Length Contraction Research Articles

Some multidimensional fixed point theorems for nonlinear contractions in C-distance spaces with applications

In this manuscript, we use the concept of multidimensional fixed point in a generalized space, namely, C-distance space with some nonlinear contraction conditions, such as Jaggi- and Dass-Gupta-type contractions. We provide results with a Jaggi-type hybrid contraction for the mentioned space. Moreover, we use control functions to get the desired results. After each theorem, we compare our results with previous ones to show that they are generalized. We provide examples to support our results. An application is also performed to solve the system of integral equations.

Derivation And Applications of The Lorentz Transformation

The Lorentz transformation is a foundational mathematical construction that has revolutionized people’s comprehension of space, time, and motion within the realm of special relativity. Crafted through the collaborative insights of Lorentz and Einstein, it offers a profound perspective on the intricate fabric of the universe. By delving into the intricate interplay between space and time, the Lorentz transformation transcends traditional Newtonian notions, providing a groundbreaking framework to grasp the behavior of matter and energy in the cosmos. This paper embarks on a meticulous exploration, deriving the Lorentz transformation equations from first principles. It subsequently delves into a captivating journey through its multifaceted applications. From unraveling the mysteries of high-energy particle interactions to enhancing the accuracy of global navigation systems, the reach of the Lorentz transformation extends across diverse scientific and technological domains. The applications include elucidating counterintuitive effects like time dilation and length contraction, elucidating the mass-energy equivalence principle, and facilitating precise measurements in high-speed scenarios like medical imaging. Moreover, the Lorentz transformation guides people’s understanding of electromagnetic fields and underpins advancements in nuclear energy and aerospace technology.

Rational Contraction in Metric Space and Common Fixed Point Theorems

The study of contraction mappings in fixed-point theory is a fascinating and crucial field of mathematics. The concept of contraction plays a vital role in proving the existence and uniqueness of fixed points. Banach's contraction theory offers a fixed point theorem that is widely accepted as unique in most analyses. By using rational expressions in metric spaces, we can achieve unique results in general contraction mapping. These results are based on several innovative ideas stemming from the latest research. The delivered results upgrade and federate many existing outcomes on the topic in the literature Bhardwaj, R. et al. (2007) Chouhan et al. (2014) and Garg and Priyanka (2016). Also gives some suitable examples for verifying our results.

SPECIAL THEORY OF RELATIVITY UNDER THREAT

In optics experiments are conducted on the wave nature of light. Interferometers are used for measurements and two waves are superimposed to generate interference fringes. In such an experiment Michelson and Morley used Michelson interferometer to measure the velocity of earth in a stationary ether. The null results of this experiment could not be explained, various other experiments were conducted such as kennedy-Thorndik experiment, Dayton- Miller experiment, Raleigh and Brace experiment, Michelson- Gale- Pearson experiment, but all experiments gave null results. While interpreting the result of these experiments a very important role of wavefront of light has been ignored. Present work describes the role of wavefront of light in these experiments conducted on the wave nature of light and describes that actually no length contraction and time dilation like phenomenons exist as stipulated by special theory of relativity.

On a Recent Question from the Orthogonal b-metric-like Spaces

In this paper, we announce a beautiful and important application of the results of the paper of two Romanian mathematicians: R. Miculescu, A. Mihail, New fixed point theorems for set-valued contractions in b-metric spaces, J. Fixed Point Theory Appl., 19, 2153-2163 (2017). Namely, the method they gave in the mentioned paper for b-metric spaces can be applied without difficulty to b-metric-like spaces as well. Thus, the results on b-metric spaces established in the papers: Hassen Aydi, Monica-Felicia Bota, Erdal Karapinar and Slobodanka Mitrović, A fixed point theorem for set-valued quasi-contraction in b-metric spaces, Fixed Point Theory and Applications 2012, 2012:88, and Nicola Fabiano, Zoran Kadelburg, Stojan Radenović, and Milica Savatović, Remarks on ”On new orthogonal contractions in b-metric spaces” and ”On orthogonal partial b-metric spaces with an application”, can be extended to b-metric-like spaces as well. Also, using the significant paper at the beginning of the mentioned Romanian professors, a positive answer can be directly given to the question given at the end of the recent paper: M. Gardašević-Filipović, K. Kukić, D. Gardašević, Z. D. Mitrović, Some best proximity point results in the orthogonal 0-complete b-metric-like spaces, J. Contemp. Math. Anal. Armen. Acad. Sci., 58 (2), 105–115 (2023). In the same way, the interval [0,1s ) belonging to the coefficient of some type of contraction within the b-metric-like space can be extended to the entire interval [0,1).

