Band 70, Heft 3

Given a finite graph H , the n th member G n of an H - linear sequence is obtained recursively by attaching a disjoint copy of H to the last copy of H in G n −1 by adding edges or identifying vertices, always in the same way. The genus polynomial Γ G ( z ) of a graph G is the generating function enumerating all orientable embeddings of G by genus. Over the past 30 years, most calculations of genus polynomials Γ G n ( z ) for the graphs in a linear family have been obtained by partitioning the embeddings of G n into types 1, 2, …, k with polynomials Γ Gnj$\begin{array}{} \Gamma_{G_n}^j \end{array}$ ( z ), for j = 1, 2, …, k ; from these polynomials, we form a column vector Vn(z)=[Γ Gn1(z),Γ Gn2(z),… ,Γ Gnk(z)]t$\begin{array}{} V_n(z) = [\Gamma_{G_n}^1(z), \Gamma_{G_n}^2(z), \ldots, \Gamma_{G_n}^k(z)]^t \end{array}$ that satisfies a recursion V n ( z ) = M ( z ) V n −1 ( z ), where M ( z ) is a k × k matrix of polynomials in z . In this paper, the Cayley-Hamilton theorem is used to derive a k th degree linear recursion for Γ n ( z ), allowing us to avoid the partitioning, thereby yielding a reduction from k 2 multiplications of polynomials to k such multiplications. Moreover, that linear recursion can facilitate proofs of real-rootedness and log-concavity of the polynomials. We illustrate with examples.

Inspired by the open problems “How to define the notions of fantastic filters and states in EQ-algebras” in [LIU, L. Z.—ZHANG, X. Y.: Implicative and positive implicative prefilters of EQ-algebras , J. Intell. Fuzzy Syst. 26 (2014), 2087–2097], we introduce the notions of fantastic filters and investigate the existence of Bosbach states and Riečan states on EQ-algebras by use of fantastic filters. Firstly, we prove that a residuated EQ-algebra has a Bosbach state if and only if it has a fantastic filter. We also establish that a good EQ-algebra has a state-morphism if and only if it has a prime fantastic filter. Furthermore, we introduce the notion of QI-EQ-algebras and obtain the necessary and sufficient condition for a residuated QI-EQ-algebra having Riečan states. Finally, we introduce the notion of semi-divisible EQ-algebras and give an example of a semi-divisible residuated EQ-algebra, which is not a semi-divisible residuated lattice. We also prove that every semi-divisible residuated EQ-algebra admits Riečan states. These works generalize a series of existing results about existence of states in several algebras, such as residuated lattices, NM-algebras, MTL-algebras, BL-algebras and so on.

In this paper, I will construct three families of Sidon sequences of certain subsets of ℝ, in particular I will study Farey sequences, square roots, and reciprocals. It will be shown that Sidon sequences over them have cardinality of between c1N3/4logN$\begin{array}{} \displaystyle c_1\frac{N^{3/4}} {\log{N}} \end{array}$ and c 2 N 3/4 , c 3 N , and c4NloglogNlogN.$\begin{array}{} \displaystyle c_4 \frac{N\log{\log{N}}}{\log{N}}. \end{array}$

Let { B n } n ≥0 be the sequence of Balancing numbers defined by B 0 = 0, B 1 = 1, and B n +2 = 6 B n +1 − B n for all n ≥ 0. In this paper, we find all repdigits in base 10 which can be written as a sum of three Balancing numbers.

We denote the local “little” and “big” Lipschitz functions of a function f : ℝ → ℝ by lip f and Lip f . In this paper we continue our research concerning the following question. Given a set E ⊂ ℝ is it possible to find a continuous function f such that lip f = 1 E or Lip f = 1 E ? In giving some partial answers to this question uniform density type (UDT) and strong uniform density type (SUDT) sets play an important role. In this paper we show that modulo sets of zero Lebesgue measure any measurable set coincides with a Lip 1 set. On the other hand, we prove that there exists a measurable SUDT set E such that for any G δ set E͠ satisfying ∣ E Δ E͠ ∣ = 0 the set E͠ does not have UDT. Combining these two results we obtain that there exist Lip 1 sets not having UDT, that is, the converse of one of our earlier results does not hold.

