Nonstandard Null Lagrangians and Gauge Functions for Newtonian Law of Inertia

New null Lagrangians and gauge functions are derived and they are called nonstandard because their forms are different than those previously found. The invariance of the action is used to make the Lagrangians and gauge functions exact. The first exact nonstandard null Lagrangian and its gauge function for the law of inertia are obtained, and their physical implications are discussed.

Set-Valued SU-Type Fixed Point Theorems via Gauge Function with Applications

In this article, we have designed two existence of fixed point theorems which are regarding to set-valued SU-type θ η -contraction and Γ α -contraction via gauge function in the setting of metric spaces. An extensive set of nontrivial example will be given to justify our claim. At the end, we will give an application to prove the existence behavior for the system of functional equation in dynamical system and integral inclusion.

Higher Order of Convergence with Multivalued Contraction Mappings

In this paper, we establish some theorems of fixed point on multivalued mappings satisfying contraction mapping by using gauge function. Furthermore, we use Q - and R -order of convergence. Our main results extend many previous existing results in the literature. Consequently, to substantiate the validity of proposed method, we give its application in integral inclusion.

Parametric estimation of a gauge function and a measure of technical efficiency

PurposeThis paper proposes the use of a gauge function as a measure of technical efficiency. The measure of technical inefficiency from a gauge function is desirable as the estimation of a gauge function is not subject to the endogeneity problem under the behavioral assumption of profit maximization in the competitive market.Design/methodology/approachThe authors address three important properties of a gauge function, i.e. linear homogeneity, monotonicity and convexity in inputs and outputs, and show how such properties are utilized in its estimation. Then, the authors apply the estimation of a gauge function to US Blacksmiths in 1850 and 1880 to show that a failure to satisfy such properties may lead to an incorrect inference on the technical efficiency.FindingsThe authors find that the Blacksmiths in the 1850s were technically more efficient than the ones in the 1880s, indicating technical regress in Blacksmithing when the properties are satisfied.Originality/valueThis paper introduces a measure of technical inefficiency from a gauge function and shows how to estimate the gauge function parametrically for the measure. The authors show McFadden's gauge function and its properties, which differ from the properties of other distance functions. The authors emphasize linear homogeneity as well as monotonicity and convexity in inputs and outputs, which must be satisfied in the estimation of a gauge function.

Triebel-Lizorkin capacity and hausdorff measure in metric spaces

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.

On multivalued Suzuki-type θ-contractions and related applications

In this study, we develop the concept of multivalued Suzuki-type θ-contractions via a gauge function and established two new related fixed point theorems on metric spaces. We also discuss an example to validate our results.

Behavior of a vacuum and naked singularity under a smooth gauge function in Lyra geometry

Lyra geometry is a conformal geometry that originated from Weyl geometry. In this article, we derive the exterior field equation under a spherically symmetric gauge function x0(r) and metric in Lyra geometry. When we impose a specific form of the gauge function x0(r), the radial differential equation of the metric component g00 will possess an irregular singular point (ISP) at r = 0. Moreover, we can apply the method of dominant balance to get the asymptotic behavior of the new space–time solution. The significance of this work is that we can use a series of smooth gauge functions x0(r) to modulate the degree of divergence of the singularity at r = 0, which will become a naked singularity under certain conditions. Furthermore, we investigate the physical meaning of this novel behavior of space–time in Lyra geometry and find out that no spaceship with finite integrated acceleration can arrive at this singularity at r = 0. The physical meaning of the gauge function and integrability is also discussed.

n-Tupled Coincidence Point Theorems for Probabilistic ψ-Contractions in Menger Spaces

We introduced n-tupled coincidence point for a pair of maps T:Xn→X and A:X→X in Menger space. Utilizing the properties of the pseudometric and the triangular norm, we will establish n-tupled coincidence point theorems under weak compatibility as well as n-tupled fixed point theorems for hybrid probabilistic ψ-contractions with a gauge function. Our main results do not require the conditions of continuity and monotonicity of ψ. At the end of this paper, an example is given to support our main theorem.