Behavior Of Function Research Articles

Stability analysis for a Rao-Nakra sandwich beam equation with time-varying weights and frictional dampings

<abstract><p>In this paper, we studied the asymptotic behavior of solutions for a Rao-Nakra sandwich beam equation with time-varying weights and frictional damping terms acting complementarily in the domain. We studied the effect of the three damping on the asymptotic behavior of the energy function. Under nonrestrictive on the growth assumption on the frictional damping terms, we established exponential and general energy decay rates for this system by using the multiplier approach. The results generalized some earlier decay results on the Rao-Nakra sandwich beam equation.</p></abstract>

WarmSPy: a numerical study of cosmological perturbations in warm inflation

We present WarmSPy, a numerical code in Python designed to solve for the perturbations' equations in warm inflation models and compute the corresponding scalar power spectrum at CMB horizon crossing. In models of warm inflation, a radiation bath of temperature T during inflation induces a dissipation (friction) rate of strength Q ∝ Tc /ϕm in the equation of motion for the inflaton field ϕ. While for a temperature-independent dissipation rate (c = 0) an analytic expression for the scalar power spectrum exists, in the case of a non-zero value for c the set of equations can only be solved numerically. For c > 0 (c < 0), the coupling between the perturbations in the inflaton field and radiation induces a growing (decaying) mode in the scalar perturbations, generally parameterized by a multiplicative function G(Q) which we refer to as the scalar dissipation function. Using WarmSPy, we provide an analytic fit for G(Q) for the cases of c = {3,1,-1}, corresponding to three cases that have been realized in physical models. Compared to previous literature results, our fits are more robust and valid over a broader range of dissipation strengths Q ∈ [10-7,104]. Additionally, for the first time, we numerically assess the stability of the scalar dissipation function against various model parameters, inflationary histories as well as the effects of metric perturbations. As a whole, the results do not depend appreciably on most of the parameters in the analysis, except for the dissipation index c, providing evidence for the universal behaviour of the scalar dissipation function G(Q).

On Tangential Boundary Behavior of Functions from Hardy Type Spaces

On Tangential Boundary Behavior of Functions from Hardy Type Spaces

On Logarithmic Holder Condition and Local Extrema of Power Takagi Functions

This paper studies one class of real functions, which we call Takagi power functions. Such functions have one positive real parameter; they are continuous, but nowhere differentiable, and are given on a real line using functional series. These series are similar to the series defining the continuous, nowhere differentiable Takagi function described in 1903. For each parameter value, we derive a functional equation for functions related to Takagi power functions. Then, using this equation, we obtain an accurate two-sides estimate for the functions under study. Next, we prove that for parameter values not exceeding 1, Takagi power functions satisfy the Holder logarithmic condition, and find the smallest value of the constant in this condition. As a result, we get the usual Hölder condition, which follows from the logarithmic Hölder condition. Moreover, for parameter values ranging from 0 to 1, we investigate the behavior of Takagi power functions in the neighborhood of their global maximum points. Then we show that the functions under study reach a strict local minimum on the real axis at binary-rational points, and only at them. Finally, we describe the set of points at which our functions reach a strict local maximum. The benefit of our research lies in the development of methods applicable to continuous functions that cannot be differentiated anywhere. This can significantly expand the set of functions being studied.

Corner transfer matrix approach to the Yang-Lee singularity in the two-dimensional Ising model in a magnetic field.

We study the two-dimensional (2D) Ising model in a complex magnetic field in the vicinity of the Yang-Lee edge singularity. By using Baxter's variational corner transfer matrix method combined with analytic techniques, we numerically calculate the scaling function and obtain an accurate estimate of the location of the Yang-Lee singularity. The existing series expansions for susceptibility of the 2D Ising model on a triangular lattice by Chan, Guttmann, Nickel, and Perk allowed us to substantially enhance the accuracy of our calculations. Our results are in excellent agreement with the Ising field theory calculations by Fonseca, Zamolodchikov, and the recent work by Xu and Zamolodchikov. In particular, we numerically confirm an agreement between the leading singular behavior of the scaling function and the predictions of the M_{2/5} conformal field theory.