In this paper, we shall establish the co-ordinated convex function. It can connect to the right-hand side of Fejér inequality in two variables and thus a new refinement can be found. In addition, some applications to estimates for Euler’s Beta function are also given in the end.

In this present investigation, we will concern with the family of normalized analytic error function which is defined by Erf(z)=π z2erf(z)=z+∑ n=2∞ (− 1)n− 1(2n− 1)(n− 1)!zn.$$\begin{array}{} \displaystyle E_{r}f(z)=\frac{\sqrt{\pi z}}{2}\text{er} f(\sqrt{z})=z+\overset{\infty }{\underset {n=2}{\sum }}\frac{(-1)^{n-1}}{(2n-1)(n-1)!}z^{n}. \end{array}$$ By making the use of the trigonometric polynomials U n ( p , q , e i θ ) as well as the rule of subordination, we introduce several new classes that consist of 𝔮-starlike and 𝔮-convex error functions. Afterwards, we derive some coefficient inequalities for functions in these classes.

We consider a family of analytic and normalized functions that are related to the domains ℍ( s ), with a right branch of a hyperbolas H ( s ) as a boundary. The hyperbola H ( s ) is given by the relation 1ρ =2cos⁡ φ ss(0< s≤ 1,|φ |< (π s)/2).$\begin{array}{} \frac{1}{\rho}=\left( 2\cos\frac{\varphi}{s}\right)^s\quad (0 \lt s\le 1,\, |\varphi| \lt (\pi s)/2). \end{array}$ We mainly study a coefficient problem of the families of functions for which zf ′/ f or 1 + zf ″/ f ′ map the unit disk onto a subset of ℍ( s ) . We find coefficients bounds, solve Fekete-Szegö problem and estimate the Hankel determinant.

We provide a upper bound for Triebel-Lizorkin capacity in metric settings in terms of Hausdorff measure. On the other hand, we also prove that the sets with zero capacity have generalized Hausdorff h -measure zero for a suitable gauge function h .

In this paper, we investigate a class of semilinear fractional differential equations with non-instantaneous impulses and integral boundary value conditions. By the method of upper and lower solutions combined with Amann three-solution theorem, existence results of at least three solutions are obtained.

In this paper we give some theoretical explanations related to the representation for the general solution of the system of the higher-order rational difference equations xn+1=1+2yn− k3+yn− k,yn+1=1+2zn− k3+zn− k,zn+1=1+2xn− k3+xn− k,$$\begin{array}{} \displaystyle x_{n+1} = \dfrac{1+2y_{n-k}}{3+y_{n-k}},\qquad y_{n+1} = \dfrac{1+2z_{n-k}}{3+z_{n-k}},\qquad z_{n+1} = \dfrac{1+2x_{n-k}}{3+x_{n-k}}, \end{array}$$ where n , k ∈ ℕ 0 , the initial values x − k , x − k +1 , …, x 0 , y − k , y − k +1 , …, y 0 , z − k , z − k +1 , …, z 1 and z 0 are arbitrary real numbers do not equal −3. This system can be solved in a closed-form and we will see that the solutions are expressed using the famous Fibonacci and Lucas numbers.

