Let $(A,\Delta)$ be a {\it weak multiplier Hopf algebra} as introduced in [VD-W3] (see also [VD-W2]). It is a pair of a non-degenerate algebra $A$, with or without identity, and a coproduct $\Delta$ on $A$, satisfying certain properties. If the algebra has an identity and the coproduct is unital, then we have a Hopf algebra. If the algebra has no identity, but if the coproduct is non-degenerate (which is the equivalent of being unital if the algebra has an identity), then the pair would be a multiplier Hopf algebra. If the algebra has an identity, but the coproduct is not unital, we have a weak Hopf algebra. In the general case, we neither assume $A$ to have an identity nor do we assume $\Delta$ to be non-degenerate and so we work with a {\it genuine} weak multiplier Hopf algebra. It is called {\it regular} if its antipode is a bijective map from $A$ to itself. \snl In this paper, we {\it continue the study of weak multiplier Hopf algebras}. We recall the notions of the source and target maps $\varepsilon_s$ and $\varepsilon_t$, as well as of the source and target algebras. Then we investigate these objects further. Among other things, we show that the canonical idempotent $E$ (which is eventually $\Delta(1)$) belongs to the multiplier algebra $M(B\ot C)$ where $B=\varepsilon_s(A)$ and $C=\varepsilon_t(A)$ and that it is a {\it separability idempotent} (as studied in [VD4.v2]). If the weak multiplier Hopf algebra is regular, then also $E$ is a {\it regular} separability idempotent. \snl We also consider {\it special cases and examples} in this paper. In particular, we see how for any weak multiplier Hopf algebra $(A,\Delta)$, it is possible to make $C\ot B$ (with $B$ and $C$ as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the 'Hopf algebra part' of the original weak multiplier Hopf algebra and only remembers the source and target algebras. It is in turn generalized to the case of any pair of non-degenerate algebras $B$ and $C$ with a separability idempotent $E\in M(B\ot C)$. We get another example using this theory associated to any discrete quantum group (a multiplier Hopf algebra of discrete type with a normalized cointegral). Finally we also consider the well-known 'quantization' of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras).

The super-weak interaction includes three simple extensions of the standard model: gauge extension, fermionic extension, and scalar extension. All of these extensions are strongly influenced by their complex phenomenology. They can explain a number of unresolved questions in particle physics and cosmology, including the genesis of dark matter, cosmic inflation, asymmetry of matter and antimatter, neutrino masses, and vacuum stability, if combined into a single structure. This is an extension of the gauge group of the standard model $G_{\rm SM}$ by $G_{\rm SM}\otimes U(1)_Z$ without any anomalies. We investigate the implications of the development of a general Majorana mass renormalization group for neutrinos with masses in the range of 0.03 $eV$ and 0.1 $eV$, which fall within the recently published range as well as the range to be explored in future planned experiments.

A method for measuring the real part of the weak value of spin for non-zero rest mass atoms is presented using a variant on the original Stern-Gerlach apparatus. The experiment utilises helium in the metastable 2³S₁ state. A full simulation for observing the real part of the weak value using the impulsive approximation has been carried out and it predicts a displacement of the beam, Δ_w, that is within the resolution of our detector. It also indicates how this shift might be increased. The full analysis also indicated that there is a limit, L, to the applicability of the weak value approximation and has been evaluated for our apparatus. This experiment has the possibility to be expanded to utilise other nobal gas species such as neon and argon in the ³P₂ metastable state, but we shall restrict this paper to metastable helium only.

There has been a recent revival of interest in the notion of a `trajectory' of a quantum particle. In this paper we detail the relationship between Dirac's ideas, Feynman paths and the Bohm approach. The key to the relationship is the weak value of the momentum which Feynman calls a transition probability amplitude. With this identification we are able to conclude that a Bohm `trajectory' is the average of an ensemble of actual individual stochastic Feynman paths. This implies that they can be interpreted as the mean momentum flow of a set of individual quantum processes and not the path of an individual particle. This enables us to give a clearer account of the experimental two-slit results of Kocsis {\em et al.}}

In the present paper, we introduced a quadratically convergent Newton’s like normal S2 iteration method free from the second derivative for the solution of nonlinear equations permitting 3 f'(x) = 0 at some points in the neighborhood of the root. Our proposed method works well 4 when the Newton method fails. Numerically it has been verified that the Newton’s like normal 5 S-iteration method converges faster than Fang et al. method [A cubically convergent Newton-type 6 method under weak conditions, J. Compute. and Appl. Math., 220 (2008), 409-412]. We studied 7 different aspects of normal S-iteration method. Lastly, fractal patterns support the numerical 8 results and explain the convergence, divergence, and stability of method.

In this review, various researches on finding the bending angle of light deflected by a massive gravitating object which regard the Gauss-Bonnet theorem as the premise have been revised. Primarily, the Gibbons and Werner method is studied apropos of the gravitational lensing phenomenon in the weak field limits. Some exclusive instances are deliberated while calculating the deflection angle, beginning with the finite-distance corrections on non-asymptotically flat spacetimes. Effects of plasma medium is then inspected to observe its contribution to the deflection angle. Finally, the Jacobi metric is explored as an alternative method, only to arrive at similar results. All of the cases are probed in three constructs, one as a generic statement of explanation, one for black holes, and one for wormholes, so as to gain a perspective on every kind of influence.

