Behavior Of Function Research Articles

Analytical solution for Ξ-type qutrit interacting nonlinearly with a cavity field through Kerr-like medium with intrinsic noise present

This paper presents an analytical solution for the Milburn equation describing the interaction between a 3-level atom in a [Formula: see text]-configuration, a cavity field, and a nonlinear Kerr-like medium. The Hamiltonian of the model is presented in the rotating wave approximation (RWA) under specific conditions. The obtained analytical solution accounts for the impact of intrinsic decoherence and nonlinear terms on the dynamics of various physical phenomena. Specifically, we explore the behavior of atomic population, entropy functions, entanglement, negativity, and correlation, highlighting their dependence on the physical parameters of the [Formula: see text]-level atom-cavity system.

Existence of static wormhole solutions in [formula omitted] gravity

This study explores wormhole solutions by analyzing energy conditions within the framework of f(R,A) modified theory of gravity, which is derived from the inverse of the Ricci tensor. For this purpose, we consider a linear model, defined as f(R,A)=R+αA, where R denotes the Ricci scalar, α is the coupling parameter, and A represents the anticurvature scalar. We further introduce a specific form of shape function to examine the characteristics of energy conditions for the distribution of anisotropic matter. Moreover, we examine the behavior of shape function and energy conditions for both isotropic and barotropic matter content as well. Through the selection of suitable parameter values, it has been shown that energy conditions are breached in some specified regions, indicating the existence of exotic matter. The thorough analysis confirms the presence of wormhole geometries in f(R,A) gravity.

Thermal Casimir effect for a Dirac field on flat space with a nontrivial circular boundary condition

This work investigates the thermal Casimir effect associated with a massive spinor field defined on a four-dimensional flat space with a circularly compactified spatial dimension whose periodicity is oriented along a vector in the xy plane. We employ the generalized zeta function method to establish a finite definition for the vacuum free energy density. This definition conveniently separates into the zero-temperature Casimir energy density and additional terms accounting for temperature corrections. The structure of existing divergences is analyzed from the asymptotic behavior of the spinor heat kernel function and removed in the renormalization by subtracting scheme. The only non-null heat coefficient is the one associated with the Euclidean divergence. We also address the need for a finite renormalization to treat the ambiguity in the zeta function regularization prescription associated with this Euclidean heat kernel coefficient and ensure that the renormalization procedure is unique. The high- and low-temperature asymptotic limits are also explored. In particular, we explicitly show that free energy density lacks a classical limit at high temperatures, and the entropy density agrees with the Nernst heat theorem at low temperatures. Published by the American Physical Society 2024

Exact equilibrium properties of square-well and square-shoulder disks in single-file confinement.

This study investigates the (longitudinal) thermodynamic and structural characteristics of single-file confined square-well and square-shoulder disks by employing a mapping technique that transforms the original system into a one-dimensional polydisperse mixture of nonadditive rods. Leveraging standard statistical-mechanical techniques, exact results are derived for key properties, including the equationof state, internal energy, radial distribution function, and structure factor. The asymptotic behavior of the radial distribution function is explored, revealing structural changes in the spatial correlations. Additionally, exact analytical expressions for the second virial coefficient are presented. The theoretical results for the thermodynamic and structural properties are validated by our Monte Carlo simulations.

Temporal scaling theory for bursty time series with clusters of arbitrarily many events.

Long-term temporal correlations in time series in a form of an event sequence have been characterized using an autocorrelation function that often shows a power-law decaying behavior. Such scaling behavior has been mainly accounted for by the heavy-tailed distribution of interevent times, i.e., the time interval between two consecutive events. Yet, little is known about how correlations between consecutive interevent times systematically affect the decaying behavior of the autocorrelation function. Empirical distributions of the burst size, which is the number of events in a cluster of events occurring in a short time window, often show heavy tails, implying that arbitrarily many consecutive interevent times may be correlated with each other. In the present study, we propose a model for generating a time series with arbitrary functional forms of interevent time and burst size distributions. Then, we analytically derive the autocorrelation function for the model time series. In particular, by assuming that the interevent time and burst size are power-law distributed, we derive scaling relations between power-law exponents of the autocorrelation function decay, interevent time distribution, and burst size distribution. These analytical results are confirmed by numerical simulations. Our approach helps to rigorously and analytically understand the effects of correlations between arbitrarily many consecutive interevent times on the decaying behavior of the autocorrelation function.

