Exponent Terms Research Articles

Positive Solution for an Elliptic System with Critical Exponent and Logarithmic Terms

Positive Solution for an Elliptic System with Critical Exponent and Logarithmic Terms

Seismogram Synthesis of Multi-Scale Layered Transversely Isotropic Saturated Half-Space Using a Revised Stiffness Matrix Method

Seismogram synthesis for a multi-scale transversely isotropic (TI) saturated layered half-space subjected to seismic dislocation source is studied by a revised stiffness matrix method in this paper. First, based on Biot’s theory of wave propagation in a fluid-saturated porous solid, the governing equations of motion in TI saturated medium are transformed by Fourier–Hankel integral transform. Next, the contribution of the seismic dislocation source can be imposed in the form of body forces. Then, the global stiffness matrix with well-conditioned properties is obtained by assembling each layer’s exact stiffness matrix, which can avoid accumulating errors in the matrix transmission. By extracting the divergence exponent terms of the stiffness matrix concerning thickness and wavenumber, the bottleneck of simulating the wave propagation process in multi-scale half-space, where the layer thickness and velocity vary sharply from the near-surface region to the crustal zone, could be overcome, and a fast and accurate seismogram synthesis could be achieved. The accuracy of the proposed method is verified by comparing the calculated results with those of published literature. Finally, the method is applied to the multi-scale system with realistic superficial fine layers. Numerical examples regarding thickness, TI properties and porosity are presented to illustrate the effects of superficial layers on ground motion. The numerical results show that the surface properties affect the wave propagation and energy distribution, leading to variable impacts on the surface response in different directions; this phenomenon is more pronounced in the case of high-frequency ground motions.

Dynamics of Heat Transfer in Magneto-Micropolar Fluids Flow Past a Vertically Expanding Sheet Featuring Prescribed Wall Temperature (PWT)

This study assesses the motion and the dynamics of heat propagation in magneto-micropolar fluid along a sheet which vertically stretches on a two-dimensional plane in a porous material. The heat distribution is developed and evaluated under the condition of the prescribed wall temperature, constant magnetic field, thermal radiation, variable heat source and viscous dissipation. The main equations are re-formulated from partial to ordinary derivatives using similarity tools and consequently solved numerically by shooting and the Runge-Kutta Fehlberg approach. The parameters of interest are presented graphically to demonstrate their reactions on the velocity profiles, thermal field and heat transfer mechanism of the problem. The outcomes of the current investigation reveal that the heat transfer appreciates in the presence of higher Prandtl number, temperature exponent term and material parameter but decreases as the magnetic field term soars.Besides, the heat boundary structure expands and heat spread occurs as the thermal radiation, magnetic field and Eckert number terms escalates but a reverse trend is encountered as the Prandtl number, material micropolar term, Grashof number and heat exponent terms grows in magnitude. Under some limiting scenarios, the obtained data strongly correspond to the published studies in the open literature.

On a nonlinear system of plate equations with variable exponent nonlinearity and logarithmic source terms: Existence and stability results

<abstract><p>In this paper, we consider a coupling non-linear system of two plate equations with logarithmic source terms. First, we study the local existence of solutions of the system using the Faedo-Galerkin method and Banach fixed point theorem. Second, we prove the global existence of solutions of the system by using the potential wells. Finally, using the multiplier method, we establish an exponential decay result for the energy of solutions of the system. Some conditions on the variable exponents that appear in the coupling functions and the involved constants that appear in the source terms are determined to ensure the existence and stability of solutions of the system. A series of lemmas and theorems have been proved and used to overcome the difficulties caused by the variable exponent and the logarithmic nonlinearities. Our result generalizes some earlier related results in the literature from the case of only constant exponent of the nonlinear internal forcing terms to the case of variable exponent and logarithmic source terms, which is more useful from the physical point of view and needed in several applications.</p></abstract>

Climatic and hydrogeomorphic controls on sediment characteristics in the southern Sierra Nevada