The vibration feature of weak gear fault is often covered in strong background noise, which makes it necessary to establish weak feature enhancement methods. Among the enhancement methods, stochastic resonance (SR) has the unique advantage of transferring noise energy to weak signals and has a great application prospection in weak signal extraction. But the traditional SR potential model cannot form a richer potential structure and may lead to system instability when the noise is too great. To overcome these shortcomings, the article presents a periodic potential underdamping stochastic resonance (PPUSR) method after investigating the potential function and system signal-to-noise ratio (SNR). In addition, system parameters are further optimized by using ant colony algorithm. Through simulation and gear experiments, the effectiveness of the proposed method was verified. We concluded that compared with the traditional underdamped stochastic resonance (TUSR) method, the PPUSR method had a higher recognition degree and better frequency response capability.

It is known that ammonium has a higher permeability through anion-exchange and bipolar membranes compared to K+ cation that has the same mobility in water. However, the mechanism of this high permeability is not clear enough. We develop a mathematical model based of the Nernst-Planck and Poisson equations for diffusion of ammonium chloride through an anion-exchange membrane; proton exchange reactions between ammonium, water and ammonia are taken into account. It is assumed that ammonium, chloride and OH− ions can only pass through membrane hydrophilic pores, while ammonia can also dissolve in membrane matrix fragments not containing water and diffuse through these fragments. It is found that due to the Donnan exclusion of H+ ions as coions, the pH in the membrane internal solution increases when approaching the membrane side facing distilled water. Consequently, there is a change in the principal nitrogen-atom carrier in the membrane: in the part close to the side facing the feed NH4Cl solution (pH<8.8), it is the NH4+ cation, and in the part close to distilled water, NH3 molecules. The concentration of NH4+ reaches almost zero at a point close to the middle of the membrane cross-section, which approximately halves the effective thickness of the diffusion layer for the transport of this ion. When NH3 takes over the nitrogen transport, it only needs to pass through the other half of the membrane. Leaving the membrane, it captures an H+ ion from water, and the released OH− goes towards the membrane side facing the feed solution to meet the NH4+ ions. The comparison of the simulation with experiment shows a satisfactory agreement.

In this paper, we study the gravitational lensing by some black hole classes within the non-linear electrodynamics in weak field limits. First, we calculate an optical geometry of the non-linear electrodynamics black hole then we use the Gauss-Bonnet theorem for finding deflection angle in weak field limits. The effect of non-linear electrodynamics on the deflection angle in leading order terms is studied. Furthermore, we discuss the effects of the plasma medium on the weak deflection angle.

In this article, we demonstrate the weak gravitational lensing in the context of Bocharova-Bronnikove-Melnikov-Bekenstein (BBMB) black hole. To this desire, we derive the deflection angle of light in a plasma medium by BBMB black hole using the Gibbons and Werner method. First, we obtain the Gaussian optical curvature and implement the Gauss-Bonnet theorem to investigate the deflection angle for spherically symmetric spacetime of BBMB black hole. Moreover, we also analyze the graphical behavior of deflection angle by BBMB black hole in the presence of plasma medium.

The main goal of this paper is to study the weak gravitational lensing by Horndeski black hole in weak field approximation. In order to do so, we exploit the Gibbons-Werner method to the optical geometry of Horndeski black hole and implement the Gauss-Bonnet theorem to accomplish the deflection angle of light in weak field region. Furthermore, we have endeavored to extend the scale of our work by comprising the impact of plasma medium on the deflection angle as properly. Later, the graphical influence of the deflection angle of photon on Horndeski black hole in plasma and non-plasma medium is examined.

In this paper, we discuss the weak gravitational lensing in the context of stringy black holes. Initially, we examine the deflection angle of photon by charged stringy black hole. For this desire, we compute the Gaussian optical curvature and implement the Gauss-Bonnet theorem to investigate the deflection angle for spherically balanced spacetime of stringy black hole. We also analyze the influence of plasma medium in the weak gravitational lensing for stringy black hole. Moreover, the graphical impact of coupling constant $\alpha$, impact parameter $b$ , black hole charge $Q$ on deflection angle by charged stringy black hole has been studied in plasma as well as non-plasma medium.

In this paper, we introduced many new notions and new contraction named as S-weakly contraction after that we obtained the p-common best proximity point theorems for different types of contractions in the setting of complete metric spaces by using weak P_p-property and proved the uniqueness of these points. Also we presented some examples to prove the validity of our results.

In an historic perspective on the development of the Standard Model of particle physics it is shown how mathematically driven axioms have masked the merits of a physically comprehensible structural view. It is concluded that the difference between the two approaches can be traced back to two major issues. Whereas in the Standard Model the quark is a Dirac particle with a single real dipole moment, the quark in the structural model, in confinement with other quarks, is a Dirac particle with two real dipole moments. The second issue is the view that empty space does not exist, but that space is filled with a polarisable energetic fluid. It is shown how recognition of these two issues pave a road to reconcile particle physics with gravity, in which the quark can be seen as a magnetic electron and in which the gluon, as the strong force carrier, can be seen as a massive photon.

