Generalized Power Function Research Articles

Synthesis of intelligent fuzzy control in the space of bounded piecewise-continuous functions based on the combined maximum principle

It is shown that the solution of the problem of control synthesis using the Hamilton-Ostrogradskii principle leads to a variational inequality, from which the conditions for the maximum of the the generalized power function in the space of bounded piecewise continuous functions follow. It allows to find a feedback structure up to a synthesis function. Using the methods of the structure construction the nonlinear structures of the relay and continuous control laws are obtained. The proposed control method allows to avoid the mode with frequented switching. It consists of the following two stages. At the first stage, the control object is brought into the vicinity of the terminal state using the relay control law. In the second stage, the quasi-optimal continuous control is used. The uncertainty of the transition area size is resolved using fuzzy logic. The efficiency of the intelligent controls is demonstrated by the example of mathematical modeling of the system dynamics.

Properties of Drained Shear Strength of Expansive Soil Considering Low Stresses and Its Influencing Factors

The shallow failure of most expansive soil slopes undergoing wet–dry cycles occurs during or after long-term rainfall. According to the actual failure state of an expansive soil slope, the conventional direct shear equipment with low normal stresses was used to evaluate the effects of initial dry density, initial moisture content, and number of wet-dry cycle with or without loading on drained shear strength for the compacted expansive soil from Nanning, China. The saturated drained direct shear tests were also performed on the natural expansive soil specimens from Nanning by considering the influence of wet–dry cycles with loading and low normal stresses. Furthermore, consolidated drained triaxial compression tests on compacted specimens were conducted. The test results show that the drained shear strength of expansive soil was primarily affected by the magnitude of normal stresses, and also influenced by the initial moisture content, initial dry density, and number and type of wet–dry cycles. It is more realistic to consider the effect of loading during wet–dry cycles than that without loading. The effective drained shear strength envelopes with low normal stresses were plotted and well fitted by the generalized power function. At low normal stress levels, the drained shear strength still decreased with the increasing number of wet–dry cycles, and the effective cohesion intersects tended to be zero after being subjected to eight wet–dry cycles. This provided a theoretical basis for properly interpreting the shallow failure of expansive soil slopes.

Stochastic resonance in overdamped systems with fractional power nonlinearity

The stochastic resonance phenomenon in overdamped systems with fractional power nonlinearity is thoroughly investigated. The first kind of nonlinearity is a general fractional power function. The second kind of nonlinearity is a fractional power function with deflection. For the first case, the response is clearly divergent for some fractional exponent values. The curve of the spectral amplification factor versus the fractional exponent presents some discrete regions. For the second case, the response will not be divergent for any fractional exponent value. The spectral amplification factor decreases with the increase in the fractional exponent. For both cases, the nonlinearity is the necessary ingredient to induce stochastic resonance. However, it is not the sufficient cause to amplify the weak signal. On the one hand, the noise cannot induce stochastic resonance in the corresponding linear system. On the other hand, the spectral amplification factor of the nonlinear system is lower than that of the corresponding linear system. Through the analysis carried out in this paper, we are able to find that the system with fractional deflection nonlinearity is a better stochastic resonance system, especially when an appropriate exponent value is chosen. The results in this paper might have a certain reference value for signal processing problems in relation with the stochastic resonance method.

Shallow Sliding Failure Prediction Model of Expansive Soil Slope based on Gaussian Process Theory and Its Engineering Application

Numerous studies on the instability and shear strength of expansive soil slopes have suggested that the fundamental reason for shallow sliding failures is insufficient shear strength caused by repeated dry-wet cycles under low stress. Therefore, this paper studied the effects of low stress and dry-wet cycles on the shear strength of expansive soil slopes. A nonlinear curve that can be well fitted using generalized power functions was observed for the shear strength of expansive soil, and a numerical model with nonlinearly distributed shear strength taken into consideration was built using FISH. In this way, the dynamic distribution of shear strength as a function of vertical stress was obtained. Compared to conventional models, the proposed model was more accurate in terms of theoretical aspects, and the obtained results exhibited improved consistency with the actual mode. In order to improve the applicability of this model, a mapping model of atmospheric action and engineering design of safety factors of slopes was established using a Gaussian Process Regression (GPR) model, which is suitable for cases characterized by high dimension, small sample population, and nonlinearity. Herein, the numerical calculations were replaced with output responses. In this way, fast prediction of the stability of expansive soil slopes in terms of a shallow sliding failure prediction model was achieved. To verify the feasibility of the proposed approach, dynamic predictions of shallow sliding failures of expansive soil slopes were carried out for Nanyou Highway under different rainfall conditions. Meanwhile, safety factor responses corresponding to different slope design schemes were also obtained using the proposed model, providing reference for slope stability analysis in expansive soil zones.