The magnitude of sediment yield from headwater catchments is controlled by the interactions among hydrology, geomorphology, and soil disturbance. In montane regions like the Sierra Nevada, snow is one of the main factors controlling the timing and magnitude of hydrological fluxes. However, the role of snow on modulating spatial and temporal variation of sediment yield remains unclear. Using 120-site years of sediment yield data from 10 headwater catchments (drainage area 50–475 ha), we examined the sediment yield characteristics across an elevational gradient (1,777–2,381 m elevation) in the southern Sierra Nevada. Across space and time, we calculated an average annual suspended sediment yield of 62 ± 147 Mg/km2. In contrast, the measured mean annual bedload yield from the study catchments was small, 1.1 ± 2.4 Mg/km2. A linear mixed-effects model showed that maximum annual discharge alone can only explain 24 % (marginal R2 = 0.24) of the variance in sediment yield. Similarly, the hypsometric integral, which is often used as a metric for erosion susceptibility, showed no predictive power (marginal R2 = 0.005). As much as 65 % of the variance in sediment yield can be explained by fixed effects when snow related drivers (i.e., center of flow timing and aspect) were included in the model along with maximum annual discharge, suggesting a strong influence of snowmelt. Moreover, the relationship between area normalized suspended sediment yield (Qs) and unit discharge (Q) was significantly different between rain and snow events (p = 0.001). Both average slope (α) and exponent (β) terms of the Qs=αQβ relationship across the ten catchments were higher for rainfall (α=5.8,β=1.84) than those for snowmelt (α=3.1,β=1.77) events. As the erosion severity and power were higher during rainfall than the snowmelt events, a shift in the precipitation form from snow to rain under a warming climate will likely increase sediment yield. These results provide critical insights on background sediment yield in the southern Sierra Nevada and likely changes under future climate.

Fixed-time projective synchronization of delayed memristive neural networks via aperiodically semi-intermittent switching control

This paper studies the fixed-time projective synchronization problem for a class of delayed memristive neural networks via aperiodically semi-intermittent switching control. Instead of using the common traditional controller containing two power exponent terms, a novel aperiodically semi-intermittent switching controller is designed, containing only one power exponent term which serves as a function of the state of the error system. With the aid of the extended differential inclusion framework, the analysis method, and the inequality technique, some new sufficient conditions are derived to guarantee fixed-time projective synchronization for the considered systems. Compared with periodically semi-intermittent control methods, the control time width of each section in aperiodically semi-intermittent control can be adjusted. Especially, the settling time is directly reckoned, which is closely related to the number of neurons and the maximum ratio of the second subinterval span in each non-periodic span to all non-periodic time spans rather than the initial value. Additionally, the projection synchronization has a strong practicality, as the projection coefficient can be adjusted for different needs instead of being fixed. Meanwhile, the study improves some previous related works. Ultimately, a numerical example is given to substantiate the correctness of the obtained results.

Blow-up for a viscoelastic von Karman equation with strong damping and variable exponent source terms

In this article, we deal with a strongly damped von Karman equation with variable exponent source and memory effects. We investigate blow-up results of solutions with three levels of initial energy such as non-positive initial energy, certain positive initial energy, and high initial energy. Furthermore, we estimate not only the upper bound but also the lower bound of the blow-up time.

Homogenous shear modulus reconstruction at multiple frequencies using the DICE method

The non-invasive technique of Digital Image Correlation Elastography was used to estimate the viscoelastic properties of a tissue-mimicking silicone phantom. Harmonic motion fields were generated over a frequency range of 40–90 ​Hz and measured at the phantom surface by 4 stereoscopic camera pairs. The storage modulus (G’) and the loss modulus (G”) were successfully reconstructed from the measured surface displacement data. Low variations were found for repeated measurements and the frequency dependent viscoelastic behavior was highly consistent across multiple excitation waveforms. Power-law analysis of the two viscoelastic properties provided exponent terms consistent with tissue-like materials.

