Abstract This paper introduces the concept of lacunary p -distance convergence for sequences of complex uncertain variables. This generalization allows the study of convergence over irregular index sets under uncertainty, extending existing convergence frameworks in uncertainty theory. Various properties, algebraic operations, and geometric aspects of the related function space are discussed.
This paper provides a survey of several results on alternative functional equations, related to the Cauchy equation and to the quadratic equation, and involving one or more unknown functions.
In this paper, we introduce new inequalities for weighted sums of powers. By utilizing known Fibonacci identities alongside our generalized inequalities, we derive new sequences of inequalities for Fibonacci numbers.
The concept of l-gap convex functions is defined, which is more general than convex functions and allows some non-convex parts of the function. A Jensen type inequality is established and some examples are discussed. As an application and generalization, we prove that the majorization theorem also holds for l-gap convex functions. Then we use the conclusions from the above sections to establish a Hermite-Hadamard type inequality for l-gap convex functions.
This paper addresses the problem of obtaining a strict ranking (i.e., a ranking without equally ranked items) of n items based on a pairwise comparisons matrix. The basic structures are described and a heuristic approach based on a condition, the & Rscr;-condition is proposed. The limitations of this ranking procedure are analyzed.
The purpose of this work is to introduce some new results about the relations between powers, roots and Moore–Penrose inverses of square matrices satisfying a cubic matrix equation and the generalized Fibonacci numbers. The results can also be used for rectangular matrices. Moreover, we give some numerical examples to verify theoretical results.
Let S be a semigroup and K be a field. In a recent article we introduced a new cosine functional equation g ( xyz ) − g ( x ) g ( yz ) − g ( y ) g ( xz ) − g ( z ) g ( xy ) + 2 g ( x ) g ( y ) g ( z ) = 0 for an unknown function g : S → K . It was shown that this equation is closely connected to the sine addition formula, and for K = ℂ its solutions are expressible in terms of multiplicative functions. Here we solve the more general functional equation f ( xyz )+ g ( x ) g ( yz )+ g ( y ) g ( xz )+ g ( z ) g ( xy ) + h ( x ) h ( y ) h ( z ) = 0 for three unknown functions f, g, h : S → ℂ, where S is a monoid. The solutions are linear combinations of two multiplicative functions.
In this paper, we use ( m +4)-convex functions to derive an estimate for Jensen’s inequality in the context of divided differences. In addition, we extend these results for ( h, g ; α − n )-convex functions. Finally, we present some results for g -convex functions, ( h, g )-convex functions and provide a discussion and examples concerning h -convex functions.
We introduce the generalized notion of semicontinuity of a function defined on a topological space and derive the useful classification of the so-called Lipschitz derivatives of functions defined on a metric space. Secondly, we investigate some connections of the Lipschitz derivatives defined on normed spaces to the Fréchet derivative and relations between little, big and local Lipschitz derivatives (denoted by lip f , 𝕃ip f and Lip f respectively) in terms of Baire limit functions. In particular, we prove that lip f is ℱ σ -lower, Lip f is ℱ σ -upper, 𝕃ip f is upper semicontinuous. Moreover, for a function f defined on an open or convex subset of a normed space, the upper Baire limit function of functions lip f and Lip f are equal to 𝕃ip f .
Generalized commutative quaternions generalize elliptic, parabolic and hyperbolic quaternions, bicomplex numbers, complex hyperbolic numbers and hyperbolic complex numbers. In this paper, we use the Mersenne numbers and polynomials in the theory of these quaternions. We introduce and study generalized commutative Mersenne quaternion polynomials and generalized commutative Mersenne–Lucas quaternion polynomials.
In this paper we will try to answer what conditions must be met by the fuzzy Xor to be used for cryptographic purposes. We will also show that defining the fuzzy Xor using other fuzzy connectives is not suitable for this purpose.
We determine the solutions of the conditional Drygas equation for functions f1 and f2 that satisfy (y2 + y)f1(x) = (x2 + x)f2(y) for all (x, y) ∈ ℝ2 under the additional conditions y = x2, or y = log(x), x > 0 or y = exp(x).
In this paper, the bidimensional extensions of the Fibonacci numbers are explored, along with a detailed examination of their properties, characteristics, and some identities. We introduce and study the matrices with bidimensional Fibonacci numbers, focusing in particular on their recurrence relation, key properties, determinant, and various other identities. It is our purpose to study the matrix version of bidimensional Fibonacci numbers and provide new results and sometimes extensions of some results existing in the literature. We aim to introduce these matrices using the bidimensional Fibonacci numbers and to give the determinant of these matrices.
This article examines integral inequalities dealing with functions of the form “a function raised to the power of another function” under varying monotonicity and convexity assumptions. First, we assess the validity of a referenced theorem on the subject. Specifically, we present a counterexample and identify a gap in its proof. We then propose an alternative version of the theorem with more flexible convexity assumptions. In addition, we establish new lower and upper bounds for the same integral using refined Hermite–Hadamard integral inequalities. A complementary variant is also discussed. Thus, our results fill gaps in the literature and extend existing results on integral inequalities under classical assumptions.
In this paper we introduce a new kind of generalized Jacobsthal numbers in a distance sense. We give the identities and matrix representations for them and their connections with the Fibonacci and the Pell numbers. We also describe the interpretations of these numbers in terms of some kind of ( k 1 A 1 , k 2 A 2 , k 3 A 3 )-edge colouring and quasi colouring.
We use the approach from Czudek and Szarek (see [1]) to prove the central limit theorem for a stationary Markov chain generated by an iterative function system for a family of increasing, injective functions on [0, 1] with “contractive” properties. We introduce a new approach to prove existence of an unique invariant measure using e-property (see [2]).
In this work we focus on a dynamical system with jumps, where the intensity of the jumps depends on the system's state. By verifying the assumptions of the theorem from [4], we show that our model satisfies the central limit theorem.
In this paper, by Schauder's fixed point theorem and the Banach contraction principle, we consider the existence, uniqueness, and stability of convex solutions of a nonhomogeneous iterative functional differential equation. Finally, some examples were considered by our results.
Let P be a Markov operator on a general state space (S, Σ) with an invariant probability measure m, assumed to be ergodic. We study conditions which yield that for every centered non-zero f ∈ L2(m) a non-degenerate annealed CLT and an L2-normalized CLT hold.
We generalize a classical result about derivation pairs on function algebras. Specifically, we describe the forms of derivation pairs on rings and rngs (non-unital rings) which are not assumed to be commutative. The proofs are based on knowledge of the solutions of the sine addition formula on a semigroup. Examples are given to illustrate the results.