Let x = {xn}n=1∞ $\begin{array}{} \displaystyle \{x_n\}_{n=1}^{\infty} \end{array}$ be a sequence of positive numbers, and 𝓙 x be the collection of all subsets A ⊆ ℕ such that ∑ k∈ A$\begin{array}{} \displaystyle \sum_{k\in A} \end{array}$ x k < +∞. The aim of this article is to study how large the summable subsequence could be. We define the upper density of summable subsequences of x as the supremum of the upper asymptotic densities over 𝓙 x , SUD in brief, and we denote it by D * ( x ). Similarly, the lower density of summable subsequences of x is defined as the supremum of the lower asymptotic densities over 𝓙 x , SLD in brief, and we denote it by D * ( x ). We study the properties of SUD and SLD, and also give some examples. One of our main results is that the SUD of a non-increasing sequence of positive numbers tending to zero is either 0 or 1. Furthermore, we obtain that for a non-increasing sequence, D * ( x ) = 1 if and only if lim infk→ ∞ nxn=0,$\begin{array}{} \displaystyle \liminf_{k\to\infty}nx_n=0, \end{array}$ which is an analogue of Cauchy condensation test. In particular, we prove that the SUD of the sequence of the reciprocals of all prime numbers is 1 and its SLD is 0. Moreover, we apply the results in this topic to improve some results for distributionally chaotic linear operators.

In 2018, Das et al. [ Characterization of rough weighted statistical statistical limit set , Math. Slovaca 68 (4) (2018), 881–896] (or, Ghosal et al. [ Effects on rough 𝓘- lacunary statistical convergence to induce the weighted sequence , Filomat 32 (10) (2018), 3557–3568]) established the result: The diameter of rough weighted statistical limit set (or, rough weighted 𝓘-lacunary limit set) of a sequence x = { x n } n ∈ℕ is 2rlim infn∈ Atn$\begin{array}{} \frac{2r}{{\liminf\limits_{n\in A}} t_n} \end{array}$ if the weighted sequence { t n } n ∈ℕ is statistically bounded (or, self weighted 𝓘-lacunary statistically bounded), where A = { k ∈ ℕ : t k < M } and M is a positive real number such that natural density (or, self weighted 𝓘-lacunary density) of A is 1 respectively. Generally this set has no smaller bound other than 2rlim infn∈ Atn$\begin{array}{} \frac{2r}{{\liminf\limits_{n\in A}} t_n} \end{array}$. We concentrate on investigation that whether in a θ -metric space above mentioned result is satisfied for rough weighted 𝓘-limit set or not? Answer is no. In this paper we establish infinite as well as unbounded θ -metric space (which has not been done so far) by utilizing some non-trivial examples. In addition we introduce and investigate some problems concerning the sets of rough weighted 𝓘-limit points and weighted 𝓘-cluster points in θ -metric space and formalize how these sets could deviate from the existing basic results.

In this note, we obtain a Tauberian theorem for a class of regular lower triangular matrices operating on cosine series with coefficients tending to zero. As corollaries we obtain Tauberian theorems for weighted mean, Nörlund, and Hausdorff matrices.

In this paper, we generalize the concepts of convexity, strict convexity and uniform convexity in the sense of non-Newtonian calculus. The main aim of this study is to obtain the non-Newtonian convexity, non-Newtonian strict convexity and non-Newtonian uniform convexity properties of the non-Newtonian sequence spaces l p ( N ).

In this article we introduce Tribonacci sequence spaces ℓ p ( T ) (1 ≤ p ≤ ∞) derived by the domain of a newly defined regular Tribonacci matrix. We give some topological properties, inclusion relation, obtain the Schauder basis and determine the α -, β - and γ -duals of the new spaces. We characterize the matrix classes on ℓ p ( T ). Finally, we give some geometric properties of the space ℓ p ( T ).

Let X and Y be compact Hausdorff spaces, E be a real or complex Banach space and F be a real or complex locally convex topological vector space. In this paper we study a pair of linear operators S , T : A ( X , E ) → C ( Y , F ) from a subspace A ( X , E ) of C ( X , E ) to C ( Y , F ), which are jointly separating, in the sense that Tf and Sg have disjoint cozeros whenever f and g have disjoint cozeros. We characterize the general form of such maps between certain classes of vector-valued (as well as scalar-valued) spaces of continuous functions including spaces of vector-valued Lipschitz functions, absolutely continuous functions and continuously differentiable functions. The results can be applied to a pair T : A ( X ) → C ( Y ) and S : A ( X , E ) → C ( Y , F ) of linear operators, where A ( X ) is a regular Banach function algebra on X , such that f ⋅ g = 0 implies Tf ⋅ Sg = 0, for all f ∈ A ( X ) and g ∈ A ( X , E ). If T and S are jointly separating bijections between Banach algebras of scalar-valued functions of this class, then they induce a homeomorphism between X and Y and, furthermore, T −1 and S −1 are also jointly separating maps.