1D electromagnetic response modeling with arbitrary source-receiver geometry based on vector potential and its implementation in MATLAB

The calculation of 1D electromagnetic (EM) responses of dipole sources is considered as a fundamental computation kernel in several geophysical applications, ranging from EM sensitivity analysis to high-dimensional EM forward modeling and inversion. To solve the EM responses caused by electric and magnetic dipoles with different orientations, we have developed an open-source MATLAB code for a complete treatment, which allows for an arbitrary transmitter-receiver geometry in the multilayered earth. A complete derivation of the EM field expressions with simplified exponent terms for all traditional dipole sources is accomplished based on the vector potential differential equations. The quadrature-with-extrapolation algorithm is adapted for calculating the integration of Bessel functions to produce EM responses with a desired accuracy. The MATLAB digit functions are used to alleviate the instability problem caused by the exponent terms. The accuracy of our code is verified by comparing the results with the published data and self-checks on an infinite thin sheet. A layered model is designed to test the capability of our code for all traditional dipole sources by placing the receiver at different depths. In the end, we apply our code to marine and airborne controlled-source EM explorations. The results indicate that our code is accurate and can potentially be used to calculate 1D EM responses for general EM problems.

Nonlocal Elliptic Systems Involving Critical Sobolev-Hardy Exponents and Concave-convex Nonlinearities

In this paper, a system of fractional elliptic equation is investigated, which involving fractional critical Sobolev-Hardy exponent and concave-convex terms. By means of variational methods and analytic techniques, the existence and multiplicity of positive solutions to the system is established.

On nonhomogeneous p-Laplacian elliptic equations involving a critical Sobolev exponent and multiple Hardy-type terms

On nonhomogeneous p-Laplacian elliptic equations involving a critical Sobolev exponent and multiple Hardy-type terms

Positive Radial Solutions for a Class of Semilinear Elliptic Problems Involving Critical Hardy-Sobolev Exponent and Hardy Terms

In this paper, we investigate the solvability of a class of semilinear elliptic equations which are perturbation of the problems involving critical Hardy-Sobolev exponent and Hardy singular terms. The existence of at least a positive radial solution is established for a class of semilinear elliptic problems involving critical Hardy-Sobolev exponent and Hardy terms. The main tools are variational method, critical point theory and some analysis techniques.

Continued Logarithms and Associated Continued Fractions

ABSTRACTWe investigate some of the connections between continued fractions and continued logarithms. We study the binary continued logarithms as introduced by Bill Gosper and explore two generalizations of the continued logarithm to base b. We show convergence for them using equivalent forms of their corresponding continued fractions. Through numerical experimentation, we discover that, for one such formulation, the exponent terms have finite arithmetic means for almost all real numbers. This set of means, which we call the logarithmic Khintchine numbers, has a pleasing relationship with the geometric means of the corresponding continued fraction terms. While the classical Khintchine's constant is believed not to be related to any naturally occurring number, we find surprisingly that the logarithmic Khintchine numbers are elementary.

A new method for estimating parent rock trace element concentrations from zircon

Zircon/bulk rock REE partition coefficients from natural samples correlate with REE concentration in zircon. The correlation is the strongest for the LREE and diminishes with decreasing ionic radius. The relationship between partition coefficient and REE concentration in zircon can be modeled as a power law and the coefficient (α) and exponent (β) terms for each of the REE are empirically determined using new and previously published data. A series of independent tests show that using variable partition coefficients based on the reported α and β terms commonly results in more accurate estimates of bulk rock REE concentrations than average partition coefficients, particularly for the LREE that have previously been difficult to constrain. These results provide a way to account for highly variable REE concentrations in zircon, which may be controlled by numerous processes such as surface enrichment, complex substitution mechanisms, or accidental sampling of sub-microscopic inclusions. The proposed method for estimating bulk rock REE concentrations is especially well-suited to detrital zircon investigations where there is no information available on the parent rock composition.

