In this article we compute many topological indices for the family of Möbius ladder. At first we give general closed form of M-polynomial of this family and recover many degree based topological indices out of it. We also compute Zagreb indices and Zagreb polynomials of this family.

Falls at height are among the major causes of fatal accidents at work, especially in the construction sector. Maintaining balance while performing a tool with both hands generates physical efforts and mental loading that needs to be managed for workers’ safety and health. The purpose of the study is to compare perceived (dis)comfort (overall and localized), the ergonomic risk and the perceived effort through the acquired postures that subjects assumed during the execution of simple and common tasks both on a ladder and on the ground. 26 subjects were asked to perform four tasks in both conditions: top screwing, frontal screwing, object picking from the bottom forward, and object picking from the bottom sideways. Subjective and objective data were gathered for assessing subjective experience (discomfort and effort, perceived) and ergonomic risk. Results highlighted few ergonomic risk differences among “on ladder” and “on ground” tasks, while subjective data reveal significant differences in different scenarios. Main results regard knees and neck areas as the most critical ones.

Textile heaters are made of knitted conductive yarns integrated into the fabric, making them stretchable, washable, breathable, and suitable for close-to-skin wear. The non-zero resistance in the lead wires, however, causes non-uniform power distribution, presenting a design challenge. To address this, the electrical performance of the heaters is modelled as an n-ladder resistor network. Using the finite difference method, simple, closed-form expressions are derived for networks with the power source connected to input terminals A1B1 and A1Bn, respectively. The results are exact and are used to derive approximations and design criteria. Solutions for ladder networks as presented in this paper apply to a wider class of physical problems, such as irrigation systems, transformer windings, and cooling fins.

The spin-1/2 quantum transverse Ising model, defined on a ladder structure, with nearest-neighbor and four-spin interaction on a plaquette, is studied by using exact diagonalization and finite-size-scaling procedures. The quantum phase transition between the ferromagnetic and paramagnetic phases has been obtained in the thermodynamic limit. The critical transverse field decreases as the antiferromagnetic four-spin interaction increases, and reaches a multicritical point. However, the exact diagonalization approach was not able to capture the essence of the dimer phase beyond the multicritical transition.

A signed graph is an undirected graph such that all of its edges are labeled positive or negative. Let D be an embedding of a signed graph G in the Euclidean line. If in D for all of the vertices their positive neighbors are closer than the negative neighbors, then D is called a valid drawing of G. In this paper we show that signed trees, unicyclic graphs, and ladder graphs have a valid drawing in the Euclidean line.

This research delved into exploring the impact of ladder training on improving the athletic performance of adolescent students, particularly focusing on boosting explosive power and speed. The study involved thirty boys aged 15-17, selected randomly from an Indian school in Doha, Qatar. The variables measured were Explosive Power (EP) and speed (S). EP was assessed using the Standing Broad Jump (SBJ), while Speed (S) was gauged through the 50-yard dash. Divided into two groups - the Experimental Group (EG) and the Control Group (CG), each consisting of fifteen students - the EG underwent dedicated ladder training for 12 weeks while the CG continued their routine. Analysis of pre-test and post-test results indicated significant enhancements in EP and speed within the EG, with notable shifts from 2.15±.27 to 2.40±.24and 7.52±.54to 6.97±.66, respectively, exhibiting significant 't' values (EP: -2.39; speed: 2.23) at P˂.05. Conversely, the CG showed minimal changes in EP (pre: 2.22±.22; post: 2.2±.21) and speed (pre: 7.51±.43; post: 7.58±.72), with 't' values of 0.295 and 0.842, respectively, failing to reach significance at P˂.05. These outcomes highlight the efficacy of ladder training in improving explosive power and speed among adolescent students, suggesting its potential as a safe and advantageous component in athletic training programs.

