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In this paper, we introduce the concept of Totally Irregular Hesitancy Fuzzy Graphs and study the basic properties of totally irregular Hesitancy Fuzzy Graphs. We establish the relationship between the totally irregular, neighbourly totally irregular and highly totally irregular Hesitancy Fuzzy Graph..

We propose a general method for constructing confidence sets and hypothesis tests that have finite-sample guarantees without regularity conditions. We refer to such procedures as “universal.” The method is very simple and is based on a modified version of the usual likelihood-ratio statistic that we call “the split likelihood-ratio test” (split LRT) statistic. The (limiting) null distribution of the classical likelihood-ratio statistic is often intractable when used to test composite null hypotheses in irregular statistical models. Our method is especially appealing for statistical inference in these complex setups. The method we suggest works for any parametric model and also for some nonparametric models, as long as computing a maximum-likelihood estimator (MLE) is feasible under the null. Canonical examples arise in mixture modeling and shape-constrained inference, for which constructing tests and confidence sets has been notoriously difficult. We also develop various extensions of our basic methods. We show that in settings when computing the MLE is hard, for the purpose of constructing valid tests and intervals, it is sufficient to upper bound the maximum likelihood. We investigate some conditions under which our methods yield valid inferences under model misspecification. Further, the split LRT can be used with profile likelihoods to deal with nuisance parameters, and it can also be run sequentially to yield anytime-valid P values and confidence sequences. Finally, when combined with the method of sieves, it can be used to perform model selection with nested model classes.

Fuzzy graph models enjoy the ubiquity of being present in nature and man-made structures, such as the dynamic processes in physical, biological, and social systems. As a result of inconsistent and indeterminate information inherent in real-life problems that are often uncertain, for an expert, it is highly difficult to demonstrate those problems through a fuzzy graph. Resolving the uncertainty associated with the inconsistent and indeterminate information of any real-world problem can be done using a vague graph (VG), with which the fuzzy graphs may not generate satisfactory results. The limitations of past definitions in fuzzy graphs have led us to present new definitions in VGs. The objective of this paper is to present certain types of vague graphs (VGs), including strongly irregular (SI), strongly totally irregular (STI), neighborly edge irregular (NEI), and neighborly edge totally irregular vague graphs (NETIVGs), which are introduced for the first time here. Some remarkable properties associated with these new VGs were investigated, and necessary and sufficient conditions under which strongly irregular vague graphs (SIVGs) and highly irregular vague graphs (HIVGs) are equivalent were obtained. The relation among strongly, highly, and neighborly irregular vague graphs was established. A comparative study between NEI and NETIVGs was performed. Different examples are provided to evaluate the validity of the new definitions. A new definition of energy called the Laplacian energy (LE) is presented, and its calculation is shown with some examples. Likewise, we introduce the notions of the adjacency matrix (AM), degree matrix (DM), and Laplacian matrix (LM) of VGs. The lower and upper bounds for the Laplacian energy of a VG are derived. Furthermore, this study discusses the VG energy concept by providing a real-time example. Finally, an application of the proposed concepts is presented to find the most effective person in a hospital.

The total irregularity of a graph G is defined by irr t ( G ) = 1 2 ∑ ( u , v ) ε V 2 ( G ) | d G ( u ) − d G ( v ) | . irr t ( G ) = 1 2 ∑ ( u , v ) ε V 2 ( G ) | d G ( u ) − d G ( v ) | For an ordered degree sequence of V(G) = {v1,v2, ..., vn} with d(v1) ≤ d(v2) ≤ ... ≤ d(vn), irrt(G) can be expressed in the form irr t ( G ) = ∑ i > j ( d ( v i ) − d ( v j ) ) i r r t ( G ) = ∑ i > j ( d ( v i ) − d ( v j ) ) where dG(x) is the degree of x∈V(G). An edge e∈E(G) is said to be total irregular positive (negative, stable) inner edge if irrt>irrt(G) (irrt,irrt= irrt(G) respectively). Total irregular positive inner edge is denoted by TIPI edge. Similarly we use the notations TINI, TISI suitably. A graph G is called total irregular positive (negative, stable) inner graph if all the edges e∈E(G) are total irregular positive (negative, stable) inner edges; otherwise G is called a total irregular mixed inner graph.Total irregular positive inner graph is denoted by TIPI graph. Similarly we use the notations TINI, TISI suitably. In this paper, we prove that the Complete graph Kmn,m ≠ n and m,n ≥ 2 is a TIPI graph and the Star graph Sn n ≥ 3 is a TINI graph.

