C(X) are partitioned into equivalence classes — called coherence classes — defined by declaring ideals to be equivalent if their pure parts are identical. In this paper we consider a similar partitioning of the lattice of ideals of a normal bounded distributive lattice. We then apply results obtained herein to augment some of those of Jenkins and McKnight. In particular, for Tychonoff spaces, new results include the following: (a)all members of any coherence class have the same annihilator(b)every ideal is alone in its coherence class if and only if the space is a P-space.

We prove that the extent partitions of a formal context $$\mathbb{K}: = (G,M,I)$$ can be constructed from the box extents of it, which form a complete atomistic lattice. $$\mathbb{K}$$ is called a one-object extension of the subcontext (H, M, J) if it is obtained by adding a new element with attributes in M to the set H. We investigate the interplay between the box extents of (H, M, J) and those of its one-object extension $$\mathbb{K}$$, and describe those extent partitions of (H, M, J) which can be extended to $$\mathbb{K}$$.

In the paper we formulate a criterion for the nonsingularity of a bilinear form on a direct sum of finitely many invertible ideals of a domain. We classify these forms up to isometry and, in the case of a Dedekind domain, up to similarity.

This note deals with Ramanujan sums c m(n) over the ring ℤ[i], in particular with asymptotics for sums of c m(n) taken over both variables m, n.

In this paper, we construct eight infinite families of binary linear codes associated with double cosets with respect to a certain maximal parabolic subgroup of the special orthogonal group SO −(2n, 2r). And we obtain four infinite families of recursive formulas for the power moments of Kloosterman sums and four those of 2-dimensional Kloosterman sums in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of “Gauss sums” for the orthogonal groups O −(2n, 2r).

In this paper we introduce the notion of TN-groupoids in an arbitrary groupoid, and discuss some properties in Bin(X), and obtain several results related with non-negativity of norms.

In this paper, we study the extremal solutions of Cauchy problems for abstract fractional differential equations. Some definitions such as L 1-Lipschitz-like, L 1-Carathéodory-like and L 1-Chandrabhan-like are introduced. By virtue of the singular integral inequalities with several nonlinearities due to Medved’, the properties of solutions are given. By using a hybrid fixed point theorem due to Dhage, existence results for extremal solutions are established. Finally, we present an example to illustrate our main results.

Let f: ℝ2 → ℝ be a function with upper semicontinuous and quasi-continuous vertical sections f x(t) = f(x, t), t, x ∈ ℝ. It is proved that if the horizontal sections f y(t) = f(t, y), y, t ∈ ℝ, are of Baire class α (resp. Lebesgue measurable) [resp. with the Baire property] then f is of Baire class α + 2 (resp. Lebesgue measurable and sup-measurable) [resp. has Baire property].

In this paper, some existence theorems are obtained for periodic solutions of second order dynamical system with (q, p)-Laplaician by using the least action principle and the saddle point theorem. Our results improve Pasca and Tang’ results.

By means of the shift operators we introduce a new periodicity concept on time scales. This new approach will enable researchers to investigate periodicity notion on a large class of time scales whose members may not satisfy the condition there exists a P > 0 such that $$t \pm P \in \mathbb{T}$$ for all $$t \in \mathbb{T}$$, which is being currently used. Therefore, the results of this paper open an avenue for the investigation of periodic solutions of q-difference equations and more.

In this paper we introduce some certain new sequence spaces via ideal convergence, λ-sequence and an Orlicz function in n-normed spaces and study different properties of these spaces and also establish some inclusion results among them.

The main aim of this paper is to investigate the (H p, L p)-type inequality for the maximal operators of Riesz and Nörlund logarithmic means of the quadratical partial sums of Walsh-Fourier series. Moreover, we show that the behavior of Nörlund logarithmic means is worse than the behavior of Riesz logarithmic means in our special sense.

The main purpose of this work is to extend the sequence spaces which are defined in [KARAKAYA, V.—POLAT, H.: Some new paranormed sequence spaces defined by Euler and difference operators, Acta Sci. Math. (Szeged) 76 (2010), 87–100] and [POLAT, H.—BASAR, F.: Some Euler spaces of difference sequences of order m, Acta Math. Sci. Ser. B Engl. Ed. 27 (2007), 254–266] by using difference operator of order m, and to give their alpha, beta and gamma duals. Furthermore, we characterize some classes of the related matrix transformations.

We give a negative answer to Problem 2 posed by R. A. McCoy in his paper [McCoy R. A.: Spaces of lower semicontinuous set-valued maps II, Math. Slovaca 60 (2010), 541–570]. Some topological properties of the space L −(X) introduced in [McCOY R. A.: Spaces of lower semicontinuous set-valued maps I, Math. Slovaca 60 (2010), 521–540] equipped with the Vietoris topology are also investigated.

In [18], Matthews introduced a new class of metric spaces, that is, the concept of partial metric spaces, or equivalently, weightable quasi-metrics, are investigated to generalize metric spaces (X, d), to develop and to introduce a new fixed point theory. In partial metric spaces, the self-distance for any point need not be equal to zero. In this paper, we study some results for single map satisfying (ψ,φ)-weakly contractive condition in partial metric spaces endowed with partial order. An example is given to support the useability of our results.

We consider quotients of complex Stiefel manifolds by finite cyclic groups whose action is induced by the scalar multiplication on the corresponding complex vector space. We obtain a description of their tangent bundles, compute their mod p cohomology and obtain estimates for their span (with respect to their standard differentiable structure). We compute the Pontrjagin and Stiefel-Whitney classes of these manifolds and give applications to their stable parallelizability.

Let 〈h n〉 denote a sequence of positive real numbers. We show that for every set A ⊆ ℝ there exists a function f: ℝ → ω such that A = {x ∈ ℝ: (∈〈h n〉)[h n ↘ 0 & (∀n ∈ ℕ)(f(x − h n) = f(x + h n) = f(x))]}. This solves a problem of K. Ciesielski, K. Muthuvel and A. Nowik.