Exact solution for a two-phase Stefan problem with power-type latent heat

A two-phase Stefan problem with latent heat a general power function of position is investigated. The background of the problem can be found in the soil-freezing process during the application of the artificial ground-freezing technique. After introducing the specific engineering condition, the governing equations of a two-phase Stefan problem are developed. An exact solution for the problem is established using the similarity transformation technique and the theory of the Kummer functions. It is proved that the coefficients in the solution can be appropriately determined if certain inequality is satisfied. Special cases of the solution are discussed, and several solutions reported in the literature are recovered. A similar two-phase Stefan problem involving first type of boundary condition is also introduced and solved. The coefficients in the solution can always be properly determined with no additional requirement. In the end, computational examples of the solution are presented and discussed. The exact solution provides a useful benchmark that can be used for verifying general numerical algorithms of Stefan problems, and it is also advantageous in the context of the inverse problem analysis.

Power function inflation potential analysis for cosmological model with Gauss-Bonnet term

Inflation is still an interesting topic in the study of our universe. An interesting inflation scenario named Gauss-Bonnet inflation proposed without inflation potential has been shown unstable. In this work, we consider the general power function inflation potential, V(ϕ) = mϕn in the model, then the solution is analyzed according to the inflation scenario. Using the stability condition, inflation potential with m positive and 0 ≤ n < 5 give proper solution for the inflation scenario.

A generalized model for bacterial disinfection: Stochastic approach

This work proposes a novel, generalized model for bacterial disinfection formulated in light of a stochastic paradigm. The model⿿s formulation is based on an intensity of transition that is proportional to the product of general power functions of the bacteria⿿s number concentration and time; thus, the generalized stochastic model embodies the results obtained from our earlier models. The proposed model gives rise to linear and non-linear cases of the master equation whose solution can be obtained analytically as well as numerically via Monte Carlo simulation. Moreover, the generalized stochastic model has been validated with a specific instance of bacterial disinfection. The model⿿s analytical and numerical results are in excellent accord among themselves as well as with those obtained from our earlier models; in addition, the model⿿s results tend to describe the available experimental data reasonably well.

Satiation meal and the effects of meal and body sizes on gastric evacuation rate in brook trout Salvelinus fontinalis fed commercial pellets.

Gastric evacuation (GE) experiments were performed on brook trout Salvelinus fontinalis fed commercial food pellets. The experiments included small fish (36 g; 15 cm total length, LT ) fed meals of 0·2, 0·4 and 0·8 g and large fish (152 g; 23 cm) fed meals of 0·8, 2·0 and 4·0 g at temperatures ranging from 15·1 to 18·2° C. The stomach contents were thereafter sampled and weighed at 3 h intervals until the first empty stomach was observed. The course of GE was examined by use of a general power function of the data that revealed that the square-root function described the GE rate (GER) by the current stomach content mass independently of original meal size. Using the square-root function, the relationship between GER and fish size was described by a power function of fish length, whereas the effect of temperature was described by a simple exponential function. GER of the commercial pellets fed to S. fontinalis could thus be described by dStdt=-0·000464L1·31e0·052TSt (g h(-1) ), where St is stomach mass (g) at time t (h), L is total fish length (cm) and T is temperature (° C). The result of this study should provide a useful tool for planning of feeding regimes in production of S. fontinalis by optimizing growth and minimizing food waste.

Synthesis of Phase-locked Loop System Structure with Adaptation Based on Combined-maximum Principle

The synthesis of the structure of the phase-locked loop system is performed based on the generalized power function maximum condition. The system effectiveness in the lock-on mode is studied via numerical simulation. Compared to the classical one phase-locked loop system with RC-filter this solution is characterized by the higher stability with respect to external influences and by reduced phase error in the steady-state.

The Stochastic Synthesis of the Adaptive Filter for Estimating the Controlled Systems State Based on the Condition of Maximum of the Generalized Power Function

The proposed procedure for the synthesis of the filter of the state estimation is based on a new mathematical model of the dynamic controlled system. It is based on the maximum condition of the function of the generalized power. The optimization boundary problem can be solved using the invariant embedding procedure. The result differs from the known ones as it has lower dimensionality, non-linear structure and provides a more accurate estimation.

