Volume 73, Issue 3

ABSTRACT We study various combinatorial parameters in the set of inversion sequences of length n . First, we provide generating functions for 1-successions and then generalise this to p -successions. Thereafter we find the formulae for the total number of successions by extracting coefficients. We then do the same for runs of length r and the length of the i th run. Next, we derive a generating function for the number of inversions in inversion sequences themselves and show that inversion sequences with no inversions are counted by the Catalan numbers. Finally, viewing inversion sequences as bargraphs, we develop a generating function for their area.

ABSTRACT Order-theoretic explorations of algebraic structures are known to lead to hitherto hidden insights. Two such relations that have stood out are those of Mitsch and Clifford – the former for the generality in its application and the latter for the insights it offers. In this work, our motivation is to study the converse: we want to explore the extent of the utility of Mitsch’s order and the applicability of Clifford’s order. Firstly, we show that if the Mitsch’s poset is either bounded or a chain, arguably a richer order theoretic structure, the semigroup reduces to one of a simple band. Secondly, noting that the special semigroups on which Clifford’s relation does give rise to an order has not been characterised so far, we solve this problem by proposing a property called Quasi-Projectivity that is essential in this context and also give necessary and sufficient conditions for the Clifford’s relation to give a total and compatible order, even if the semigroup is not commutative. Further, by showing some interesting connections between this relation and the orders obtained by Green’s relations, we further reaffirm the importance and naturalness of the order proposed by Clifford. Finally, by discussing the Clifford’s relations on ordered semigroups, we present some novel perspectives and also show that some of the assumptions in the often cited results of Clifford’s are not necessary. On the whole, our study argues favourably towards Clifford’s than that of the Mitsch’s relation, in so far as the structural information gained about the underlying semigroup.

ABSTRACT Motivated by applications of lattice-valued functions (lattice-valued fuzzy sets) in the theory of ordered structures, we investigate a special kind of posets and lattices induced by these mappings. As a framework, we use the Formal Concept Analysis in which these ordered structures can be naturally observed. We characterize the lattice of cut sets and the Dedekind-MacNeille completion of the set of images of a lattice valued function by suitable concept lattices and we give necessary and sufficient conditions under which these lattices coincide. In addition, we give conditions under which the lattice of cuts is completely distributive.

ABSTRACT Several trigonometric sums are examined by means of partial fraction decompositions. Their generating functions will be evaluated in closed forms and the related asymptotic formulae in terms of Riemann zeta function will be established.

ABSTRACT Let d ( n ) denotes the number of positive integers dividing the positive integer n , and let φ ( n ) denotes Euler’s function representing the number of numbers less than and prime to n . In this paper, we present some notes on the equation d ( n ) = d ( φ ( n )). Several results on the related inequalities are also obtained.

ABSTRACT We define a new matrix whose entries are defined by the idea of combining two different asymmetric rules. We will derive explicit formulæ for the matrices L and U come from LU -decomposition, L −1 , U −1 , inverse of the main matrix as well as its determinant. To prove the claimed results, we use backward induction method.

ABSTRACT Let M be a set of positive integers. We study the maximal density μ ( M ) of the sets of nonnegative integers S whose elements do not differ by an element in M . In 1973, Cantor and Gordon established a formula for μ ( M ) for | M | ≤ 2. Since then, many researchers have worked upon the problem and found several partial results in the case | M | ≥ 3, including some results in the case when M is an infinite set. In this paper, we study the maximal density problem for the families M = { a , a +1, 2 a +1, n } and M = { a , a +1, 2 a +1, 3 a +1, n }, where a and n are positive integers and n is sufficiently large. In most of the cases, we find bounds for the parameter kappa , denoted by κ ( M ), which actually serves as a lower bound for μ ( M ). The parameter κ ( M ) has already got its importance due to its rich connection with problems such as the “lonely runner conjecture” in Diophantine approximation and coloring parameters such as “circular coloring” and “fractional coloring” in graph theory. We also give some partial results for the general family M = { a , a +1, 2 a + 1,…, ( s – 2) a + 1, n }, where s ≥ 5 and mention related problems in the remaining cases for future work.

ABSTRACT In this paper, using Fink’s identity and Green’s function, we obtain several extensions of Jensen’s inequality, Jensen–Steffensen inequality, and the converse of Jensen’s inequality for diamond integrals. Functions involved in these extensions are n -convex functions, n∈ℤ+ . Some bounds for related identities are also part of the discussion. An improved Hölder’s inequality is obtained as an application of improved Jensen’s inequality.

ABSTRACT In this paper, an equality is proved for twice differentiable convex functions involving Riemann–Liouville fractional integral. With the help of this equality, there are established several fractional Simpson type inequalities for functions whose second derivatives in absolute value are convex. By using special cases of the main results, previously obtained Simpson type inequalities are found for the Riemann–Liouville fractional integral.

ABSTRACT A Bromwich-Hansen series is extended to derive the infinite sum of the addition of two incomplete gamma functions in terms of the Hurwitz-Lerch Zeta function. This formula is then used to derive infinite sums and products involving elementary and trigonometric functions. In some cases the infinite products are highly-oscillatory and difficult to evaluate. Symmetric plots are produced to showcase interesting features of special cases of infinite product and summation forms.

