Suitable Counterexamples Research Articles

Characterizations of generalized Fuzzy γ∗-closed sets

Generalized fuzzy open sets are playing a vital role in the study of fuzzy topological space as well as that of fuzzy bitopological space since its inception. More often, it is reported that fuzzy closed sets are always included in the family of generalized fuzzy closed sets. Very recently, it has appeared that fuzzy γ∗ -open sets are incomparable with fuzzy open sets. This paper aims to present three different kinds of fuzzy generalized closed sets in the light of fuzzy γ∗ -open set and associated closure operators with the terminologies- generalized fuzzy γ∗ -closed set, γ∗ -generalized fuzzy closed set and γ∗ -generalized fuzzy γ∗-closed set and it is found that the relation between any two concepts is not necessarily linear. Also, the interrelationships among them are established along with suitable counter examples which are properly placed to make the paper self-sufficient.

Isomorphic Copies of ell ^infty in the Weighted Hardy Spaces on the Unit Disc

It is still unclear whether the density of analytic polynomials in an H-admissible space is sufficient to the minimality of the space? This question has a purely foundational background, relating fundamental concepts from the theory of H^p spaces. We hypothesize that there is no general relationship between the density of analytic polynomials and the R-admissibility of an H-admissible space. We solve this problem by finding suitable counterexamples of Hardy spaces built upon some weighted Lebesgue spaces. In particular, we provide a direct construction of weights from Szegő class, which guarantees the existence of isomorphic copies of the space of bounded sequences in weighted Hardy spaces on the unit disc.

Weak well-posedness for a class of degenerate Lévy-driven SDEs with Hölder continuous coefficients

In this article, we study the effects of the propagation of a non-degenerate Lévy noise through a chain of deterministic differential equations whose coefficients are Hölder continuous and satisfy a weak Hörmander-like condition. In particular, we assume some non-degeneracy with respect to the components which transmit the noise. Moreover, we characterize, for some specific dynamics, through suitable counter-examples, the almost sharp regularity exponents that ensure the weak well-posedness for the associated SDE. As a by-product of our approach, we also derive some Krylov-type estimates for the density of the weak solutions of the considered SDE.

A Novel Framework for Generalizations of Soft Open Sets and Its Applications via Soft Topologies

Soft topological spaces (STSs) have received a lot of attention recently, and numerous soft topological ideas have been created from differing viewpoints. Herein, we put forth a new class of generalizations of soft open sets called “weakly soft semi-open subsets” following an approach inspired by the components of a soft set. This approach opens the door to reformulating the existing soft topological concepts and examining their behaviors. First, we deliberate the main structural properties of this class and detect its relationships with the previous generalizations with the assistance of suitable counterexamples. In addition, we probe some features that are obtained under some specific stipulations and elucidate the properties of the forgoing generalizations that are missing in this class. Next, we initiate the interior and closure operators with respect to the classes of weakly soft semi-open and weakly soft semi-closed subsets and look at some of their fundamental characteristics. Ultimately, we pursue the concept of weakly soft semi-continuity and furnish some of its descriptions. By a counterexample, we elaborate that some characterizations of soft continuous functions are invalid for weakly soft semi-continuous functions.

Ulam Stability Results of Functional Equations in Modular Spaces and 2-Banach Spaces

In this work, we investigate the refined stability of the additive, quartic, and quintic functional equations in modular spaces with and without the Δ2-condition using the direct method (Hyers method). We also examine Ulam stability in 2-Banach space using the direct method. Additionally, using a suitable counterexample, we eventually demonstrate that the stability of these equations fails in a certain case.

Stabilities and Non-Stabilities of a New Reciprocal Functional Equation

Abstract The intention of this study is to present some stronger results by investigating Ulam-JRassias product stability and Ulam-JRassias mixed-type sum-product stability of a new reciprocal functional equation. Also, a suitable counter-example is presented to show the failure of stability result for the singular case.

