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## Abstract

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⁡φss(0 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.

## Abstract

Let x = ${xn}n=1∞$ be a sequence of positive numbers, and 𝓙x be the collection of all subsets A ⊆ ℕ such that $∑k∈A$ xk < +∞. 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 𝓙x, 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 𝓙x, 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 $lim infk→∞nxn=0,$ 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.

## Abstract

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.

## Abstract

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.

## Abstract

In this present investigation, we will concern with the family of normalized analytic error function which is defined by

$Erf(z)=πz2erf(z)=z+∑n=2∞(−1)n−1(2n−1)(n−1)!zn.$

By making the use of the trigonometric polynomials Un(p, q, eiθ) 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.

## Abstract

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

## Abstract

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 fg = 0 implies TfSg = 0, for all fA(X) and gA(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.

## Abstract

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 Gδ set satisfying ∣EΔ∣ = 0 the set 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.

## Abstract

For a closed set A ⊂ ℝn 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.