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A lattice L is slim if it is finite and the set of its join-irreducible elements contains no three-element antichain. Slim, semimodular lattices were previously characterized by G. Czédli and E. T. Schmidt as the duals of the lattices consisting of the intersections of the members of two composition series in a group. Our main result determines the number of (isomorphism classes of) these lattices of a given size in a recursive way. The corresponding planar Hasse diagrams, up to similarity, are also enumerated. We prove that the number of diagrams of slim, distributive lattices of a given length n is the n th Catalan number. Besides lattice theory, the paper includes some combinatorial arguments on permutations and their inversions.

We introduce a non-commutative generalization of quasi-MV algebra, called quasipseudo-MV algebra. We present some properties of quasi-pseudo-MV algebras and investigate the direct product decomposition of them. Further, we generalize quasi l -group to the non-commutative case and prove the interval of a non-commutative quasi l -group with a strong quasi-unit is a quasipseudo-MV algebra.

A lattice ordered group A will be said to be a lexico group if there exists a convex ℓ-subgroup A 0 of A with A 0 ≠ A such that for each a ∈ A \ A 0 we have either a > a 0 for each a 0 ∈ A 0 , or a < a 0 for each a 0 ∈ A 0 . We prove the following result. Let A be a convex ℓ-subgroup of a lattice ordered group ∈ such that (i) A is a lexico group, and (ii) A fails to be upper bounded in G . Then A is a direct factor of G .

We consider limit periodic sequences and functions with values in pseudometric spaces. We construct limit periodic sequences and functions with given values. For any totally bounded countable set, we find a limit periodic sequence which attains each value from this set periodically. A corresponding result concerning such a construction of limit periodic functions is proved as well. For any totally bounded countable set which is dense in itself, we construct a limit periodic bijective map from the integers into this set. As corollaries, we obtain new results about non-almost periodic solutions of almost periodic transformable difference systems.

In this paper, we give the definitions of crossed quasi-Hopf π -coalgebra H and a crossed left π - H -modules, and show that the category of crossed left π - H -modules is a monoidal category. Finally, we show that a family R = { R α , β ∈ H α ⊗ H β } of elements is a quasitriangular structure of a crossed quasi-Hopf π -coalgebra H if and only if the category of crossed left π - H -modules over H is a braided monoidal category with braiding defined by R .

The q -Bernstein basis used to construct the q -Bernstein polynomials is an extension of the Bernstein basis related to the q -binomial probability distribution. This distribution plays a profound role in the q -boson operator calculus. In the case q > 1, q -Bernstein basic polynomials on [0, 1] combine the fast increase in magnitude with sign oscillations. This seriously complicates the study of q -Bernstein polynomials in the case of q > 1. The aim of this paper is to present new results related to the q -Bernstein polynomials B n , q of discontinuous functions in the case q > 1. The behavior of polynomials B n , q ( f ; x ) for functions f possessing a logarithmic singularity at 0 has been examined.

We introduce some family of functions f : ℝ → ℝ modifying Darboux property analogously as it was done in GRANDE, Z.: On a subclass of the family of Darboux functions, Colloq. Math. 117 (2009), 95-104, and changing approximate continuity with 𝓘-approximate continuity, i.e. continuity with respect to the 𝓘-density topology. We prove that our family is a strongly porous set in the space of Darboux functions having the Baire property and that each function from our family is quasi-continuous.

In this paper we characterize the pairs ( A 0 , A 1 ) and A – , A + ) of disjoint subsets of ℝ which can be separated by a peripherally continuous function.

In this paper, we study a stochastic delay predator-prey model in a polluted environment. Sufficient criteria for extinction and non-persistence in the mean of the model are obtained. The critical value between persistence and extinction is also derived for each population. Finally, some numerical simulations are provided to support our main results.

In the paper some algebraic properties of functions which are preponderantly continuous in O'Malley sense and functions satisfying property A 1 are studied. The maximal additive class, the maximal multiplicative class and the maximal class with respect to minimums for these families of functions are described.

In this paper we present some new inequalities for the normalized Jensen functional which will much improve the famous Jensen’s inequality. We also give some applications in Analysis and Information Theory. Our results refine and generalize the corresponding ones announced in [DRAGOMIR, S. S.: Bounds for the normalised Jensen functional , Bull. Aust. Math. Soc. 74 (2006), 471–478] and [MITANI, K. I. et al.: On sharp triangle inequalities in Banach spaces , J. Math. Anal. Appl. 336 (2007), 1178–1186].

The main object of this present paper is to investigate the problem of majorization of certain class of analytic functions of complex order defined by the Dziok-Raina linear operator. Moreover we point out some new or known consequences of our main result.

In this note, we investigate the Fekete-Szegö problem for a class 𝓢𝒽 of functions f analytic in the open unit disc Δ = {z: |z| < 1} (and which is related to a shell-like curve associated with Fibonacci numbers) satisfying the conditions that f(0)=0,  f′(0)=1  and  zf′(z)f(z)≺1+τ2z21−τz−τ2z2 (z∈Δ),$$f(0) = 0,\qquad f'(0) = 1\qquad {\text{and}}\qquad \frac{{zf'(z)}}{{f(z)}} \prec \frac{{1 + {\tau ^2}{z^2}}}{{1 - \tau z - {\tau ^2}{z^2}}}\quad (z \in \Delta ),$$ where ≺ denotes the subordination and the number τ=(1−5)/2$\tau = (1 - \sqrt 5 )/2$ is such that |τ| fulfils the golden section of the segment [0,1].