Stability of Quartic Functional Equation in Modular Spaces via Hyers and Fixed-Point Methods

In this work, we introduce a new type of generalised quartic functional equation and obtain the general solution. We then investigate the stability results by using the Hyers method in modular space for quartic functional equations without using the Fatou property, without using the Δb-condition and without using both the Δb-condition and the Fatou property. Moreover, we investigate the stability results for this functional equation with the help of a fixed-point technique involving the idea of the Fatou property in modular spaces. Furthermore, a suitable counter example is also demonstrated to prove the non-stability of a singular case.

Relation between electrical resistance and conductance using multifarious functional equations and applications to parallel circuit

In this paper, we introduce a new generalized p-dimensional multifarious radical reciprocal functional equation combining three classical means: arithmetic, geometric, and harmonic. Mainly, we find its general solution and stability related to the Ulam problem in modular spaces by using the fixed point method with suitable counterexamples. Importantly, in this paper, we illustrate the geometrical interpretation and applications of the introduced Pythagorean means multifarious functional equation in connection with the parallel circuit. Furthermore, we provide a formula for finding the equivalent resistance R_{mathrm{eq}} of parallel electrical circuit using functional equations, which relates the electrical resistances and conductances with suitable examples.

Optimal derivative loss for abstract wave equations

We consider an abstract wave equation with a propagation speed that depends only on time. We assume that the propagation speed is differentiable for positive times, continuous up to the origin, but with first derivative that is potentially singular at the origin. We examine the derivative loss of solutions, and in particular we investigate which conditions on the modulus of continuity and on the behavior of the derivative in the origin yield, respectively, no derivative loss, an arbitrarily small derivative loss, a finite derivative loss, or an infinite derivative loss. As expected, we obtain that stronger assumptions on the modulus of continuity can compensate weaker assumptions on the growth of the derivative, and viceversa. Suitable counterexamples show that our results are sharp. We prove indeed that, for every set of conditions, the class of propagation speeds that satisfy the given conditions, and for which the corresponding equation exhibits a derivative loss as large as possible, is nonempty and actually also residual in the sense of Baire category.

Sufficient conditions for extremum of fractional variational problems

Sufficient conditions for extremum of fractional variational problems are formulated with the help of Caputo fractional derivatives. The Euler–Lagrange equation is defined in the Caputo sense and Jacobi conditions are derived using this. Again, Wierstrass integral for the considered functional is obtained from the Jacobi conditions and the transversality conditions. Further, using the Taylor’s series expansion with Caputo fractional derivatives in the Wierstrass integral, the Legendre’s sufficient condition for extremum of the fractional variational problem is established. Finally, a suitable counterexample is presented to justify the efficacy of the fresh findings.

New Stability Results of Multiplicative Inverse Quartic Functional Equations

The purpose of this investigation is to introduce different forms of multiplicative inverse functional equations, to solve them and to establish the stability results of them in the framework of matrix normed spaces. A suitable counter-example is also provided to prove the instability of the results for a singular case. The applications of the equations dealt in this study are discussed with the fluid resistances of blood vessels and also an important concept in Raman spectroscopy.
 2010 Mathematics Subject Classification. 39B82, 39B72.

Approximation of Mixed Euler-Lagrange σ -Cubic-Quartic Functional Equation in Felbin’s Type f-NLS

In this research paper, the authors present a new mixed Euler-Lagrange σ -cubic-quartic functional equation. For this introduced mixed type functional equation, the authors obtain general solution and investigate the various stabilities related to the Ulam problem in Felbin’s type of fuzzy normed linear space (f-NLS) with suitable counterexamples. This approach leads us to approximate the Euler-Lagrange σ -cubic-quartic functional equation with better estimation.

Stabilities of various multiplicative inverse functional equations

This study deals with the validation of classical stabilities of various multiplicative inverse functional equations pertinent to Ulam stability theory of functional equations in the setting of non-Archimedean fields. Suitable counter-examples are demonstrated to disprove the invalidity of stability results of these equations for singular cases.

Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result

We investigate the effects of the propagation of a non-degenerate Brownian noise through a chain of deterministic differential equations whose coefficients are rough and satisfy a weak like H{\"o}rmander structure (i.e. a non-degeneracy condition w.r.t. the components which transmit the noise). In particular we characterize, through suitable counterexamples , almost sharp regularity exponents that ensure that weak well posedness holds for the associated SDE. As a by-product of our approach, we also derive some density estimates of Krylov type for the weak solutions of the considered SDEs.

