Volume 67, Issue 4

In the present paper, the notion of an outer measure m * on lattice ordered effect algebras L is introduced and investigated. Carathéodory’s criterion for the outer measure m * is given and established on L . Properties of an induced outer measure m * defined on a lattice ordered effect algebra L are studied and finally, Choquet theorem is proved for the induced outer measure m * on a σ -complete lattice ordered effect algebra L .

We introduce the notion of Hamiltonian ordered algebra and explore its relationships with various congruence extension properties for ordered algebras. Furthermore, we give sufficient conditions for extending order-congruences or compatible quasiorders from subalgebras.

All considered groups are torsion or do not contain infinitely generated subgroups. If such a group G acts transitively on a connected algebra A , then all elements of A have the same stabilizer, so this stabilizer is a normal subgroup (it is also shown that these facts are not true for arbitrary groups). Hence the automorphism group Aut( A ) is a homomorphic image of G . In particular, if the action of G is, in addition, faithful, then G is isomorphic to Aut( A ). By these results we first obtain that for each (not necessarily connected) algebra A , if all of its Aut( A )-orbits are connected, then Aut( A ) is a subdirect product of automorphism groups of these orbits. Secondly, connected components of Aut( A )-orbits have automorphism extension property. Next, we show that if A is a connected algebra such that Aut( A ) acts transitively on it, then the group structure of Aut( A ) may be transported on A such that the left multiplications are all automorphisms of A and the right multiplications are all unary term operations of A . Hence all the unary term operations of A are bijective, so they generate a subgroup of the group of all bijections of the carrier of A . It is shown that this group is anti-isomorphic to Aut( A ). Thus the Birkhoff’s construction of an algebra with a given group of automorphisms is unique in some sense when restricted to groups we consider here. The Birkhoff’s construction can be slightly modified so as to obtain a smaller set of operations. In fact, it is enough to take the right multiplications by generators. Moreover, we show that this is the best possible lower bound for the number of unary operations in the case of groups considered here. If we admit non-unary operations, then for finite and countable groups we can reduce the number of operations to one binary operation. On the other hand, if A is a connected algebra such that Aut( A ) is torsion and acts on A transitively, then each element generates A . Hence if A is such an algebra with an uncountable group Aut( A ), then the cardinality of the set of operations of A is greater or equal than the cardinality of Aut( A ).

Diophantine triples taking values in recurrence sequences have recently been studied quite a lot. In particular the question was raised whether or not there are finitely many Diophantine triples in the Tribonacci sequence. We answer this question here in the affirmative. We prove that there are only finitely many triples of integers 1 ≤ u < v < w such that uv + 1, uw + 1, vw + 1 are Tribonacci numbers. The proof depends on the Subspace theorem.

The notion of *-normal idempotents is introduced. The structure theorem for abundant semigroups with a *-normal idempotent is obtained. As its applications, we establish the construction theorem of naturally ordered abundant semigroups with a greatest idempotent.

In this paper, we shall present some uniqueness and Ulam’s type stability concepts for the Darboux problem of partial functional differential equations with not instantaneous impulses and state-dependent delay in Banach spaces. Some examples are also provided to illustrate our results.