Magnetoelectric (ME) sensor is a new type magnetic sensor with ultrahigh sensitivity, low power consumption, and suitable for the measurement of low frequency weak magnetic field. In this study, metglas/PZT-5B ME sensor with a mechanical resonance frequency f_res of 60.041 kHz was prepared. It is interesting to note that its magnetic field resolution reaches 0.20 nT at f_res and 0.34 nT for DC field, respectively. In order to measure the ultralow frequency AC magnetic fields, a frequency upconversion technique was employed. Under this technique, a limit of detection (LOD) of AC magnetic field lower than 1 nT at 8 Hz is obtained, and the minimum LOD of 0.51 nT is achieved at 20 Hz. The high resolution ME sensor with sub-pT level is promising in the field of low frequency weak magnetic field measurement technology.

Identifying ambient noise-based (ANb) signatures of streams can help in the estimation of their erosive potential (EP) that promotes reverie landslides and soil losses in the fluvial valleys. This is particularly imperative on flooding or rainy days, leading to stronger erosion-prone conditions (colluvium and boulders) of the valley beds inferred from georadar attribute analysis. Developing such research direction can benefit the local communities, as is the case with the Cerrado region of Brazil, where these phenomena have high destructive potential with social, economic, and climatic implications. For the present study, a seasonal stream in the Federal District of Brazil was investigated by ANb monitoring supported by Ground Penetration Radar (GPR) for site characterization. The ANb monitoring was conducted (at a safe distance) with a seismometer over several durations of dry and rainy conditions. The power spectral density (PSDs) was computed as a function of several variables, including weather conditions (rainfall, wind speed, and pressure), time-frequency spectrograms, and ambient noise displacement root mean square (RMS). This analysis also considered the single station horizontal-to-vertical spectral ratio (HVSR), where rain, wind, pressure, river flow and anthropogenic signatures were evident (at selective frequency ranges). Multi-peaks that emerged on the HVSR curve were further analyzed to identify amplitude and frequency changes, and the three peaks shift on average to a lower position during the rainy period. The GPR amplitude and waveform variation features were attributed to the stratigraphy (i.e., the boundary between valid and invalid regions and coherence value) of the floodplain and regions susceptible to erosion (erosion-prone lithological spots). This approach provides the basis for non-destructive monitoring tools enabling the detection of 'seismic signatures' and weak spots of the fluvial channels for improving environmental management.

In this paper, by considering Einstein-Hilbert-Bumblebee (EHB) gravity around global monopole field, we derive exactly a black hole spacetime metric. To test the effect of global monopole field and bumblebee field, which causes the spontaneous Lorentz symmetry breaking, we calculate the weak deflection angle using the Gauss-Bonnet theorem.

In this paper, we use a new asymptotically flat and spherically symmetric solution in the generalized Einstein-Cartan-Kibble-Sciama (ECKS) theory of gravity to study the weak gravitational lensing and its shadow cast. To this end, we first compute the weak deflection angle of generalized ECKS black hole using the Gauss–Bonnet theorem in plasma medium and in vacuum. Next by using the Newman-Janis algorithm without complexification, we derive the rotating generalized ECKS black hole and in the sequel study its shadow. Then, we discuss the effect of the ECKS parameter on the shadow of the black hole and weak deflection angle. In short, the goal of this paper is to give contribution to the ECKS theory and look for evidences to understand how the ECKS parameter effects the gravitational lensing.

Ultra-weak photon emissions (UPEs) including those from human have been experimented for decades. The photo-genic origin of UPE has also been attributed to the oxidative stress or free radical production that is unique to metabolically active states of biological organisms. However, there are considerable gaps in quantitative understanding of UPE. In this work, I propose an analytical framework of hypothesis for the initial objective of modeling a few superficial presentations of UPE of human, including the systematic dependency on age, the diurnal variation, and the geometric asymmetry associated with serious asymmetrical pathological conditions. The hypothesis which is currently limited to human assumes a new form of energy state, termed vivo-nergy, which resides only in metabolically active organisms that are also under neuronal control. The hypothesis projects a decrease of the vivo-nergy in human during growth beyond puberty, with the rate of decrease dictated by a critical time-scale---the age of the first opposite-sex sexual intercourse (FOSSI). The hypothesis also proposes a modification of the vivo-nergy by the phases of systematic or homeostatic physiology. The hypothesis further postulates that the deviation of the physiology-modified vivo-nergy from the pre-puberty level is deteriorated by acquired organ-specific pathological conditions. Any reduction of vivo-nergy from the pre-puberty level is hypothesized to proportionally cause oxidative stress that functions as the physical source of UPE. The resulted steady-state diffusion of the photon emitted from a photo-genic source of UPE in a human geometry simplified as a homogeneous spherical domain is modeled by photon diffusion principles incorporating an extrapolated zero-boundary condition. The age and systematic physiology combined determines the intensity of the centered physiological photo-genic source. The acquired (single) pathology sets both the intensity and the off-center position of the (single) pathological photo-genic source. When the age-related, physiology-commanded, and pathology-controlled modifications of the photo-genetic sources are implemented in the photon diffusion model, the photon fluence rate at the surface of the simplified human-representing spherical domain reveals the dependency on age, the temporal variation corresponding to systematic physiology, and the geometric asymmetry associated with significant asymmetric pathological condition as reported previously for UPE. The hypothesis, as it provides analytical conveniences for quantitative estimation of UPE patterns, may be useful to further model-based interpretation of the spatial-temporal characteristics of UPE.