With the rapid development of China's economy, construction industrialization has become the main development trend of China's construction industry, and the convenient, efficient, environmental protection has become the new fashion, therefore the application of prefabricated building is more and more common. Stair serves as the weak link in prefab type building, the connecting node has an important influence on its failure. In this paper, pseudo-static tests are carried out on the joints in different forms of ladder holes to study the mechanical properties of flexible hinge points filled with flexible materials, and the hysteresis curve, skeleton curve and bearing capacity of each specimen are analyzed. The test results show that the flexible hinge can give full play to its flexible characteristics to ease the stress of ladder beam, pin bolt reinforcement and ladder hole, and the bearing capacity of flexible hinge points is moderate, which can not only mitigate the inclined bracing effect under the action of small and medium earthquakes, but also ensure the stair does not fall off under the action of large earthquakes. Through the experiment comparison, the ladder hole adopts node form 2, which can give full play to the advantage of flexible material.

Abstract: Background: Cow’s milk allergy (CMA) is one of the most common and complex food allergies affecting children worldwide and with few exceptions, presents in the first few months of life. Baked milk containing diets are well tolerated in the majority of milk allergic children and allow dietary restrictions to be relaxed. In addition, the early introduction of tolerated forms of allergenic foods to the infant’s diet in small amounts may enhance the outgrowth of milk allergy through oral tolerance induction. The methods of milk introduction vary widely across the globe. Methods: We convened an expert group to develop a comprehensive milk ladder based on calculated milk protein content using Indian foods. To validate the milk ladder, foods chosen for the ladder were analyzed and the ladder was re-evaluated based on amount cooked milk protein content. Results: Combining expert consensus and validation of milk protein content, we created the world’s first milk ladder containing Indian foods. This is the first ladder that provides information on timing and temperature of cooking, with validated milk protein content. Conclusions: This is the first milk ladder based on unique features of Indian food habits built by consensus of Indian experts along with international collaboration with quantified of milk protein in each step. We believe the “The Indian Milk Ladder” will be a very helpful tool for pediatricians helping manage CMA in children as well as their parents and caregivers, not only in India but in countries world-wide where these foods are commonly consumed.

Pain is frequently reported during cancer disease, and still remains poorly controlled in 40% of patients. Recent developments in oncology have helped to better control pain. Targeted treatments may cure cancer disease and significantly increase survival. Thereby, a novel population of patients (cancer survivors) has emerged, also enduring chronic pain (27.6% moderate to severe pain). The present review discuss the different options currently available to manage pain in (former) cancer patients in the light of progress made in the last decade. Major progress in the field are recent development of a chronic cancer pain taxonomy now included in International Classification of Diseases (ICD-11) and update of WHO analgesic ladder. Until recently, cancer pain management has mostly relied on pharmacotherapy, opioids being considered as mainstay. The opioids crisis has prompted the reassessment of opioids use, both in cancer patients and cancer survivors. The review focuses on the current utilization of opioids, on the neuropathic pain component often neglected and on techniques and non-pharmacological strategies available which help to personalize patient’s treatment. Cancer pain management is now closer to the management of chronic non-cancer pain i.e. “an integrative pain care” aiming to improve patient’s quality of life.

The expansion of the universe yields two consistent differing values of a Hubble constant depending on the methods of measurement. The aim of this work was to explain the Hubble tension by properties of space corresponding to a radially inhomogeneous metrics. A cosmological model described by this perturbed FLRW metric has zero pressure and exponentially expanding space. But the observed space appears to be compressed in the radial direction. We consider the dependence of the change rate of photon energy on the direction of its motion. It is found applying the principle of the photon's energy integral extremum and the relationship between it and the energy of a material particle obtained using Lagrange mechanics. The change rate of a photon's energy varies depending on whether it moves in a radial direction or has an angular component. The metric coefficients are determined by the difference in measurements of the Hubble constant using gravitational lensing and a distance ladder.