We report laboratory experiments of long-crested irregular water surface waves propagating over a shoal. For a sufficiently shallow shoal we find that the surface elevation can have a local maximum of skewness and kurtosis above the shallower part of the shoal close to the edge on the incoming side, and a local minimum of skewness over the downward slope on the lee side of the shoal. We find that the horizontal fluid velocity can have a local maximum and minimum of skewness at the same locations as those for the surface elevation. However, the kurtosis of the horizontal fluid velocity can have a local maximum over the downward slope on the lee side of the shoal, different from the location of the maximum of kurtosis of the surface elevation.

We determine irregular set chromatic number of comb, friendship, K-ary tree, n-sunlet, coconut tree and jelly fish graph.

Aiming at the phenomenon of a large amount of roof subsidence and rib caving in the gob-side roadway under the special “umbrella” coal pillar in isolated panel No. 325 of Yangcun Coal Mine, the stress distribution regularity of the gob-side roadway under a “umbrella” coal pillar was studied in detail by establishing mathematical models, theoretical analysis, and numerical simulations. The results show that the following: (1) The stress distribution regularity of the irregular coal pillar is actually the mechanical structure of irregular gob. (2) The abutment pressure of the gob is always distributed vertically along the edge of the gob. When the edge of the gob intersects diagonally with the gob roadway, the stress distribution of the surrounding rock of the roadway cannot be calculated only by the width of the coal pillar—the angle between the edge of the gob and the roadway should also be considered. (3) The abutment pressure at each cusp position in irregular gob extends in the opposite direction of the bisection cusp line, which may lead to peak stress concentration area in gob roadways with solid coal on both sides, and even roof caving and sidewall deviation disasters.

As an attractive collector medium for hypervelocity particles, combined with outstanding physical properties and suitable compositional characteristics, SiO2 aerogel has been deployed on outer space missions and laser shock-loaded collection experiments. In this paper, impact experiments were conducted to understand the penetration process of irregular grains, irregular Al2O3 grains with two different sizes and speeds (~110 μm@7 km/s, ~251 μm@2.3 km/s) at various density silica aerogels. By classifying the shapes of projectile residues and tracks, the morphology of tracks was analyzed. It was observed that there were several kinds of typical tracks in the penetration of irregular grains, accompanied by residues with the shapes of near-sphere, polyhedron, streamlined body wedge, and rotator. The rotational behavior was demonstrated by the final status of one flake projectile as direct evidence. In addition, there was no obvious relationship between the track length and experimental parameters, which may be caused by the uncertain interaction between aerogels and irregular particles. In addition, it confirmed the existence of fragmentation, melting situation by observing the shape of the impact entrance hole. At the same time, optical coherence tomography was used to observe the detail of tracks clearly, which provided a method to characterize the tracks nondestructively.

In this narrative review, we examined what level of research evidence is available that shift workers' sleep-wake disturbances can be minimized through ergonomie shift scheduling. We classified the pertinent studies conducted on real shift workers in field conditions by the type of shift system and study design (ie, whether the shift systems were modified or not— \"treatment\" versus \"no treatment\"). The results of the observational studies in which no changes to the shift system were made (ie, no treatment) showed that, irrespective of the shift system, night and early-morning shifts and quick returns are associated with short sleep and increases in sleepiness. The same is true for very long shifts (> 16 hours) and extremely long weekly working hours (> 55 hours). For all categories of shift systems, there were a lack of controlled intervention studies, limiting the possibility to provide solution-focused recommendations for shift scheduling. Most of the controlled intervention studies had been conducted on workers under regular 3-shift systems. These studies suggested that a change from slowly backward-rotating shifts to rapidly forward-rotating shifts is advantageous for alertness and, to some degree, sleep. We also found that a change from an 8-to 12-hour shift system does not necessarily result in impairments in the sleep-wake pattern. The level of research evidence was affected by many of the studies' frequent methodological limitations in measuring sleep and sleepiness. In all, to have reliable and solution-focused recommendations for shift scheduling, methodologically sound controlled intervention studies are required in different categories of shift systems.