Abstract: During firing from a mortar, an important issue is the parameters of compressibility of the ground on which the mortar is placed. This affects the operation of the mortar (including safety). During the qualification tests of the mortar, the influence of different types of substrate on its strength and work during shooting should be examined. Until now in the Polish standardization documents was no clear description about the substrate parameters used for this kind of tests. Analysis of the literature also did not allow to determine the dependence of the mortars displacement in the function of the type of substrate and its geotechnical parameters. Therefore, the authors carried out analytical and experimental analysis of the use of geotechnical methods to determine the parameters of the substrate for mortar testing. The paper presents the proposed types of standardized soil and their research methods. Preliminary tests were also carried out using the prescribed measurement methods and a comparative mortar firing test. Subsequently, an analysis of the results was carried out and the possibilities of using the proposed methods of measuring the parameters of soil compressibility were determined.

We give an exact mathematical construction of a spacelike order $<$, which is dual to the standard chronological order $\ll$ in the $n$-dimensional Minkowski space $M^n$ and we discuss its order-theoretic, geometrical as well as its topological implications, conjecturing a possible extension to curved spacetimes.

In this paper, we introduce a new iteration scheme, named as the S**-iteration scheme, for approximation of fixed point of the nonexpansive mappings. This scheme is faster than Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, Thakur, and Ullah iteration schemes. We show the stability of our instigated scheme and give a numerical example to vindicate our claim. We also put forward some weak and strong convergence theorems for Suzuki's generalized nonexpansive mappings in the setting of uniformly convex Banach spaces. Our results comprehend, improve, and consolidate many results in the existing literature.

In this paper, we study the weak gravitational deflection angle of relativistic massive particles by the Kerr-like black hole in the bumblebee gravity model. In particular, we focus on weak field limits and calculate the deflection angle for a receiver and source at a finite distance from the lens. To this end, we use the Gauss-Bonnet theorem of a two-dimensional surface defined by a generalized Jacobi metric. The spacetime is asymptotically non-flat due to the existence of a bumblebee vector field. Thus the deflection angle is modified and can be divided into three parts: the surface integral of the Gaussian curvature, the path integral of a geodesic curvature of the particle ray and the change in the coordinate angle. In addition, we also obtain the same results by defining the deflection angle. The effects of the Lorentz breaking constant on the gravitational lensing are analyzed. We then consider the finite-distance correction for the deflection angle of massive particles.

Maxwell's fish eye has been known to be a perfect lens in optics. In this letter, using the Gibbons-Werner method, namely Gauss-Bonnet theorem on optical geometry of black hole, we extend the calculation of the weak gravitational lensing within the Maxwell's fisheye as a perfect lensing in medium composed of an isotropic refractive index that near-field information can be obtained from far-field distances. Moreover, these results provide an excellent tool to observe compact massive object by weak gravitational lensing within the dark matter medium and to understand the nature of the dark matter that may effect the gravitational waves. Moreover, we show that Gauss-Bonnet theorem is a global effect and this method can be used as a new tool on any optical geometry of compact objects in dark matter medium.

At nuclear scale, we present three heuristic relations pertaining to strong and electroweak coupling constants. With these relations, close to beta stability line, it is possible to study nuclear binding energy with a single energy coefficient of magnitude $\left(\frac{1}{{\alpha }_s}\right)\left[\frac{e^2}{4\pi {\epsilon }_0{R}_0}\right]\approx 10.0\text{MeV}\text{.}$ With reference to up and down quark masses, it is also possible to interpret that, nuclear binding energy is proportional to the mean mass of $\left[\left(2m_u+m_d\right)\text{and}\left(m_u+2m_d\right)\right]\approx 10.0\text{MeV}.$

The principal objective of this project is to investigate the gravitational lensing by asymptotically flat black holes in the framework of Horndeski theory in weak field limits. To achieve this objective, we utilize the Gauss-Bonnet theorem to the optical geometry of asymptotically flat black holes and applying the Gibbons-Werner technique to achieve the deflection angle of photons in weak field limits. Subsequently, we manifest the influence of plasma medium on deflection of photons by asymptotically flat black holes in the context of Horndeski theory. We also examine the graphical impact of deflection angle on asymptotically flat black holes in the background of Horndeski theory in plasma as well as non-plasma medium.

In this work, we investigate the weak deflection angle of light from exact black hole within the non-linear electrodynamics. First we calculate the Gaussian optical curvature using the optical spacetime geometry. With the help of modern geometrical way popularized by Gibbons and Werner, we examine the deflection angle of light from exact black hole. For this desire, we determine the optical Gaussian curvature and execute the Gauss-Bonnet theorem on optical metric and calculate the leading terms of deflection angle in the weak limit approximation. Furthermore, we likewise study the plasma medium's effect on weak gravitational lensing by exact black hole. Hence we expose the effect of the non-linear electrodynamics on the deflection angle in the weak gravitational field.