Structure and reversibility of cross-link junctions play pivotal roles in deter- mining the nature of thermoreversible gelation and dynamic mechanical prop- erties of the produced polymer networks. We attempt to explore new types of sol–gel transitions with mechanical sharpness by allowing cross-links to grow without upper bound. We consider thermoreversible gelation of the primary molecules R{Af} carrying the number f of low molecular weight functional groups (gelators) A. Gelators A are assumed to form supramolecular assemblies. Some examples are: telechelic polymers (f = 2) carrying π–π stacking benzene derivatives at their both ends, trifunctional star molecules (f = 3) bearing mul- tiple hydrogen-bonding gelators. The sol–gel transition of the primary molecules becomes sharper with the cooperativity parameter of the stepwise linear growth of the cross-links. There is a polymerization transition (crossover without singu- larity) of the junctions in the postgel region after the gel point is passed. If the gelator A tends to form supramolecular rings competitively with linear chains, there is another phase transition in the deep postgel region where the average molecular weight of the rings becomes infinite (Bose-Einstein condensation of rings). As a typical example of binary cross-links where gelators A and B form mixed junctions, we specifically consider metal-coordinated binding of ligands A by metal ions B. Two types of multi-nuclear supramolecular complexes are studied: (i) linear stacking (ladder) of the sandwich A2B units, (ii) linear train of egg-box A4B units. The average molecular weight, the gel fraction, the av- erage length of the cross-link junctions are calculated for all of these models as functions of the functionality f, the concentration of the solute molecules, and temperature. Potential candidates for the realization of these new types of thermoreversible gelation are discussed.

The rigged Hilbert spaces (RHS) are the right mathematical context which includes many tools used in quantum physics, or even in some chaotic classical systems. It is particularly interesting that in RHS coexist discrete and continuous basis, abstract basis along basis of special functions and representations of Lie algebras of symmetries by continuous operators. This is not possible in Hilbert spaces. In the present paper, we study a model showing all these features, based on the one dimensional P\"oschl-Teller Hamiltonian. Also, RHS support representations of all kinds of ladder operators as continuous mappings. We give an interesting example based on one dimensional Hamiltonians with an infinite chain of SUSY partners, in which the factorization of Hamiltonians by continuous operators on RHS plays a crucial role.

The ocean of the planet Earth, which is as thin as a cicada's wings, play a role of catalyst to the evolving of the ancient plates. The ocean and the ancient plates of the planet Earth formed a magic Whirlpool of the Cross “十” shape with double whirl arms. The central core of the Cross “十” shape Ocean-Plate Whirlpool is the Hint Black Hole (HBH) located at the Northwestern Pacific Ocean floor which is decorated with invisible geological age distribution of clam shell. The double whirl arms of ancient plates are distinguished to be left and right at Bering Strait and just formed the ring of fire of the Pacific Ocean.The Cross “十” shape Ocean-Plate Whirlpool of the planet Earth was born at the equatorial area about 66 Ma ago and is rotating anticlockwise and drifting northward by a mini footprint of spiral at s speed of as slow as a snail moves. Obviously, it would come to an end at the North Pole about 100 Ma in the future. So, the lifespan of the Cross “十” shape Ocean-Plate Whirlpool would be about 166 Ma, from birth to death on the planet Earth.What is hidden behind of the evolving of the oceans and the ancient plates is the golden law of geometrical progression of ladder pagoda of attributes evolving of them, which is “1➨2➨4➨2➨1”, including upgrading and downgrading process. It seems to hint that what is happening on the planet Earth, including the top-level species of human race, is actually unique in the Milky Way galaxy in essence.The property roles of expanding and shrinking are existing in the Cross “十” shape Ocean-Plate Whirlpool by face to face and the exchange of property roles took place tens of millions of years ago. And that situation would happen again tens of millions of years in the future. It seems a giant circle of evolving of the exchange of property roles would be drawn by the Cross “十” shape Ocean-Plate Whirlpool in her lifetime.