Ruminal acidosis and its definition: A critical review

Ruminal acidosis occurs as a continuum of disorders, stemming from ruminal dysbiosis and disorders of metabolism, of varying severity. The condition has a marked temporal dynamic expression resulting in cases expressing quite different rumen concentrations of VFA, lactic acid, ammonia, and rumen pH over time. Clinical ruminal acidosis is an important condition of cattle and subclinical ruminal acidosis (SRA) is very prevalent in many dairy populations with estimates between 10 to 26% of cows in early lactation. Estimates of the duration of a case suggest the lactational incidence of the condition may be as high as 500 cases per 100 cows in the first 100 d of lactation. Historical confusion about the etiology and pathogenesis of ruminal acidosis led to definitions that are not fit for purpose as acidic ruminal conditions solely characterized by ruminal pH determination at a single point fail to reflect the complexity of the condition. Use of a model, based on integrated ruminal measures including VFA, ammonia, lactic acid, and pH, for evaluating ruminal acidosis is fit for purpose, as indicated by meeting postulates for assessing metabolic disease, but requires a method to simplify application in the field. While it is likely that this model, that we have termed the Bramley Acidosis Model (BAM), will be refined, the critical value in the model is that it demonstrates that ruminal acidosis is much more than ruminal pH. Disease, milk yield and milk composition are more associated with the BAM than rumen pH alone. Two single VFA, propionate and valerate are sensitive and specific for SRA, especially when compared with rumen pH. Even with the use of such a model, astute evaluations of the condition whether in experimental or field circumstances will be aided by ancillary measures that can be used in parallel or in series to enhance diagnosis and interpretation. Sensing methods including rumination detection, behavior, milk analysis, and passive analysis of rumen function have the potential to improve the detection of SRA; however, these may advance more rapidly if SRA is defined more broadly than by ruminal pH alone.

The weakly bound states in Gaussian wells: From the binding energy of deuteron to the electronic structure of quantum dots

AbstractGaussian potentials serve as a valuable tool for the comprehensive modeling of short‐range interactions, spanning applications from Nuclear Physics to the artificial confinement of electrons within quantum dots. This study focuses on examining the lowest states within Gaussian wells, with particular emphasis on the weakly bound regime. The analysis delves into the asymptotic behavior of the exact wave function at both small and large distances, motivating the development of a few parametric Ansatz which is locally accurate and yields to a fast convergent basis set. To validate its efficacy, we assess its convergence rate using a toy model of Nuclear Physics, specifically for Deuteron. Furthermore, we employ the expansion of the energy close to the threshold to derive an analytical formula for the binding energy of the Deuteron whose accuracy improves as the effective parameter approaches the critical. To conclude our study, we test the ability of our Ansatz to describe relativistic corrections into a quantum dot and its performance as an orbital in the exploration of the electronic structure of a two‐electron quantum dot.

Analysis of different objective functions in petroleum field development optimization

Oilfield development optimization plays a vital role in maximizing the potential of hydrocarbon reservoirs. Decision-making in this complex domain can rely on various objective functions, including net present value (NPV), expected monetary value (EMV), cumulative oil production (COP), cumulative gas production (CGP), cumulative water production (CWP), project costs, and risks. However, EMV is often the main function when optimization is performed under uncertainty. The behavior and performance of different objective functions has been investigated in this paper, when EMV is the primary criterion for optimization under reservoir and economic uncertainty. One of the goals of this study is to provide insights into the advantages and limitations of employing EMV as the sole objective function in oil field development decision-making. The designed optimization problem included sequential optimization of design variables including well positions, well quantity, well type, platform capacity, and internal control valve placements. A comparative analysis is presented, contrasting the outcomes obtained from optimizing the EMV-based objective function against traditional objective functions. The study underscores the importance of incorporating multiple objective functions alongside EMV to guide decision-making in oilfield development. Potential benefits in minimizing CGP and CWP are revealed, aiding in the mitigation of environmental impact and optimization of resource utilization. A strong correlation between EMV and COP is identified, highlighting EMV’s role in improving COP and RF.