In this paper, we calculate the deflection angle of photon by Casimir wormhole in the weak field limit approximation. First we calculate Gaussian optical curvature with the help of optical spacetime geometry and so we use the Gauss-Bonnet theorem on the Gaussian optical metric. Then we find the deflection angle of photon by Casimir wormhole. Moreover, we calculate the photon's deflection angle in the presence of plasma medium and we also see the graphical nature of deflection angle in both cases. Second, we move towards the shadow of Casimir wormhole. After the observations of Event Horizon Telescope, the study of shadow become very important so that we plot the shapes of shadow of Casimir wormhole, and we calculate the photon geodesic around the Casimir wormhole.

In this article, we analyze the weak gravitational lensing in the context of Einstein-non-linear Maxwell-Yukawa black hole. To this desire, we derive the deflection angle of light by Einstein-non-linear Maxwell-Yukawa black hole using the Gibbons and Werner method. For this purpose, we obtain the Gaussian optical curvature and implement the Gauss-Bonnet theorem to investigate the deflection angle of Einstein-non-linear Maxwell-Yukawa black hole. Moreover, we derive the deflection angle of light in the presence of plasma medium. We also analyze the graphical behavior of deflection angle by Einstein-non-linear Maxwell-Yukawa black hole in the presence of plasma as well as non-plasma medium.

The aim of this work is to study generalizations of the notion of mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the Weak Law of Large Numbers in the same way that the ordinary mean relates to the Strong Law. We propose a further generalization, also based on an improper integral, called the doubly weak mean, applicable to heavy-tailed distributions such as the Cauchy distribution and the other symmetric stable distributions, We also consider generalizations arising from Abel-Feynman type mollifiers that damp the behavior at infinity and alternative formulations of the mean in terms of the cumulative distribution and the characteristic function.

In training a deep learning system to perform audio transcription, two practical problems may arise. Firstly, most datasets are weakly labelled, having only a list of events present in each recording without any temporal information for training. Secondly, deep neural networks need a very large amount of labelled training data to achieve good quality performance, yet in practice it is difficult to collect enough samples for most classes of interest. In this paper, we propose factorising the final task of audio transcription into multiple intermediate tasks in order to improve the training performance when dealing with this kind of low-resource datasets. We evaluate three data-efficient approaches of training a stacked convolutional and recurrent neural network for the intermediate tasks. Our results show that different methods of training have different advantages and disadvantages.

This paper attempts to describe the large-scale (solar system-like, astronomical-scale) systems of the known world using the physical models and mathematical tools of relativistic quantum mechanics. The value of H_x can be introduced and approximated as an analogue of the constant ℏ. Based on the quantum mechanical approach, the proper time scale of the solar system can be determined.

In our recently published papers, considering Planck mass based light speed growing black hole universe and scaled Hawking’s black hole temperature formula, we have developed a simple procedure for estimating the current Hubble parameter and current cosmic mass. In addition to that, without considering galactic dark matter, starting from 10 km/sec to 500 km/sec, we have tried to fit the observed galactic flat rotation speeds with a concept of super gravity of galactic baryonic mass. To estimate the equivalent mass of currently hypothecated galactic dark matter, we have introduced a timely increasing cosmological reference mass unit. Its current magnitude seems to be around 200 million solar masses. In this paper, considering 200 million solar masses as a characteristic representation of cosmic weak interaction mass unit, we have developed a simple formula for its estimation. We would like to appeal that, weak interaction boosts the gravity of galaxies in proportion with their virtual dark mass as (galactic baryonic mass)^3/2 / (200 million solar masses)^1/2. With further study- dark matter existence and physical properties can be understood in a theoretical approach.

In this paper, the notion of θ∗-weak contraction is introduced, which is utilized to prove some fixed point results. These results are helpful to give a positive response to certain open question raised by Kannan [Amer. Math. Monthly 76:1969] and Rhoades [Contemp. Math. 72:1988] on the existence of contractive definition which does not force the mapping to be continuous at the fixed point. Some illustrative examples are also given to support our results. As applications of our result, we investigate the existence and uniqueness of a solution of non-linear matrix equations and integral equations of Volterra type as well.

In this paper, the authors employ Mountain Pass Lemma, the method of weak solution regularization and Lyapunov function method to derive the unique existence of globally exponential stable positive stationary solution of a single-species model with diffusion and delayed feedback. The obtained stability criterion illuminates that under some suitable conditions, a certain internal competition is conducive to the overall stability of the population, and a certain amount of family planning is conducive to the overall stability of the population. A numerical example and three tables show the effectiveness of the proposed methods.

The purpose of this paper is twofold. First, we introduce a geometric approach to study the circular orbit of a particle in static and spherically symmetric spacetime based on Jacobi metric. Second, we apply the circular orbit to study the weak gravitational deflection of null and time-like particles based on Gauss-Bonnet theorem. By this way, we obtain an expression of deflection angle and extend the study of deflection angle to asymptotically non-flat black hole spacetimes. Some black holes as lens are considered such as a static and spherically symmetric black hole in the conformal Weyl gravity and a Schwarzschild-like black hole in bumblebee gravity. Our results are consistent with the previous literature. In particular, we find that the connection between Gaussian curvature and the radius of a circular orbit greatly simplifies the calculation.