Study of the Modules over Rings and Their Extensions

We study the problem of transfer of the properties of some types of modules under identity preserving ring homomorphisms. These studies seem to be overlooked in the literature. We have chosen several types of modules and provided proofs for these properties that are transferable and suitable counterexamples for the properties that are not transferable.

On the Baker-Campbell-Hausdorff Theorem: non-convergence and prolongation issues

ABSTRACT We investigate some topics related to the celebrated Baker-Campbell-Hausdorff Theorem: a non-convergence result and prolongation problems. Given a Banach algebra with identity I, and given , we study the relationship of different issues: the convergence of the BCH series , the existence of a logarithm of , and the convergence of the Mercator-type series which provides a selected logarithm of . We fix general results, among which we provide a non-convergence result for the BCH series, and (by suitable matrix counterexamples) we show that various pathologies can occur. These are related to some recent results, of interest in physics, on closed formulas for the BCH series: while the sum of the BCH series presents several non-convergence issues, these closed formulas can provide a prolongation for the BCH series when it is not convergent. On the other hand, we show by suitable counterexamples that an analytic prolongation of the BCH series can be singular even if the BCH series itself is convergent.

Back-and-forth systems for fuzzy first-order models

This paper continues the study of model theory for fuzzy logics by addressing the fundamental issue of classifying models according to their first-order theory. Three different definitions of elementary equivalence for fuzzy first-order models are introduced and separated by suitable counterexamples. We propose several back-and-forth conditions, based both on classical two-sorted structures and on non-classical structures, that are useful to obtain elementary equivalence in particular cases as we illustrate with several examples.

Incompressible inelasticity as an essential ingredient for the validity of the kinematic decomposition [formula omitted

The multiplicative decomposition of the total deformation F=FeFi between an elastic (Fe) and an inelastic component (Fi) is standard in the modeling of many irreversible processes such as plasticity, growth, thermoelasticity, viscoelasticty or phase transformations. The heuristic argument for such kinematic assumption is based on the chain rule for the compatible scenario (CurlFi=0) where the individual deformation tensors are gradients of deformation mappings, i.e. F=Dφ=D(φe∘φi)=(Dφe)∘φi(Dφi)=FeFi. Yet, the conditions for its validity in the general incompatible case (CurlFi≠0) has so far remained uncertain. We show in this paper that detFi=1 and CurlFi bounded are necessary and sufficient conditions for the validity of F=FeFi for a wide range of inelastic processes. In particular, in the context of crystal plasticity, we demonstrate via rigorous homogenization from discrete dislocations to the continuum level in two dimensions, that the volume preserving property of the mechanistics of dislocation glide, combined with a finite dislocation density, is sufficient to deliver F=FeFp at the continuum scale. We then generalize this result to general two-dimensional inelastic processes that may be described at a lower dimensional scale via a multiplicative decomposition while exhibiting a finite density of incompatibilities. The necessity of the conditions detFi=1 and CurlFi bounded for such systems is demonstrated via suitable counterexamples.

Detour eccentric and detour distance degree sequences in graphs

In this paper we define the detour eccentric sequences and detour distance degree sequences of graphs. Some of the properties of these sequences are analyzed. It is observed that certain of these properties behave differently to their counterpart if defined using detour distance and these facts are established by providing suitable counterexamples. The concept of detour distance distributions in graphs is also discussed.

On the role of gradient terms in coercive quasilinear differential inequalities on Carnot groups

In the sub-Riemannian setting of Carnot groups, this work investigates a priori estimates and Liouville type theorems for solutions of coercive, quasilinear differential inequalities of the type ΔGφu≥b(x)f(u)l(|∇u|). Prototype examples of ΔGφ are the (subelliptic) p-Laplacian and the mean curvature operator. The main novelty of the present paper is that we allow a dependence on the gradient l(t) that can vanish both as t→0+ and as t→+∞. Our results improve on the recent literature and, by means of suitable counterexamples, we show that the range of parameters in the main theorems are sharp.