Free boundary regularity and support propagation in mean field games and optimal transport

We study the behavior of solutions to the first-order mean field games system with a local coupling, when the initial density is a compactly supported function on the real line. Our results show that the solution is smooth in regions where the density is strictly positive, and that the density itself is globally continuous. Additionally, the speed of propagation is determined by the behavior of the cost function near small values of the density. When the coupling is entropic, we demonstrate that the support of the density propagates with infinite speed. On the other hand, for a power-type coupling, we establish finite speed of propagation, leading to the formation of a free boundary. We prove that under a natural non-degeneracy assumption, the free boundary is strictly convex and enjoys C1,1 regularity. We also establish sharp estimates on the speed of support propagation and the rate of long time decay for the density. Moreover, the density and the gradient of the value function are both shown to be Hölder continuous up to the free boundary. Our methods are based on the analysis of a new elliptic equation satisfied by the flow of optimal trajectories. The results also apply to mean field planning problems, characterizing the structure of minimizers of a class of optimal transport problems with congestion.

Contribution functions for quantitative bipolar argumentation graphs: A principle-based analysis

We present a principle-based analysis of contribution functions for quantitative bipolar argumentation graphs that quantify the contribution of one argument to another. The introduced principles formalise the intuitions underlying different contribution functions as well as expectations one would have regarding the behaviour of contribution functions in general. As none of the covered contribution functions satisfies all principles, our analysis can serve as a tool that enables the selection of the most suitable function based on the requirements of a given use case.

Holographic Einstein ring of a charged Rastall AdS black hole with bulk electromagnetic field* *Supported by the National Natural Science Foundation of China (11675140, 11705005, 12375043), the Innovation and Development Joint Foundation of Chongqing Natural Science Foundation (CSTB2022NSCQ-LZX0021), and the Basic Research Project of Science and Technology Committee of Chongqing (CSTB2023NSCQ-MSX0324)

We study the Einstein images of a charged Rastall AdS black hole (BH) within the fabric of AdS/CFT correspondence. Considering the holographic setup, we analyze the amplitude of the total response function for various values of model parameters. With an increase in parameter λ and temperature T, the amplitude of the response function decreases, while it increases with an increase in electric charge e and chemical potential μ. The influence of frequency ω also plays an important role in the bulk field, as it is found that decreasing ω leads to an increase in the periods of the waves, which means that the amplitude of the response function also depends on the wave source. The relation between T and the inverse of the horizon for various values of parameter λ is interpreted under fixed values of other involved parameters. These, in turn, affect the behavior of the response function and the Einstein ring, which may be used to differentiate the present study from previous ones. We construct the holographic images of the BH in bulk via a special optical system. The results show that the Einstein ring always appears with concentric stripes at the position of the north pole, and this ring transforms into a luminosity-deformed ring or bright light spot when the distant observer lies away from the north pole. Finally, we discuss the influence of the associated parameters on the Einstein ring radius, which is consistent with wave optics.

The scandium effect in Gd-rich BMGs: How and why does this ingredient work better than others?

Scandium is commonly known as a superior modifier for most alloys and compounds, substantially improving their phase stability and physical properties, as well as glass-forming ability. A special case is rare-earth based magnetic metallic glasses, which are an unique platform for intra-elemental substitution due to the chemical affinity of the lanthanides, yttrium and scandium. The latter, used as an alloying additive and being the smallest atom among the entire family, can drastically affect structure formation and magnetism in these metallic glasses. The main issue is that its modifying role is not well understood due to scarce information on the scandium effects in such rare-earth systems. This study is focused on thermal and magnetic analysis of some popular Gd-based BMGs redesigned with minor Sc additives. Here, we consider temperature behavior of specific heat capacity and entropy functions for both amorphous and crystalline phases to estimate the scandium effect on GFA and thermal stability of the glasses. As well, we inspect magnetic and magnetocaloric properties of these BMGs to understand whether this very expensive metal can radically improve the alloy magnetism and make these materials suitable for potential applications.