By considering three virtual gravitational constants assumed to be associated with gravitational, electromagnetic and strong interactions, Fermi’s weak coupling constant can be shown to be a natural manifestation of microscopic quantum gravity. As our approach is heuristic and completely different from the current methods of estimating the Newtonian gravitational constant, with reference to the call of ‘Ideas lab 2016’ organized by NSF, we appeal for inclusion of this theoretical work as a project under unification scheme.

There exists an implicit potential limitation in every physical discoveries that has been implemented and understood. However, the limitations can be bounded within a safe limit to prevent any constructing theory to be free from errors. As, it’s the inert nature of the humans, to go far beyond the scope of experimental findings in order to pursue any studies with the sole help of logical reasoning and mathematics, the argument can be prevailed in the form of WEAK Clampdown Effect & STRONG Clampdown Effect. More, the theories are constructed out of physical nature, more the theory gets hypothetical without any finding evidence, but that does or doesn’t actually justify the phenomenon, that too with the more increment of KARDASHEV Scale, more moderate ways of experimentation got developed curbing down the limitations within the human limit of ‘ERRORS’, that does can be neglected by approximation. Relationship being cross-judgmental on the basis of the computational limits and calculation accuracy, leading to a soft singularity, as a warning, that if computer powers cannot be checked on the basis of error approximations, then this may lead to the hitting of a hard singularity, that in phase with the forbidden gap (or after the optimum limit that arises at the core constraints of nature) to prevent the computation being carried off with respect to super-intelligence machines that are cognitive capability oriented future computers responsible for self growth & reproduction with more improvement algorithm, restricting all forms of humanity & constraints the human growth by virtue of limiting capacities of the humans as compared to computers.

In this paper, we study a three step iterative scheme to approximate fixed points of Suzuki's generalized non-expansive mappings. We establish some weak and strong convergence results for such mappings in uniformly convex Banach spaces. Further, we show numerically that iterative scheme (1.8) converges faster than some other known iterations for Suzuki's generalized non-expansive mappings. To support our claim, we give an illustrative example and approximate fixed points of such mappings using Matlab program. Our results are new and generalize several relevant results in the literature.

Magnetic nanoparticles are met across many biological species ranging from magnetosensitive bacteria, fishes, bees, bats, rats, birds, to humans. They can be both of biogenetic origin and due to environmental contamination, being either in paramagnetic or ferromagnetic state. The energy of such naturally occurring single-domain magnetic nanoparticles can reach up to 10-20 room k_BT in the magnetic field of the Earth, which naturally led to supposition that they can serve as sensory elements in various animals. This work explores within a stochastic modeling framework a fascinating hypothesis of magnetosensitive ion channels with magnetic nanoparticles serving as sensory elements, especially, how realistic it is given a highly dissipative viscoelastic interior of living cells and typical sizes of nanoparticles possibly involved.

We would like to suggest that, by considering three virtual gravitational constants assumed to be associated with gravitational, electromagnetic and strong interactions along with a strongly interacting virtual nuclear elementary charge, a workable model of final unification can be developed. In a verifiable approach, Newtonian gravitational constant and Fermi’s weak coupling constant can be interrelated via nuclear and electromagnetic gravitational constants.

The stiffness of the myosin cross-bridges is a key factor in analysing possible scenarios to explain myosin head changes during force generation in active muscles. The seminal study of Huxley and Simmons (1971: Nature 233: 533) suggested that most of the observed half-sarcomere instantaneous compliance (=1/stiffness) resides in the myosin heads. They showed with a so-called T₁ plot that, after a very fast release, the half-sarcomere tension reduced to zero after a step size of about 60Å (later with improved experiments reduced to 40Å). However, later X-ray diffraction studies showed that myosin and actin filaments themselves stretch slightly under tension, which means that most (at least two-thirds) of the half sarcomere compliance comes from the filaments and not from cross-bridges. Here we have used a different approach, namely to model the compliances in a virtual half sarcomere structure in silico. We confirm that the T₁ curve comes almost entirely from length changes in the myosin and actin filaments, because the calculated cross-bridge stiffness (probably greater than 0.4 pN/Å) is higher than previous studies have suggested. In the light of this, we present a plausible modified scenario to describe aspects of the myosin cross-bridge cycle in active muscle. In particular, we suggest that, apart from the filament compliances, most of the cross-bridge contribution to the instantaneous T₁ response comes from weakly-bound myosin heads, not myosin heads in strongly attached states. The strongly attached heads would still contribute to the T₁ curve, but only in a very minor way, with a stiffness that we postulate could be around 0.1 pN/Å, a value which would generate a working stroke close to 100 Å from the hydrolysis of one ATP molecule. The new program can serve as a tool to calculate sarcomere elastic properties for any vertebrate striated muscle once various parameters have been determined (e.g. tension, T₁ intercept, temperature, X-ray diffraction spacing results).