Fixed point theorem using contraction in complete multiplicative metric space

In our paper, we prove a fixed point theorem which extends results of Gupta and Garg and broadens a number of findings in the setting of multiplicative metric space.

Fixed points via interpolative enriched contractions in metric and quasi-metric spaces

Fixed points via interpolative enriched contractions in metric and quasi-metric spaces

Fixed point results for almost ($ \zeta -\theta _{\rho } $)-contractions on quasi metric spaces and an application

<abstract><p>This research paper investigated fixed point results for almost ($ \zeta-\theta _{\rho } $)-contractions in the context of quasi-metric spaces. The study focused on a specific class of ($ \zeta -\theta _{\rho } $)-contractions, which exhibit a more relaxed form of contractive property than classical contractions. The research not only established the existence of fixed points under the almost ($ \zeta -\theta _{\rho } $)-contraction framework but also provided sufficient conditions for the convergence of fixed point sequences. The proposed theorems and proofs contributed to the advancement of the theory of fixed points in quasi-metric spaces, shedding light on the intricate interplay between contraction-type mappings and the underlying space's quasi-metric structure. Furthermore, an application of these results was presented, highlighting the practical significance of the established theory. The application demonstrated how the theory of almost ($ \zeta -\theta _{\rho } $)-contractions in quasi-metric spaces can be utilized to solve real-world problems.</p></abstract>

Nonlinear contractions on directed graphs with applications to boundary value problems

<abstract><p>This article contains some outcomes on fixed points for a graph preserving nonlinear contraction in a metric space endued with a transitive directed graph. Our results improved, enriched, and subsumed various known fixed point theorems. To argue for reliability of our results, we presented two examples. We concluded the manuscript to investigate a unique solution of a certain first-order boundary value problem by means of our results.</p></abstract>

Istraživanje viševrednosne verovatnoće ps-kontrakcije sa orbitama u b-Mengerovim prostorima

Introduction/purpose: The paper presents a novel approach to certain well-established fixed point theorems for multivalued probabilistic contractions in b-Menger spaces, leveraging the boundedness of the orbits. The aim was to generalize and enhance the results previously derived by Fang and Hadžić. Methods: The boundedness of orbits in b-Menger spaces is used to establish their approach for multivalued probabilistic contractions. Results: The findings of the study not only generalized the existing fixed point theorems but also enhanced them significantly. The effectiveness of the approach in extending the results originally proposed by Fang and Hadžić was showcased. Moreover, the applicability of the coincidence fixed point theorem in fuzzy b-metric spaces was demonstrated. Conclusions: The study presented a novel perspective on fixed point theorems in multivalued probabilistic contractions within b-Menger spaces. By leveraging boundedness and introducing a coincidence fixed point theorem for fuzzy b-metric spaces, the work contributed to the advancement in this field.

Revisiting the Electron Radius in Light of Length Contraction

Revisiting the Electron Radius in Light of Length Contraction

General relativity and background Lorentz transformations solve Supplee's submarine paradox

A submarine moving at relativistic horizontal velocity sinks in Earth's rest frame due to length contraction while appearing to float in its own frame. Using spacetime geometry and the Lorentz transformations, we show that the resolution lies in how metric components transform between reference frames in relative motion. This solution frees us from assumptions made in previous studies on how a Newtonian gravitational force should transform. The method of background Lorentz transformations is technically simpler than previous treatments in the framework of general relativity. Moreover, we find a novel and intuitive understanding of the paradox, and correct an erroneous expression for the gravitational force obtained by Supplee and used again in the literature.

Meir–Keeler Fixed-Point Theorems in Tripled Fuzzy Metric Spaces

In this paper, we first propose the concept of a family of quasi-G-metric spaces corresponding to the tripled fuzzy metric spaces (or G-fuzzy metric spaces). Using their properties, we give the characterization of tripled fuzzy metrics. Second, we introduce the notion of generalized fuzzy Meir–Keeler-type contractions in G-fuzzy metric spaces. With the aid of the proposed notion, we show that every orbitally continuous generalized fuzzy Meir–Keeler-type contraction has a unique fixed point in complete G-fuzzy metric spaces. In the end, an example illustrates the validity of our results.