In this paper, by means of a recently obtained inequality, we study the delta fractional difference, and we obtain the following interrelated theorems, which improve recent results in the literature. Theorem A Assume that f : ℕ a → ℝ and that Δ aν f(t) $\Delta^\nu_af(t)$ ≥ 0, for each t ∈ ℕ a +2− ν , with 1 < ν < 2. If f(a+1)≥ ν k+2f(a), $f(a+1) \geq \frac{\nu}{k+2}f(a),$ for each k ∈ ℕ 0 , then Δ f ( t ) ≥ 0 for t ∈ ℕ a +1 . Theorem B Assume that f : ℕ a → ℝ and that Δ aν f(t) $\Delta^\nu_af(t)$ ≥ 0, for each t ∈ ℕ a +2− ν , with 1 < ν < 2. If f(a+2)≥ ν k+1f(a+1)+(k+1− ν )ν (k+2)(k+3)f(a) $$f(a+2)\geq\displaystyle\frac{\nu}{k+1}f(a+1)+\frac{(k+1-\nu)\nu}{(k+2)(k+3)}f(a) $$ for each k ∈ ℕ 1 , then Δ f ( t ) ≥ 0 for t ∈ ℕ a +2 . Theorem C Assume that f : ℕ a → ℝ and that Δ aν f(t) $\Delta^\nu_af(t)$ ≥ 0, for each t ∈ ℕ a +2− ν , with 1 < ν < 2. If f(a+3)≥ ν kf(a+2)+(k− ν )ν k(k+1)f(a+1)+(k+1− ν )(k− ν )ν (k+2)(k+1)kf(a) $$f(a+3)\geq\displaystyle\frac{\nu}{k}f(a+2)+\frac{(k-\nu)\nu}{k(k+1)}f(a+1)+\frac{(k+1-\nu)(k-\nu)\nu}{(k+2)(k+1)k}f(a) $$ for k ∈ ℕ 2 , then Δ f ( t ) ≥ 0, for t ∈ ℕ a +3 . In addition, we obtain the following result, which extends a recent result due to Atici and Uyanik. Theorem D Assume that f : ℕ a → ℝ, Δ N f ( t ) ≥ 0 for t ∈ ℕ a , and (−1) N − i Δ i f ( a ) ≤ 0 for i = 0, 1, …, N − 1. Then Δ aν f(t) $\Delta^\nu_af(t)$ ≥ 0 for t ∈ ℕ a + N − ν .

A special family of multigeometric series is considered from the point of view of behaviour of their sets of subsums. A sufficient condition for their sets of subsums to be M-Cantorvals is proven. The Lebesgue measure of those special M-Cantorvals is computed and it is shown to be equal to the sum of lengths of all component intervals of the M-Cantorvals. A new sufficient condition for the set of subsums of a series to be a Cantor set is formulated and it is used to demonstrate that the discussed multigeometric series always have Cantor sets as their sets of subsums for sufficiently small ratios of the series.

The purpose of this paper is to study the notion of a ψ -porosity point and ψ -porosity topology, generated by it analogously to the porosity topology on the real line. The paper includes a necessary and sufficient conditions under which two ψ -porous topologies generated by two functions ψ 1 and ψ 2 are equal. The condition is formulated in terms of the behavior of two sequences of sets Ak={x∈ (0,∞ ): ψ 1(x)< 1kψ 2(x)} and Bk={x∈ (0,∞ ): ψ 2(x)< 1kψ 1(x)}. $\begin{array}{} A_k=\{x\in(0,\infty)\colon\psi_1(x) \lt \frac{1}{k}\psi_2(x)\} \text{ and } B_k=\{x\in(0,\infty)\colon\psi_2(x) \lt \frac{1}{k}\psi_1(x)\}. \end{array}$

We introduce a convergence of weight g : ℕ → [0, ∞) where g ( n ) → ∞ and n / g ( n ) ↛ 0 with respect to a summability matrix method A for sequences (which generalizes the notion of A -convergence of order α , 0 < α ≤ 1 [BOCCUTO, A.—DAS, P.: On matrix methods of convergence of order ( α ) in ( ℓ )- groups , Filomat 29 (2015), 2069–2077] in a ( ℓ )-group. We prove some basic results including a Cauchy-type criterion. Finally a closedness result for the space of such convergent sequences is proved.

In this work we present a new geometric approach to some problems in differential subordination theory. In the paper some sufficient conditions for function to be starlike or univalent or to be in the class of Carathéodory functions are obtained. We also discuss the new results closely related to the generalized Briot-Bouquet differential subordination.

Domains with conical sections is an underlying concept in the area of complex function theory although is an interesting topic and it deserves more attention. There has been many works focusing towards this area for the past two decades. However, the concept of Hankel determinant has not been studied so far. Exploiting this, we provide an estimate for the Hankel determinant with domains bounded by conical sections. The authors sincerely hope this article will revive and encourage the other researchers to obtain similar sort of estimates for other classes connected with conical domains. The concept of conical domains was introduced by Kanas and Wiśniowska.