The spin magnetic moments and spin g factors (g_s = -2) of electron, muon and tau are explained based on the electric charges (EC) and lepton charges (LC) in terms of the three-dimensional quantized space model. The spin g factors of electron, muon and tau are g_s = -2 which is the sum of the EC g factor (g_EC = -1) and the LC g factor (g_LC = -1). The spin g factor (g_s = -2) of the electron is predicted by the Dirac’s equation. The orbit g factors of electron, muon and tau are g_L = g_EC = -1 from the EC g factor (g_EC = -1) without the contribution of the LC g factor (g_LC = -1). The spin g factors of the elementary fermions are calculated from the equation of g_s = g_EC + g_LC + g_CC where g_EC = EC/|EC|, g_LC = LC/|LC| and g_CC = CC/|CC|. For example, the spin g factors of the neutrinos and dark matters are g_s = -1. The spin g factors of the u and d quarks are g_s = 0 and g_s = -2, respectively. The g factor problem of neutrinos with the non-zero LC charges are solved by the LC Coulomb force of F_c(LC) ≈0. It is, for the first time, proposed that the binary motion (fluctuations) of the m_EC and m_LC masses for the electron, muon and tau leptons make the anomalous g factor. This binary motion could be originated from the virtual particle processes including the photons. Also, the weak force (beta) decay is closely related to the binary motion of the m_EC and m_LC for the electron, muon and tau leptons.

This work addresses the role of charge regulation (CR) and the associated fluctuations in the conformational and mechanical properties of weak polyelectrolytes. Due to CR, changes in the pH-value modifies the average macromolecular charge and conformational equilibria. A second effect is that, for a given average charge per site, fluctuations can alter the intensity of the interactions by means of correlation between binding sites. We investigate both effects by means of Monte Carlo simulations at constant pH-value, so that the charge is a fluctuating quantity. Once the average charge per site is available, we turn off the fluctuations by assigning the same average charge to every site. A constant charge MC simulation is then performed. We make use of a model which accounts for the main fundamental aspects of a linear flexible polyelectrolyte i.e. proton binding, angle internal rotation, bond stretching and bending. Steric excluded volume and differentiated treatment for short-range and long-range interactions are also included in the model. It can be regarded as a kind of "minimal'' model in the sense that contains a minimum number of parameters but still preserving the atomistic detail. It is shown that, if fluctuations are activated, gauche state bond probabilities increase, and the persistence length decreases, so that the polymer becomes more folded. Macromolecular stretching is also analyzed in presence of CR (the charge depends on the applied force) and without CR (the charge is fixed to the value at zero force). The analysis of the low force scaling behavior concludes that Pincus exponent becomes pH-dependent. Both with and without CR, a transition from 1/2 at high pH-values (phantom chain) to 3/5 to low pH-values (Pincus regime), is observed. Finally, the intermediate force stretching regime is investigated. It is found that CR induces a moderate influence in the force-extension curves and persistence length (which in this force regime becomes force-dependent). It is thus concluded that the effect of CR on the stretching curves is mainly due to changes in the average charge at zero force. It is also found that, for the cases studied, the effect of steric excluded volume is almost irrelevant compared to electrostatic interactions.

We examine data assimilation coupling between meteorology and chemistry in the stratosphere from both weak and strong coupling strategies. The study was performed with the Canadian operational weather prediction Global Environmental Multiscale (GEM) model coupled online with the photochemical stratospheric chemistry developed at the Belgian Institute for Space Aeronomy, described in Part I. Here, the Canadian Meteorological Centre’s operational variational assimilation system was extended to include errors of chemical variables and cross-covariances between meteorological and chemical variables in a 3D-Var configuration, and we added the adjoint of tracer advection in the 4D-Var configuration. Our results show that the assimilation of limb sounding observations from the MIPAS instrument on board Envisat can be used to anchor the AMSU-A radiance bias correction scheme. Also, the added value of limb sounding temperature observations on meteorology and transport is shown to be significant. Weak coupling data assimilation with ozone-radiation interaction is shown to give comparable on meteorology whether a simplified linearized or comprehensive ozone chemistry scheme is used. Strong coupling data assimilation, using static error cross-covariances between ozone and temperature in a 3D-Var context, produced inconclusive results with the approximations we used. We have also conducted the assimilation of long-lived species observations using 4D-Var to infer winds. Our results showed the added value of assimilating several long-lived species, and an improvement in the zonal wind in the Tropics within the troposphere and lower stratosphere. 4D-Var assimilation also induced a correction of zonal wind in the surf zone and a temperature bias in the lower tropical stratosphere

We propose a new theory beyond the standard model of elementary-particle physics. Employing the concept of a quantized spacetime, our theory demonstrates that the zero-point energy of the vacuum alone is sufficient to create all the fields, including gravity, the static electromagnetic field, and the weak and strong interactions. No serious undetermined parameters are assumed. Furthermore, the relations between the forces at the quantum-mechanics level is made clear. Using these relations, we quantize Einstein’s gravitational equation and explain the Dark Energy in our universe. Beginning with the zero-point energy of the vacuum, and after quantizing Newtonian gravity, we combine the energies of a static electromagnetic field and gravity in a quantum spacetime. Applying these results to the Einstein gravity equation, we substitute the energy density derived from the zero-point energy in addition to redefining differentials in a quantized spacetime. We thus derive the quantized Einstein gravitational equation without assuming the existence of macroscopic masses. This also explains the existence of the Dark Energy in the universe. For the weak interaction, by considering plane-wave electron and the zero-point energy, we obtain a wavefunction that represents a β collapse. In this process, from a different point of view than Weinberg-Salam theory, we derive the masses of the W and Z bosons and the neutrino, and we calculate the radius of the neutron. For the strong interaction, we previously reported an analytical theory for calculating the mass of a proton by considering a specific linear attractive potential obtained from the zero-point energy, which agrees well with the measurements. In the present study, we calculate the strong interaction between two nucleons, i.e., the mass of the pi-meson. The resulting calculated quantities agree with the measurements, which verifies our proposed theory.