The thermodynamic properties of diatomic gases. A rigorous analytical approach

The paper reports a closed-form analytical representation of the thermodynamic functions of diatomic molecules in the ground electronic state. For this purpose, analytical formulas for vibrational and rotational partition functions are derived. A rigorous method is proposed for factorizing the vibrational and rotational contributions to the ro-vibrational partition function using an average vibrational quantum number. The behavior of thermodynamic functions as a function of temperature is discussed for carbon monoxide as a test system. The reliability of the analytical method is verified by comparison with experimental data and with the direct summation method, which uses theoretical values of energy levels.

Proposal for Use of the Fractional Derivative of Radial Functions in Interpolation Problems

This paper presents the construction of a family of radial functions aimed at emulating the behavior of the radial basis function known as thin plate spline (TPS). Additionally, a method is proposed for applying fractional derivatives, both partially and fully, to these functions for use in interpolation problems. Furthermore, a technique is employed to precondition the matrices generated in the presented problems through QR decomposition. Similarly, a method is introduced to define two different types of abelian groups for any fractional operator defined in the interval [0,1), among which the Riemann–Liouville fractional integral, Riemann–Liouville fractional derivative, and Caputo fractional derivative are worth mentioning. Finally, a form of radial interpolant is suggested for application in solving fractional differential equations using the asymmetric collocation method, and examples of its implementation in differential operators utilizing the aforementioned fractional operators are shown.

DIRECT SOLUTION OF THE DYNAMICAL ELASTICITY PROBLEM FOR A QUARTER SPACE

The wave field of an elastic quarter space is constructed when one face is rigidly fixed and a dynamic normal compressive load is concentrated at the point on another face. The problem was solved by the direct application of the integral Laplace and Fourier transforms to the motion equations and the boundary conditions. This operation leads to the one-dimensional vector inhomogeneous boundary value problem with respect to unknown displacement’s transformants. The problem was solved using the matrix differential calculus. A fundamental matrix and a decreasing solution to the corresponding homogenous matrix equation were constructed with a basic residue theorem. A singular integral equation was obtained in the process by satisfying unrealized boundary condition. Weakly convergent part of the equation was summed up. The behavior of the unknown function had been analyzed based on its mechanical sense. The form of unknown function was expressed as a series on Laguerre polynomials. The original displacements’ field was found after an application the inverse integral transforms.

Two-step nucleation and crystal growth in a metastable solution

This study is concerned with a theory of two-step nucleation and growth of crystals in a metastable liquid. This mechanism is that crystalline nuclei formation occurs in dense liquid clusters suspended in the solution. These clusters contain higher solution concentration and viscosity, leading to a lower surface free energy barrier and faster phase transition route. The theory is based on growth laws of crystals during the two-step bulk phase transformation. At the initial stage, the crystals evolve in a diffusion-limited environment with almost unchanged supersaturation. At the second stage, they become larger, move beyond these clusters, and evolve in accordance with a hyperbolic tangent law. A generalized particle growth law joining the first and second stages is obtained by stitching the diffusion limited and hyperbolic tangent laws. On this basis, an integrodifferential model of the evolution of a polydisperse ensemble of crystals was formulated and solved. The crystal-size distribution function increases and the solution supersaturation remains practically unchanged until the particle size corresponds to a transition in the particle growth rate from a diffusion-limited branch to a hyperbolic tangent branch. This is followed by an increase in the crystal growth rate, a decrease in the distribution function and solution supersaturation. Then the distribution function increases up to the maximum size of crystals grown in the solution. A sufficiently long time interval of almost constant supersaturation and the N-shaped behavior of the distribution function are the consequences of a two-step nucleation and growth mechanisms.

Optimization of key joints

Key-joints are used in many drives and rotating elements in engineering practice. Nowadays decreasing the dead-weight of the structures is more and more important (in vehicle industry, aircrafts, etc.). This paper shows an optimization process for minimum weight design of key-joints. For the solution of the optimization problem a 3D grapho-analytical optimization process is used, based on the Kuhn–Tucker optimality criterium. During the graphic part of the process, three dimensional diagrams show the behaviour of the objective function and of the design constraints, which makes easy to read the optimum solution from the diagram. The final optimum solution is given in a table numerically too. For better demonstration, the finite element analysis of the three dimensional CAD model of the optimal joint is shown.