A proof of the power rule by Taylor's theorem

The standard way of proving the power rule for irrational exponents is to use the differentiation of exponential and logarithmic functions. We derive the power rule for irrational exponents from that for rational exponents by using Taylor's theorem, without relying on exponential and logarithmic functions. This approach is straightforward in view of extending the definition of the power function from rational exponents to irrational exponents. It also helps students to understand that Taylor's theorem is useful to investigate the local behaviour of functions.

Exploring Pointed Sets and their Role in the Category of Pointed F-Sets: A Study on Kernels in the Category F-Sets

This paper delves into the realm of pointed sets and their significance within the framework of algebraic structures, particularly focusing on their role in the category of pointed F-sets. Pointed sets, characterized by a single nullary operation identifying a base point, serve as fundamental components in algebraic reasoning, with applications in various mathematical domains. The category of pointed F-sets, comprising pointed sets as objects and pointed F-associations as morphisms, offers a rich ground for exploring the behavior of specialized functions. Central to our investigation is the demonstration of kernels within the category F-sets, elucidating the foundational properties and structural insights underlying pointed sets and their associations.

Multiple testing of interval composite null hypotheses using randomized p-values

Equivalence tests are statistical hypothesis testing procedures that aim to establish practical equivalence rather than the usual statistical significant difference. These testing procedures are frequent in “bioequivalence studies," where one would wish to show that, for example, an existing drug and a new one under development have comparable therapeutic effects. In this article, we propose a two-stage randomized (RAND2) p-value that depends on a uniformly most powerful (UMP) p-value and an arbitrary tuning parameter c∈[0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c\\in [0,1]$$\\end{document} for testing an interval composite null hypothesis. We investigate the behavior of the distribution function of the two p-values under the null hypothesis and alternative hypothesis for a fixed significance level t∈(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\in (0,1)$$\\end{document} and varying sample sizes. We evaluate the performance of the two p-values in estimating the proportion of true null hypotheses in multiple testing. We conduct a family-wise error rate control using an adaptive Bonferroni procedure with a plug-in estimator to account for the multiplicity that arises from our multiple hypotheses under consideration. The various claims in this research are verified using a simulation study and real-world data analysis.

Behavior of Correlation Functions in the Dynamics of the Multiparticle Quantum Arnol'd Cat.

The multi-particle Arnol'd cat is a generalization of the Hamiltonian system, both classical and quantum, whose period evolution operator is the renowned map that bears its name. It is obtained following the Joos-Zeh prescription for decoherence by adding a number of scattering particles in the configuration space of the cat. Quantization follows swiftly if the Hamiltonian approach, rather than the semiclassical approach, is adopted. The author has studied this system in a series of previous works, focusing on the problem of quantum-classical correspondence. In this paper, the dynamics of this system are tested by two related yet different indicators: the time autocorrelation function of the canonical position and the out-of-time correlator of position and momentum.

On the wavefunction cutoff factors of atomic hydrogen confined by an impenetrable spherical cavity

AbstractThe Schrödinger equation for the hydrogen atom enclosed by an impenetrable spherical cavity is solved through a Finite‐Differences approach to gain an insight on the actual nature and structure of the ansatz wavefunction cutoff factor widely used in an ad hoc manner in corresponding variational calculations to comply with the Dirichlet boundary conditions. The results of this work provide a theoretical foundation for the choice of the appropriate analytical cutoff functions that fulfill the boundary conditions. We find three different regions for the behavior of the cutoff functions. Small cavity radius where the cutoff function has a parabolic behavior, an intermediate region where the cutoff function is quasi‐linear, and a large cavity region where the cutoff function is a step‐like function. We deduce the traditional linear and quadratic cutoff functions used in the literature as well as its validity region for the confining radius. Finally, we provide a mathematical deduction of the exact cutoff function in terms of the nodal structure of the free hydrogenic wavefunctions and a relation to the Laguerre polynomials for some cavity radii where the free atomic energy level coincides with a confined energy level. We find that the cutoff function transit over several unconfined solutions in terms of its nodal structure as the system is compressed.