With reference to electromagnetic interaction and Abdus Salam’s strong (nuclear) gravity, 1) Square root of ‘reciprocal’ of the strong coupling constant can be considered as the strength of nuclear elementary charge. 2) ‘Reciprocal’ of the strong coupling constant can be considered as the maximum strength of nuclear binding energy. 3) In deuteron, strength of nuclear binding energy is around unity and there exists no strong interaction in between neutron and proton. $G_s\cong 3.32688\times {10}^{28}{\mathrm{m}}^3{\mathrm{kg}}^{-1}{\sec }^{-2}$ being the nuclear gravitational constant, nuclear charge radius can be shown to be, $R_0\cong \frac{2G_s{m}_p}{c^2}\cong 1.24\mathrm{fm}.$ $e_s\cong \left(\frac{G_s{m}_p^2}{\hslash c}\right)e\cong 4.716785\times {10}^{-19}\mathrm{C}$ being the nuclear elementary charge, proton magnetic moment can be shown to be, ${\mu }_p\cong \frac{e_s\hslash }{2m_p}\cong \frac{e{G}_s{m}_p}{2c}\cong 1.48694\times {10}^{-26}\mathrm{J}{.\mathrm{T}}^{-1}.$ ${\alpha }_s\cong {\left(\frac{\hslash c}{G_s{m}_p^2}\right)}^2\cong 0.1153795$ being the strong coupling constant, strong interaction range can be shown to be proportional to $\exp \left(\frac{1}{{\alpha }_s^2}\right).$ Interesting points to be noted are: An increase in the value of ${\alpha }_s$ helps in decreasing the interaction range indicating a more strongly bound nuclear system. A decrease in the value of ${\alpha }_s$ helps in increasing the interaction range indicating a more weakly bound nuclear system. From $Z\cong 30$ onwards, close to stable mass numbers, nuclear binding energy can be addressed with, ${\left(B\right)}_{A_s}\cong Z\times \left\{\left(\frac{1}{{\alpha }_s}+1\right)+\sqrt{\sqrt{30\times 31}}\right\}\left(m_n-m_p\right)c^2\approx Z\times 19.66\mathrm{MeV}.$ With further study, magnitude of the Newtonian gravitational constant can be estimated with nuclear elementary physical constants. One sample relation is, $\left(\frac{G_N}{G_s}\right)\cong \frac{1}{2}{\left(\frac{m_e}{m_p}\right)}^{10}\left[\sqrt{\frac{G_F}{\hslash c}}/\left(\frac{\hslash }{m_{e}c}\right)\right]$ where $G_N$ represents the Newtonian gravitational constant and $G_F$ represents the Fermi’s weak coupling constant. Two interesting coincidences are, ${\left(m_p/m_e\right)}^{10}\cong \exp \left(1/{\alpha }_s^2\right)$ and $2G_s{m}_e/c^2\cong \sqrt{G_F/\hslash c}.$

This paper first explains the theory of the origin of force from a new perspective, explaining that the origin of all forces is caused by the "missing of energy in the time dimension" (non-conservation in a certain period of time), and constructs a unified theory of force on this basis. Then, through the unified theory of force, the following phenomena are reinterpreted: (1) The common origin of the inverse square law of gravitation and electromagnetic force; (2) The origin of the repulsive force of electrons and protons at small distances; (3) The mystery of the rotation curve of galaxies and the new explanation of "dark matter"; (4) explain the origin of the magnetic field; (5) Predict that moving magnets have the same radiation effect as charged particles (electron magnetic moments also have radiation effects); (6) Reinterpret Maxwell's displacement current; (7) Reinterpret synchrotron radiation and bremsstrahlung; (8) Explain that the strong and weak forces are short-range forces; (9) Predict the true structure of black holes and the origin of gravitational waves;(10) Possible explanation for the origin of the speed of light.

There are unsolved problems related to inflation, gravity, dark matter, dark energy, missing antimatter, and the birth of the universe. Some of them can be better answered by assuming the existence of aether and hypoatoms. Both were created during the inflation in the very early universe. While aether forms vacuum, hypoatoms, composed of both matter and antimatter and believed to be neutrinos, form all observable matter. In vacuum, aether exists between the particle-antiparticle dark matter form and the dark energy form in a dynamic equilibrium: A + A-bar = gamma + gamma. The same reaction stabilizes hypoatoms and generates a 3-dimensional sink flow of aether that causes gravity. Based on the hypoatom structure, the singularity does not exist inside a black hole; the core of the black hole is a hypoatom star or neutrino star. By gaining enough mass, ca. 3 X 10²² M_sun, to exceed neutrino degeneracy pressure, the black hole collapses or annihilates into the singularity, thus turning itself into a white hole or a Big Bang. The universe is anisotropic. Its center, or where the Big Bang happened, is at 0.66^+0.03_-0.01 times the radius of the observable universe at Galactic coordinates (l, b) = (286⁺¹⁰_-10, -43⁺⁷_-6). If we look from the Local Group to the center of the universe, the universe is rotating clockwise.