Bridging hydraulic diffusivity from aquifer to particle-size scale: a study on loess sediments from southwest Hungary

In situ megascale hydraulic diffusivities (D) of a confined loess aquifer were estimated at various scales (10 ≤ L ≤ 1500 m) by a finite difference model, and laboratory microscale diffusivities of a loess sample by empirical formulas. A scatter plot reveals that D fits to a single power function of L, providing that microscale diffusivities are assigned to L = 1 m and that differences in diffusivity observed between micro- and megascales are assigned to medium heterogeneity appraised by variations in the curvature and slope of natural hydraulic head waves propagating through the aquifer. Subsequently, a general power relationship between D and L is defined where the base and exponent terms stand for the aquifer storage capability under a confined regime of flow, for the microscale hydraulic conductivity and specific yield of loess, and for the changes in curvature and slope of hydraulic head waves relative to values defined at unit scale.Editor Z.W. Kundzewicz

Automated metamodel generation for Design Space Exploration and decision-making – A novel method supporting performance-oriented building design and retrofitting

Design and retrofitting of buildings for high performance in terms of low consumption of energy and exergy requires the examination of a large number of design variants, including time-consuming simulation. Metamodels (surrogate models) based on the Response Surface Method (RSM) can solve this time problem by shifting computational effort for simulation from within a design process to a prior time. However, traditional metamodelling by RSM with second-order polynomials performs well only for selected problems and requires mathematical and technical understanding and manual adjustment by the user. To generate models without user interaction, the paper presents a novel method for automatically generating a higher-quality mathematical structure of the metamodel. With minimal user interaction, the method searches for all degrees of interaction and allows for simple definition of high order polynomials. The primary component of this method is an algorithm that determines the mathematical structure of the metamodel by composing an exponent matrix step-by-step while minimising the modelling error. First, we employ standard mathematical test functions to demonstrate the method’s ability to identify models with up to six interacting variables; these functions determine its performance and limitations. An important observation is that the number of simulation experiments needs to be 1.5 to 2 times the number of exponent terms. Second, we apply the method to the design decisions and respective simulation data of a parametric Design Space Exploration (DSE) for an example case of an office building retrofit. This application demonstrates that the method improves the accuracy in cross-validation to an error of 7.2% for the total energy consumption, whereas the standard static RSM leads to an error of 35.9% (26.8% with interactions). Additional analyses demonstrate the benefits and limitations of metamodels for separated heating and cooling loads, as well as exergy. One benefit of applying the method is a quick-responding performance model. The use of this model is illustrated with a tool mock-up. A second benefit is obtaining global knowledge of the design space, as derived from interpreting the mathematical structure of the metamodel, i.e., the exponent and coefficient matrices. This structure reveals the quantitative impacts of factors and their interactions, and it allows identifying different design strategies, which is valuable for high-performance building design and retrofitting.

Multiplicity result for some nonlocal anisotropic equation via nonsmooth critical point theory approach

Abstract In this paper we study quasilinear problems involving variable exponent growth conditions and nonlocal terms on the whole space R N . A multiplicity result is established. All the coefficients involved in the terms of the equation depend both on the variable x and the unknown function u. Our main argument is nonsmooth critical point theory.

Elliptic systems involving the critical exponents and potentials

In this paper, we investigate a singular elliptic system, which involves the critical Sobolev exponent and multiple Hardy-type terms. By employing variational methods, the existence of its positive solutions is established. By the Moser iteration method, some asymptotic properties of its nontrivial solutions at the singular points are verified.

Ground states of the quasilinear problems involving the critical Sobolev exponent and potentials

In this paper we study a quasilinear elliptic problem which involves the critical Sobolev exponent and multiple Hardy-type terms. By the analytic technics and variational methods, the existence and nonexistence of the ground states for the problem is established under certain assumptions.

Positive solutions for Neumann elliptic problems involving critical Hardy–Sobolev exponent with boundary singularities

In this paper we consider the existence of positive solution for some semilinear elliptic equations with Neumann boundary condition involving a critical Hardy–Sobolev exponent and Hardy terms with boundary singularities. Using mountain pass lemma without (PS) condition and the strong maximum principle, we get the existence of a positive solution.