α-decay properties of superheavy nuclei with 117 ≤ Z ≤ 120 from the systematics of decay chains and isotopic chains* *Supported by the National Natural Science Foundation of China (12075121), the Natural Science Foundation of Jiangsu Province (BK20190067), and the Fundamental Research Funds for the Central Universities (30922010312)

Recently, the synthesis of new elements above Z = 118 has been a hot topic in nuclear physics. Meanwhile, the α-decay chain is expected to be the unique tool to identify these heaviest nuclei. We have systematically calculated the α-decay energies and half-lives on the same footing for superheavy nuclei (SHN) within the cluster model along with a slightly modified Woods-Saxon (W.S.) potential as the nuclear potential. Based on the available experimental data, the key radius parameter (R) in the α-core potential is determined via the systematic trend from the α-decay and isotopic chains. The α-decay energy ( ) values and half-lives are then obtained simultaneously for those unknown SHN in the range of 117 ≤ Z ≤ 120, during which the decay width is obtained using a new treatment for the asymptotic behavior of the α-core wave function. The theoretical values and experimental data are found to be in excellent agreement for the nuclei 117 and 118 regardless of the method used to determine the R parameter. Predicting the α-decay chains for new elements Z = 119 and Z = 120 can be useful in ongoing or forthcoming experiments.

Marginal propensity to save from workers of a commercialization and distribution business of mass products in Ciudad Del Este, Paraguay

Background: Keynes was the first economist to state that consumption depends fundamentally on income. Savings is equal to income minus consumption. Objective: To determine the behavior of the savings function of the workers of a company that distributes mass consumption products in Ciudad Del Este. Method: The sample (191) was calculated according to the population. The approach is a mixed, descriptive study, using surveys, based on a non- experimental and correlational design. For the analysis, the Dispersion Diagram, Pearson's Coefficient and Simple Linear Regression by the Ordinary Least Squares Method were used with the help of GRETL software. Results: The income obtained by the workers of a mass consumption products Distribution Company surveyed was identified and it was possible to recognize the expenses incurred by the members of the households. The relationship between income - savings expenditure and income - consumption has been analyzed and the existence of correlation between the variables studied was proved. The Marginal Propensity to save and the Marginal Propensity to Consume were determined. Conclusion: There is a direct/positive relationship between the income variable and the savings variable.

Jointly determining the point sources and obstacle from Cauchy data

A numerical method is developed for recovering both the source locations and the obstacle from the scattered Cauchy data of the time-harmonic acoustic field. First of all, the incident and scattered components are decomposed from the coupled Cauchy data by the representation of the single-layer potentials and the solution to the resulting linear integral system. As a consequence of this decomposition, the original problem of joint inversion is reformulated into two decoupled subproblems: an inverse source problem and an inverse obstacle scattering problem. Then, two sampling-type schemes are proposed to recover the shape of the obstacle and the source locations, respectively. The sampling methods rely on the specific indicator functions defined on target-oriented probing domains of circular shape. The error estimates of the decoupling procedure are established and the asymptotic behaviors of the indicator functions are analyzed. Extensive numerical experiments are also conducted to verify the performance of the sampling schemes.

An asymptotic expansion for the number of two-connected chord diagrams

We derive a functional relation between the generating functions of connected chord diagrams and two-connected chord diagrams. This relation enables us to calculate an asymptotic expansion for the number of two-connected chord diagrams on n chords. The asymptotic information obtained from this expansion refines the last established results and provides a simple alternative for calculating the asymptotic behaviour of the certain functions in (zero-dimensional) quantum electro-dynamics.

Decreasing behavior of the depth functions of edge ideals

Decreasing behavior of the depth functions of edge ideals

Boundary Behavior of Conformal Maps on Domains with Corners

This paper investigates the behavior of conformal maps near corner points and other non-differentiable boundary points of planar domains. Conformal maps that preserve angles locally will exhibit singular behavior when approaching corners or other less smooth parts of the boundary. We classify types of isolated corner singularities and characterize the magnitude, derivatives, and integral properties of analytic functions near such points. Explicit mappings are constructed between model domains with cusps, wedges, slits, and logarithmic-type corner points. The behavior of the mapping functions is analyzed as the boundary coordinates approach the singular points. We establish several theorems describing the boundary limits, convergence, boundary correspondences, and boundary integrals of these conformal maps on domains with corners. The mapping properties provide insight into the effect of geometric singularities on analytic functions in application areas such as physics, fluid flow, and engineering problems involving complex mappings. The boundary behavior classifications developed here expand the mathematical understanding of conformal maps on domains with sharp corners or discontinuities.