For a delay differential equation x¨ (t)+a(t)x˙ (t)+∑ k=1mbk(t)x(gk(t))=0,gk(t)≤ t, $$\ddot{x}(t) + a(t)\dot{x}(t)+ \displaystyle\sum\limits_{k=1}^m b_k(t)x(g_k(t))=0, g_k(t)\leq t, $$ a generalized Riccati inequality is constructed for nonoscillation and exponential stability of the differential equation.

We discuss the existence of positive solutions for the nonlinear perturbation of Hammerstein integral equation. The technique we use is based on the fixed point theorem of Leggett-Williams type for strict set contractions. We conclude the paper by providing some examples that illustrate our claim.

In this paper, we prove that if a 3-dimensional cosymplectic manifold M 3 admits a Ricci soliton, then either M 3 is locally flat or the potential vector field is an infinitesimal contact transformation.

We define probabilistic convergence groups based on Tardiff’s neighborhood systems for probabilistic metric spaces and develop the basic theory. We study, as natural examples, probabilistic metric groups and probabilistic normed groups as well as probabilistic limit groups under a t -norm as defined earlier by the authors. We further show that a probabilistic convergence group induces a natural probabilistic uniform convergence structure and give a result on probabilistic metrization.

We introduce and study the concept of the super socle of C ( X ), denoted by SC F ( X ) (i.e., the set of elements of C ( X ), which are zero everywhere except on a countable number of points of X ). Using this concept we extend some of the basic results concerning C F ( X ), the socle of C ( X ), to SC F ( X ). In particular, we determine spaces X such that C F ( X ) and SC F ( X ) coincide. Spaces X such that Ann( SC F ( X )) is generated by an idempotent are fully characterized. It is shown that SC F ( X ) is an essential ideal in C ( X ) if and only if the set of countably isolated points (i.e., points with countable neighborhoods) of X is dense in X . The one-point Lindelöffication of uncountable discrete spaces is algebraically characterized via the concept of the super socle. Consequently, it is observed that whenever O x ⊆ SC F ( X ) and SC F ( X ) is a regular ideal (von Neumann), then X is either a countable discrete space or the one-point Lindelöffication of an uncountable discrete space. Consequently, in this case SC F ( X ) is a prime ideal in C ( X ) (note, C F ( X ) is never prime C ( X ))

The article concerns the geometrical theory of general systems Ω of partial differential equations in the absolute sense , i.e., without any additional structure and subject to arbitrary change of variables in the widest possible meaning. The main result describes the uniquely determined composition series Ω 0 ⊂ Ω 1 ⊂ … ⊂ Ω where Ω k is the maximal system of differential equations “induced” by Ω such that the solution of Ω k depends on arbitrary functions of k independent variables (on constants if k = 0). This is a well-known result only for the particular case of underdetermined systems of ordinary differential equations. Then Ω = Ω 1 and we have the composition series Ω 0 ⊂ Ω 1 = Ω where Ω 0 involves all first integrals of Ω, therefore Ω 0 is trivial (absent) in the controllable case. The general composition series Ω 0 ⊂ Ω 1 ⊂ … ⊂ Ω may be regarded as a “multidimensional” controllability structure for the partial differential equations. Though the result is conceptually clear, it cannot be included into the common jet theory framework of differential equations. Quite other and genuinely coordinate-free approach is introduced.

In this paper we are concerned with a three-body system with point interaction, which is called the Ter-Martirosian–Skornyakov extension. We locate the bottom of the essential spectrum of that system and establish the finiteness of the discrete spectrum below the bottom. Our work here refines the result of [MINLOS, R. A.: On point - like interaction between n fermions and another particle , Mosc. Math. J. 11 (2011), 113–127], where the semi-boundedness of the operator is obtained.

In this paper, we introduce a subclass of analytic functions in the open unit disc. This class generalizes the class of Bazilevic functions of order α . We find arc length, coefficient bounds, coefficient difference, growth result, q th Hankel determinant of this class of functions.