Variation in functional composition of reef fishes along a tropical‐to‐temperate gradient

AbstractAimsTransformation of species and functional composition on subtropical reefs are ongoing due to poleward range shifts of some tropical species, with largely unknown consequences to ecosystem functioning. Trait‐based approaches are powerful tools to quantify such changes. Here, we evaluated changes in the trait composition of coral‐associated fish assemblages along a tropical‐to‐temperate environmental gradient of ca. 1400 km in southern Japan with abundance‐weighted trait expression to assess how trait composition changes with increasing latitude.LocationRyukyu Islands and southern Pacific coast of Japan.TaxonReef fish.MethodsWe tested for shifts in trait space and functional redundancy, based on five morphological, life history and behavioural traits: maximum length, pelagic larval duration, trophic level, substrate preference and reproductive mode. Our trait database was coded with two approaches, first, by attributing a single value to each trait per species, and second, by fuzzy coding that allows more than one value per trait and hence considers some intraspecific trait variation.ResultsWe found a reduction in specialist habitat traits (coral substrate preference, nesters and herbivores) and an increase in generalist traits (predators) with increasing latitude, along with a contraction in trait space from tropical‐to‐temperate reefs. Functional redundancy declined with increasing latitude. These trends were closely linked with latitudinal gradients in temperature, along with changes in other environmental factors such as turbidity and photosynthetically active radiation.Main ConclusionFunctional turnover and contractions are thus likely due to the marginal conditions for coral‐associated fishes at higher latitudes, favouring generalist species, whereas increased resources at lower latitudes favour high redundancy and niche partitioning. Accounting for intraspecific trait variation indicated the same trends but highlighted increased functional vulnerability across all sites. We show that trait complexity in coral‐associated fish communities decreases from tropical‐to‐temperate reefs, highlighting the reduced functional scope that comes with marginal environmental conditions.

Demonstrating Einstein’s Time Dilation Using Simple Muon Detectors

Einstein’s theory of special relativity is one of the topics taught in undergraduate university physics courses. As undergraduates rarely have the opportunity to collect data and verify the concepts of Special Relativity, namely time dilation and length contraction, this paper aims to serve as a detailed instructional manual (self-contained) as well as provide insights from the undergraduates’ point of view on how to use a muon detector to collect data at various locations and altitudes in order to experimentally verify the aforementioned concepts. This experimental element enhances and complements the understanding of Special Relativity. The desktop muon detector used — the CosmicWatch Desktop Muon Detector V2 — was developed by the Department of Physics at Massachusetts Institution of Technology.

Afternoons can be a critical time of the day that reflects associations between life‐space mobility and loneliness among living alone older adults: longitudinal analyses with motion sensor data

AbstractBackgroundDaily activity patterns of life‐space mobility can serve as indicators for loneliness and subsequent cognitive change. Assessment of life‐space mobility often uses self‐reported data, and the measurement intervals are often less frequent than needed to capture change over time. This study examined whether the life‐space mobility changed when loneliness was reported weekly.MethodsParticipants were 139 older adults who lived alone (age = 78.1±8.6, 103 (74%) were female, 32 (23%) were African Americans, 19 (14%) had MCI diagnosis) and were enrolled in ORCATECH (ORegon Center for Aging & TECHnology) longitudinal aging studies and Minority Aging Research Study. Passive infrared motion sensors were placed in common rooms of each participant’s home (bathroom; bedroom; kitchen; living room). Time spent in each room across the day (morning; afternoon; evening; night) and out‐of‐home were used as indicators of life‐space mobility and were derived from home sensor data. Participants reported whether they felt lonely on a weekly basis using an online survey sent via email. Generalized estimating equations were used to correlate the time spent in each room and out‐of‐home and the likelihood of feeling lonely. Age, gender, education, race, cognitive status (MCI/ non‐MCI), and average daily time out‐of‐home were controlled in the model.ResultsWe analyzed 4994 weeks of data; 232 weeks had an incidence of loneliness (5%). Spending an additional hour in the bedroom in the afternoon (noon‐6 pm) was associated with a 21% increase in the likelihood of experiencing loneliness (p = .03). Spending an additional hour out‐of‐home in the afternoon was associated with a 13% decrease in the likelihood of experiencing loneliness (p = .02). Time spent in the kitchen, bathroom, and living room in a day was unrelated to weekly loneliness.ConclusionTime spent outside the home may suggest an expansion of life‐space, while spending time in one’s bedroom may indicate contraction of life space. Our findings suggest carefully monitoring a person’s location in the afternoon might help detect loneliness in this population. Frequent and objective measurements of life‐space mobility can produce renewed insights into the experience of loneliness among older adults who live alone, and thus, inform timely interventions for loneliness and prevent cognitive decline.