Majorana edge states in Kitaev chains of the BDI symmetry class

Majorana edge states in Kitaev chains possessing an effective time reversal symmetry with one fermionic site per unit cell are studied. For a semi-infinite chain the equations for the wave functions of Majorana zero modes can be reduced to a single Wiener-Hopf equation, which has an exact analytical solution. We use this solution to determine the asymptotic behaviors of the Majorana wave functions at large distances from the edge of the chain for several infinite-range models described in the literature with focus on a model with slow power-law falloff of pairing and hopping amplitudes. For these models we also determine the asymptotic behavior of the energy of the fermionic state composed of two Majorana modes in the limit of long (finite) Kitaev chains.

Impact of tidal environment on galaxy clustering in GAMA

ABSTRACT We constrain models of the galaxy distribution in the cosmic web using data from the Galaxy and Mass Assembly (GAMA) survey. We model the redshift-space behaviour of the 2-point correlation function (2pcf) and the recently proposed Voronoi volume function (VVF) – which includes information beyond two-point statistics. We extend the standard halo model using extra satellite degrees of freedom and two assembly bias parameters: αcen and αsat, which correlate the occupation numbers of central and satellite galaxies with their host halo’s tidal environment, respectively. We measure $\alpha _{\rm sat}=1.44^{+0.25}_{-0.43}$ and $\alpha _{\rm cen}=-0.79^{+0.29}_{-0.11}$ using a combination of 2pcf and VVF measurements, representing a detection of assembly bias at the 3.3σ (2.4σ) significance level for satellite (central) galaxies. This result remains robust to possible anisotropies in the halocentric distribution of satellites as well as technicalities of estimating the data covariance. We show that the growth rate (fσ8) deduced using models with assembly bias is about 7 per cent (i.e. 1.5σ) lower than if assembly bias is ignored. When projected on to the Ωm–σ8 plane, the model constraints without assembly bias overlap with Planck expectations, while allowing assembly bias introduces significant tension with Planck, preferring either a lower Ωm or a lower σ8. Finally, we find that the all-galaxy weak-lensing signal is unaffected by assembly bias, but the central and satellite sub-populations individually show significantly different signals in the presence of assembly bias. Our results illustrate the importance of accurately modelling galaxy formation for cosmological inference from future surveys.

Electrical resistivity and the Hall effect in the doped Mott-Hubbard material with strong spin-charge coupling

The kinetic characteristics of the doped Mott-Hubbard material are considered within the realistic spin-fermion model which takes into account the strong spin-charge coupling. The kinetic equation constructed on the basis of the mechanism of carrier scattering on the spin fluctuations is solved using the multi-moment method, which allows one to analyze the temperature behavior of nonequilibrium distribution function in the problems of electrical resistivity ρ and the Hall coefficient R H . The calculated dependences ρ(T) and R H (T) for the underdoped and optimally doped regimes demonstrate good qualitative agreement with the experimental data. In particular, the Hall coefficient calculated for the underdoped regime reproduces the experimentally observed sharp drop and even a change in sign at low temperatures.

Different glassy characteristics are related to either caging or dynamical heterogeneity.

Despite the enormous theoretical and application interests, a fundamental understanding of the glassy dynamics remains elusive. The static properties of glassy and ordinary liquids are similar, but their dynamics are dramatically different. What leads to this difference is the central puzzle of the field. Even the primary defining glassy characteristics, their implications, and if they are related to a single mechanism remain unclear. This lack of clarity is a severe hindrance to theoretical progress. Here, we combine analytical arguments and simulations of various systems in different dimensions and address these questions. Our results suggest that the myriad of glassy features are manifestations of two distinct mechanisms. Particle caging controls the mean, and coexisting slow- and fast-moving regions govern the distribution of particle displacements. All the other glassy characteristics are manifestations of these two mechanisms; thus, the Fickian yet non-Gaussian nature of glassy liquids is not surprising. We discover a crossover, from stretched exponential to a power law, in the behavior of the overlap function. This crossover is prominent in simulation data and forms the basis of our analyses. Our results have crucial implications on how the glassy dynamics data are analyzed, challenge some recent suggestions on the mechanisms governing glassy dynamics, and impose strict constraints that a correct theory of glasses must have.