Multi-technique characterization and thermal degradation study of epoxy modified resins designed for multifunctional applications

Tetraglycidyl methylene dianiline (TGMDA) was mixed with 1,4-Butanediol diglycidyl ether (BDE) (in a 4:1 mass ratio) and with a stoichiometric amount of the curing agent diaminodiphenyl sulfone which was solubilized at 120 °C for 20 min in the liquid mixture TGMDA + BDE. The so obtained unfilled epoxy resin matrix, denoted as ER, was blended with glycidyl polyhedral oligomeric silsesquioxane and carbon nanotubes in suitable proportions to obtain binary and ternary mixtures. Characterization of the formulated materials was performed using different experimental techniques, such as Dynamic mechanical analysis, Thermogravimetry (TG), Field emission scanning electron microscopy. Furthermore, the investigation of the flame behavior was carried out by the limiting oxygen index and mass loss calorimeter measurements. Direct current measurements and investigation by Tunneling atomic force microscopy of the conductive nanodomain map allowed the evaluation of the electrical properties of the developed nanofilled systems. The TG data related to thermal decomposition of ER and its binary and ternary mixtures were processed according to isoconversional kinetic analysis by assuming a non-Arrhenian behavior of the temperature function, and lifetime prediction was estimated at suitable relatively low temperatures and possible relation between the thermal stability and the presence of each component was discussed. This method of kinetic analysis paves the way for the possibility of evaluating in a more realistic way, on the basis of thermal stability, the potential application of structural resins with primary load functions in contact with hot areas of aeronautical aircraft engines.

The approximation capabilities of Durrmeyer-type neural network operators

AbstractIn this paper, a new family of neural network (NN) operators is introduced. The idea is to consider a Durrmeyer-type version of the widely studied discrete NN operators by Costarelli and Spigler (Neural Netw 44:101–106, 2013). Such operators are constructed using special density functions generated from suitable sigmoidal functions, while the reconstruction coefficients are based on a convolution between a general kernel function $$\chi $$ χ and the function being reconstructed, f. Here, we investigate their approximation capabilities, establishing both pointwise and uniform convergence theorems for continuous functions. We also provide quantitative estimates for the approximation order thanks to the use of the modulus of continuity of f; this turns out to be strongly influenced by the asymptotic behaviour of the sigmoidal function $$\sigma $$ σ . Our study also shows that the estimates we provide are, under suitable assumptions, the best possible. Finally, $$L^p$$ L p -approximation is also established. At the end of the paper, examples of activation functions are discussed.

Radial behavior of Mahler functions

Many papers have been recently devoted to the study of the radial behavior as [Formula: see text] of transcendental r-Mahler functions holomorphic in the open unit disk. In particular, Bell and Coons showed in 2017 that, in a generic sense, r-Mahler functions behave like [Formula: see text] for some [Formula: see text] and [Formula: see text] is a real analytic function of [Formula: see text] such that [Formula: see text]. They did not provide a formula for [Formula: see text] which was made explicit only in a few examples of r-Mahler functions of orders 1 and 2, and for specific values of r. In this paper, we first provide an explicit expression of [Formula: see text] as an exponential of a Fourier series in the variable [Formula: see text] for every r-Mahler function of order 1. Then, extending to a large setting a method introduced by Brent–Coons–Zudilin in 2016 to compute [Formula: see text] associated to the Dilcher–Stolarsky function (a 4-Mahler function of order 2 in [Formula: see text]), we provide an explicit expression of [Formula: see text] for every r-Mahler function of order 2 under mild assumptions on the coefficients in [Formula: see text] of the underlying r-Mahler equations. This applies in particular to the generating function of the Baum–Sweet sequence. We do the same for r-Mahler functions solutions of inhomogeneous Mahler equations of order 1.