ABSTRACT The aim of the paper is to study the oscillation behavior for third-order differential equations with several delays and with p -Laplacian operator. Our technique bases essentially on the presentation of sharper estimates of positive solutions of the equation studied using an iterative method. What is more, the iterative nature of the new criteria allows us to check for oscillation of all solutions, even if the known findings relevant to this fail to apply.

ABSTRACT In this paper, the functional differential equation (DCa+αx)(t)+∑i=0m(Tix(i))(t)=f(t),t∈[a,b], with Caputo fractional derivative DCa+α is studied. The operators T i act from the space of continuous to the space of essentially bounded functions. They can be operators with deviations (delayed and advanced), integral operators and their various linear combinations and superpositions. Such equations could appear in various applications and in the study of systems of, for example, two fractional differential equations, when one of the components can be presented from the first equation and substituted then to another. For two-point problems with this equation, assertions about negativity of Green’s functions and their derivatives with respect to t are obtained. Our technique is based on an analog of the Vallée-Poussin theorem for differential inequalities, which is proven in our paper and gives necessary and sufficient conditions of negativity of Green’s functions and their derivatives for two-point problems: there exists a positive function v satisfying corresponding boundary conditions and the inequality (DCa+αν)(t)+∑i=0m(Tiv(i))(t)<0, t∈[ a, b ] . Choosing the function v , we obtain explicit sufficient tests of sign-constancy of Green’s functions and its derivatives. It is demonstrated that these tests cannot be improved in a general case. Influences of delays on these sufficient conditions are analyzed. It is demonstrated that the tests can be essentially improved for “small” deviations.

ABSTRACT The authors examine the oscillatory behavior of solutions to a class of third order half-linear delay differential equations. The results are obtained by a comparison with first-order linear delay differential equations whose oscillatory characters are known. An example is provided to illustrate the results.

ABSTRACT This article discusses differential equations in infinite-dimensional spaces. We introduce a general version of the Lipschitz-type continuity criterion without the notion of a distance and prove an existence and uniqueness result for ordinary differential equations in locally convex topological linear spaces. It is shown that remarkable techniques of classical analysis, such as the method of successive approximations, are still adaptable tools in the study of the abstract models.

ABSTRACT Let d ∈ {1, 2, 3, . . .} and Ω ⊂ ℝ d be open bounded with Lipschitz boundary. Consider the reaction-diffusion parabolic problem P { ut|x|4+Δ(|Δu|m−2Δu)=k(t)|u|p−1u(x,t)∈Ω×(0,T),u(x,t)= ∂xju(x,t)=0,(x,t)∈∂Ω×(0,T), j∈{1,…,d},u(x,0)=u0(x),x∈Ω, where T > 0, m ∈ [2, ∞), p ∈ (1, ∞) and 0≠u0∈W02,m(Ω)∩Lp+1(Ω) $0 \ne u_0 \in W^{2,m}_0(\Omega) \cap L^{p+1}(\Omega)$. We investigate the upper and lower bounds on the blow-up time of a weak solution to ( P ).

ABSTRACT In this paper, we represent that the following three-dimensional system of difference equations xn+1=αyn+aynyn−βzn−1, yn+1=βzn+bznzn−γxn−1, zn+1=γxn+cxnxn−αyn−1,  n∈ℕ0, $$\matrix{{{x_{n + 1}} = \alpha {y_n} + {{a{y_n}} \over {{y_n} - \beta {z_{n - 1}}}},\quad {y_{n + 1}} = \beta {z_n} + {{b{z_n}} \over {{z_n} - \gamma {x_{n - 1}}}},\quad {z_{n + 1}} = \gamma {x_n} + {{c{x_n}} \over {{x_n} - \alpha {y_{n - 1}}}},\qquad n \in {{\mathbb N}_0},} \cr} $$ where the parameters a, b, c, α, β, γ and the initial values x − i , y − i , z − i , i ∈ {0, 1}, are real numbers, can be solved in closed form by using transformation. We analyzed the solutions in 10 different cases depending on whether the parameters are zero or nonzero. It is noteworthy to depict that the solutions of some particular cases of this system are presented in terms of generalized Fibonacci numbers. Note that our results considerably extend and improve some recent results in the literature.

ABSTRACT In this paper, the families of functions continuous with respect to some family of subsets of the real line as a domain are considered. There is proved, among others, that if the family of subsets has some special property, then each function continuous with respect to this family is quasi-continuous (Corollary 2.2.1). We say that two families of subsets of the real line are equivalent if the families of functions which are continuous in generalized sense with respect to these families are equal. Limited considerations to the families which are generalized topologies with additional properties, we find in equivalence classes the smallest and the largest family with respect to inclusion (see Theorem 3.5).

ABSTRACT In this work, a new family of distributions referred to as type II exponentiated half-logistic-Gompertz-G (TIIEHL-Gom-G) family of distributions is introduced and studied. Some of the main statistical properties of these family of distributions are derived. The model parameters are estimated using the maximum likelihood estimation technique and consistency of maximum likelihood estimators is evaluated by performing a simulation study. The importance and versatility of the TIIEHL-Gom-G family of distributions is demonstrated in an application to two real data sets from different fields.

ABSTRACT In this paper, we establish some results on strong consistency for the least square estimator in a simple linear errors-in-variables regression models based on widely orthant dependent errors. The results obtained in the paper generalize or improve the corresponding ones for dependent random variables. We also carry out some simulation studies to verify the validity of our theoretical results.