In this paper we answer a question of Bednarz and Sokól concerning concave univalent functions. We prove that there exist central functions for the classes Co( p ) of concave univalent functions with pole at the point z = p ∈ (0, 1). Further, we construct a generalized neighborhood of this central function such that the whole class Co(p) is contained in such neighborhood. We also consider similar questions for the class of functions that are analytic and univalent in the unit disc and for some of its important subclasses.

In the paper we define classes of functions associated with conic domains. Some characterizations and inclusion properties of these classes of functions are given.

In the paper, we study the uniqueness of meromorphic functions when certain nonlinear differential polynomials share a nonzero polynomial. The results of the paper improves two recent results due to [LI, X. M.—YI, H. X.: Uniqueness of meromorphic functions whose certain nonlinear differential polynomials share a polynomial , Comput. Math. Appl. 62 (2011), 539–550].

A formula for the solution of a differential-difference equation is given, and then the corresponding semigroup theory is developed, with a detailed description of the infinitesimal generator and some of its spectral properties. Results in Sections 2 and 3 are used in particular to show how the equation can be rewritten as a classical Cauchy problem. This, in Section 4, allows to study its exact null-controllability, by using some techniques from Functional Analysis and Operator Theory. The control u is constrained to lie in a separable weakly compact subset Ω of an arbitrary Banach space U .

In this paper, we consider the oscillation for the second-order quasi-linear neutral dynamic equation (r(t)|ZΔ(t)|α−1ZΔ(t))Δ+q(t)|x(δ(t))|β−1x(δ(t))=0.$${(r(t)|{Z^\Delta }(t{)|^{\alpha - 1}}{Z^\Delta }(t))^\Delta } + q(t)|x(\delta (t{))|^{\beta - 1}}x(\delta (t)) = 0.$$ on time scale 𝕋, where Z ( t ) = x ( t ) + p ( t ) x ( τ ( t )), α , β > 0 are constants. We establish some new oscillation criteria and give sufficient conditions to insure that all solutions of quasi-linear neutral dynamic equation are oscillatory on time scale 𝕋. The new oscillation criteria are presented that improve some known results in the literature.

In this paper, we study the time periodic solution of a sixth order nonlinear parabolic equation, which arises in oil-water-surfactant mixtures based on Leray-Schauder’s fixed point theorem, we prove the existence of time-periodic solutions.

In the present work, we proved a refined direct theorem and an exact inverse theorem of trigonometric approximation in Orlicz spaces with weights satisfying some Muckenhoupt’s A p condition. As a consequence, refined Marchaud inequality and its inverse are obtained.

We obtain Voronovskaya-type theorems for the partial sums of Fourier series using the second order Cesáro method of summation. Then we obtain two versions of Voronovskaya-type theorems for Fejér operators and finally we deduce an integral identity.

Assume that L ψ , L ψ 1 , …, L ψ n are Orlicz spaces. We consider the size of the set Eψ(ψ1,…,ψn)={(f1,…,fn)∈Lψ1×…×Lψn: f1⋯fn∈Lψ}$E_\psi ^{({\psi _1}, \ldots, {\psi _n})} = \{ ({f_1}, \ldots, {f_n}) \in {{\mathbf{L}}^{{\psi _1}}} \times \ldots \times {{\mathbf{L}}^{{\psi _n}}}:\;{f_1} \cdots {f_n} \in {{\mathbf{L}}^\psi }\} $. We show that either it is very small (meager or σ -porous), or it is equal to L ψ 1 × … × L ψ n . This work is a continuation of our previous papers, where we considered a similar problem for L p spaces and Lorentz spaces.

For special universal C *-algebras associated to k -semigraphs we present the universal representations of these algebras, prove a Cuntz–Krieger uniqueness theorem, and compute the K -theory. These C *-algebras seem to be the most universal Cuntz–Krieger like algebras naturally associated to k -semigraphs. For instance, the Toeplitz Cuntz algebra is a proper quotient of such an algebra.

Sets of operators which have a given set of numbers in the numerical range are studied. We prove an interesting result which characterizes the set of operators that have 0 in the convex hull of its spectrum. Along the way we give a simpler proof of the well known Hilbebrandt’s theorem.

We prove that a compact-valued multifunction F : X × Y → Z , where X is a Baire space and Y , Z are separable metrizable spaces, is quasi-continuous if and only if F is horizontally quasi-continuous and there exists an residual subset M of X such that for any x ∈ M the multifunction F x = F ( x , ·) is quasi-continuous on Y .

It is rather complicated to construct the confidence region in nonlinear regression model mainly when number of parameters is large. If the nonlinearity of the model is weak, then it is possible, after some modification, to approximate the confidence region by a confidence ellipsoid in the linearized model. The aim of the paper is to propose a solution in singular models with constraints.

We propose an SEI epidemic model for plant diseases, which incorporates disease latency, disease-caused removal, and constant recruitment in both susceptible and exposed classes. Because of the recruitment and disease-caused removal, the total population is varying. It is shown that the model only has an endemic equilibrium and the equilibrium is globally stable.