In this paper, we discuss some topological properties of graphical metric spaces and introduce the G -set metric with respect to a graphical metric. Some fixed point results are introduced which generalize the famous Nadler’s fixed point theorem.

Let Hn+$\begin{array}{} \displaystyle H_{n}^{+} \end{array}$(ℝ) be the cone of all positive semidefinite (symmetric) n × n real matrices. Matrices from Hn+$\begin{array}{} \displaystyle H_{n}^{+} \end{array}$(ℝ) play an important role in many areas of engineering, applied mathematics, and statistics, e.g. every variance-covariance matrix is known to be positive semidefinite and every real positive semidefinite matrix is a variance-covariance matrix of some multivariate distribution. Three of the best known partial orders that were mostly studied on various sets of matrices are the Löwner, the minus, and the star partial orders. Motivated by applications in statistics authors have recently investigated the form of maps on Hn+$\begin{array}{} \displaystyle H_{n}^{+} \end{array}$(ℝ) that preserve either the Löwner or the minus partial order in both directions. In this paper we continue with the study of preservers of partial orders on Hn+$\begin{array}{} \displaystyle H_{n}^{+} \end{array}$(ℝ). We characterize surjective, additive maps on Hn+$\begin{array}{} \displaystyle H_{n}^{+} \end{array}$(ℝ), n ≥ 3, that preserve the star partial order in both directions. We also investigate the form of surjective maps on the set of all symmetric real n × n matrices that preserve the Löwner partial order in both directions.

For a closed set A ⊂ ℝ n a representation theorem for locally defined operators maping the space C k , ω ( A ) consisting of all k -times continuously differentiable functions on A whose k -th derivatives have modulus of continuity ω into C 0, ω ( A ) is presented.

Let 𝓖 D (𝓗) denote the generalized effect algebra consisting of all positive linear operators defined on a dense linear subspace D of a Hilbert space 𝓗. The D -weak operator topology (introduced by other authors) on 𝓖 D (𝓗) is investigated. The corresponding closure of the set of bounded elements of 𝓖 D (𝓗) is the whole 𝓖 D (𝓗). The closure of the set of all unbounded elements of 𝓖 D (𝓗) is also the set 𝓖 D (𝓗). If Q is arbitrary unbounded element of 𝓖 D (𝓗), it determines an interval in 𝓖 D (𝓗), consisting of all operators between 0 and Q (with the usual ordering of operators). If we take the set of all bounded elements of this interval, the closure of this set (in the D -weak operator topology) is just the original interval. Similarly, the corresponding closure of the set of all unbounded elements of the interval will again be the considered interval.

Suggested is a non-parametric method and algorithm for estimating the probability distribution of a stochastic sum of independent identically distributed continuous random variables, based on combining and numerically inverting the associated empirical characteristic function (CF) derived from the observed data. This is motivated by classical problems in financial risk management, actuarial science, and hydrological modelling. This approach can be naturally generalized to more complex semi-parametric modelling and estimating approaches, e.g., by incorporating the generalized Pareto distribution fit for modelling heavy tails of the considered continuous random variables, or by considering the weighted mixture of the parametric CFs (used to incorporate the expert knowledge) and the empirical CFs (used to incorporate the knowledge based on the observed or historical data). The suggested numerical approach is based on combination of the Gil-Pelaez inversion formulae for deriving the probability distribution (PDF and CDF) from the associated CF and the trapezoidal quadrature rule used for the required numerical integration. The presented non-parametric estimation method is related to the bootstrap estimation approach, and thus, it shares similar properties. Applicability of the proposed estimation procedure is illustrated by estimating the aggregate loss distribution from the well-known Danish fire losses data.

This note presents a short proof of an internal characterization of complete regularity.