Quantum mechanical Gaussian wavepackets of single relativistic particles

We study the evolutions of selected quasi-(1+1) dimensional wavepacket solutions to the Klein–Gordon equation for a relativistic charged particle in uniform motion or accelerated by a uniform electric field in Minkowski space. We explore how good the charge density of a Klein–Gordon wavepacket can be approximately described by a Gaussian state with the single-particle interpretation. We find that the minimal initial width of a wavepacket for a good Gaussian approximation in position space is about the Compton wavelength of the particle divided by its Lorentz factor at the initial moment. This can be considered as a manifestation of relativistic length contraction, which is also observed numerically in the spreading of the wavepacket’s charge density.

Appearance and reality: Einstein and the early debate on the reality of length contraction

In 1909, Ehrenfest published a note in the Physikalische Zeitschrift showing that a Born rigid cylinder could not be set into rotation without stresses, as elements of the circumference would be contracted but not the radius. Ignatowski and Varićak challenged Ehrenfest’s result in the same journal, arguing that the stresses would emerge if length contraction were a real dynamical effect, as in Lorentz’s theory. However, no stresses are expected to arise, according to Einstein’s theory, where length contraction is only an apparent effect due to an arbitrary choice of clock synchronization. Ehrenfest and Einstein considered this line of reasoning dangerously misleading and took a public stance in the Physikalische Zeitschrift, countering that relativistic length contraction is both apparent and real. It is apparent since it disappears for the comoving observer, but it is also real since it can be experimentally verified. By drawing on his lesser-known private correspondence with Varićak, this paper shows how Einstein used the Ehrenfest paradox as a tool for an ‘Einsteinian pedagogy’. Einstein’s argumentative stance is contrasted with Bell’s use of the Dewan-Beran thread-between-spaceships paradox to advocate for a ‘Lorentzian pedagogy’. The paper concludes that the disagreement between the two ways of ‘teaching special relativity’ stems from divergent interpretations of philosophical categories such as ‘reality’ and ‘appearance’.

High-Speed Railway Network, City Heterogeneity, and City Innovation

The emergence of the time–space contraction effect from the high-speed railway (HSR) network in China has been beneficial in breaking down regional divisions, thus facilitating the circulation of resources and optimizing resource distribution and production efficiency. However, research has not adequately addressed the city disparities of the HSR network and their effects on city innovation. Through the heterogeneity perspective of ‘New’ new economic geography, this study employs the 2008–2019 panel data at the city level in China and builds a spatial Durbin model based on continuous spatial difference in differences to investigate the mechanism of the HSR network on city innovation and to analyze its agglomeration and diffusion effect of innovative factors under different city sizes and spatial perspectives. This study revealed that the HSR network could significantly increase the innovation of local cities and neighboring cities, yet there is a certain threshold of city size that affects city innovation. Large cities covered by HSRs can take advantage of gathering talent, financial capital, and industry from nearby regions, thus constructing a new spatial pattern of innovative development. This study also found that the innovation accelerative effect gradually decreases as the distance from the city covered by HSRs increases and completely disappears at the distance of 400 km. Therefore, it is necessary to optimize the HSR network and increase the mobility and agglomeration of innovative elements between cities, thus deepening the collaboration between cities through differentiated strategies. This will enhance the spatial spillover effect of innovation, thus ultimately achieving a balanced spatial pattern of city innovation.

Modelling of noise due to impulsive excitation using nonlinear time series analysis

Early acoustic machine design phases should target safety and comfort requirements, minimising the risk of hearing damage and noise exposure and ensuring a pleasant machine sound. Therefore, physical mechanisms of sound generation, propagation and radiation need to be modelled. Today, nonlinear physical relations in vibroacoustic models such as contact problems are mostly linearised. The aim of this paper is to increase the prediction quality of vibroacoustic models so that physical mechanisms behind sound generation due to complex physical, nonlinear phenomena can be better understood. The focus is on sound generation due to impulsive excitation, which is a significant sound generation mechanism in machinery, e.g. considering gears, forming processes (e.g. punching), or road loads of vehicles. We describe a model setup to simulate impulsive excitation on a laboratory scale, where structural vibrations can be analysed. Physical correlations including nonlinear correlations could be identified using nonlinear time series analysis. Recurrence plots will be used to identify features such as phase space contraction, e.g. to identify damping. These features form the basis to deepen the physical understanding of machinery noise, to contribute to enhanced acoustic machine designs and to enhance data-driven modelling using sparse regression in future research.