Translation, contraction and dilation of dual generalized order statistics

A general class of distributions has been characterized through translation of two non-adjacent dual generalized order statistics (dgos). Moreover, the characterizing results are obtained for generalized Pareto distribution through dilation of dgos and generalized power function distribution through contraction of non-adjacent generalized order statistics.

Race to idle

We study an energy conservation problem where a variable-speed processor is equipped with a sleep state. Executing jobs at high speeds and then setting the processor asleep is an approach that can lead to further energy savings compared to standard dynamic speed scaling. We consider classical deadline-based scheduling, that is, each job is specified by a release time, a deadline and a processing volume. For general convex power functions, Irani et al. [2007] devised an offline 2-approximation algorithm. Roughly speaking, the algorithm schedules jobs at a critical speed s crit that yields the smallest energy consumption while jobs are processed. For power functions P ( s ) = s α &amp; γ, where s is the processor speed, Han et al. [2010] gave an α α + 2)-competitive online algorithm. We investigate the offline setting of speed scaling with a sleep state. First, we prove NP-hardness of the optimization problem. Additionally, we develop lower bounds, for general convex power functions: No algorithm that constructs s crit -schedules, which execute jobs at speeds of at least s crit , can achieve an approximation factor smaller than 2. Furthermore, no algorithm that minimizes the energy expended for processing jobs can attain an approximation ratio smaller than 2. We then present an algorithmic framework for designing good approximation algorithms. For general convex power functions, we derive an approximation factor of 4/3. For power functions P ( s ) = β s α + γ, we obtain an approximation of 137/117 &gt; 1.171. We finally show that our framework yields the best approximation guarantees for the class of s crit -schedules. For general convex power functions, we give another 2-approximation algorithm. For functions P ( s ) = β s α + γ, we present tight upper and lower bounds on the best possible approximation factor. The ratio is exactly eW −1 (− e −1−1/ e )/( eW −1 (− e −1−1/ e )+1) &gt; 1.211, where W -1 is the lower branch of the Lambert W function.

On a characterization of generalized Erlang-truncated exponential distribution through distributional properties of dual generalized order statistics

Generalized Erlang-truncated exponential distribtion ${F(x)=[1-e^{-\beta(1-e^{-\lambda})x}]}$ has been characterized through translation of two non-adjacent dual generalized order statistics (dgos) and then the characterizing results are obtained for generalized Pareto distribution through dilation of dual generalized order statistics (dgos) and generalized power function distribution through pcontraction of non-adjacent generalized order statistics (gos). Further, the results are deduced for order statistics, lower record statistics, upper record statistics and adjacent generalized order statistics and dual generalized order statistics.

Generalized power functions

In this paper some relations for the kernels of the Carleman–Vekua equation, in particular the representations of these kernels in the form of generalized power functions completely analogous to the well-known elementary Cauchy kernel expansion, are studied. The obtained results are applied to some problems of the theory of generalized analytic functions.

Variants of the general interval power function

The article gives an overview of power function variants and compares three differently extensive versions for use in interval arithmetics. Aiming at the general power function for inclusion in interval libraries, which is defined for as many pairs of base and exponent as possible, we observe that the definition of a power for each such pair depends on the context. This problem eventually comes up at the point where reasonable doubts about the definition $$0^0$$ and powers with negative base in correlation with a non-integral exponent arise. We come up with several variants serving distinct purposes, yet also recommend a general-purpose power function, which is unique for being restricted to integral exponents on the domain of negative bases. Three different variants of interval power functions satisfy all meaningful general exponentiation needs. We provide a unified treatment to handle the variants and present valuable implementation techniques for the computation of these interval functions along with a mathematic foundation.

Risk-Averse Newsvendor Model with Strategic Consumer Behavior

The classic newsvendor problem focuses on maximizing the expected profit or minimizing the expected cost when the newsvendor faces myopic customers. However, it ignores the customer’s bargain-hunting behavior and risk preference measure of the newsvendor. As a result, we carry out the rational expectation (RE) equilibrium analysis for risk-averse newsvendor facing forward-looking customers who anticipate future sales and choose purchasing timing to maximize their expected surplus. We propose the equations satisfied by the RE equilibrium price and quantity for the risk-averse retailer in general setting and the explicit equilibrium decisions for the case where demand follows the uniform distribution and utility is a general power function. We identify the impacts of the system parameters on the RE equilibrium for this specific situation. In particular, we show that the RE equilibrium price for some risk-averse newsvendors is lower than for a risk-neutral retailer and the RE equilibrium stocking quantity for some risk-averse newsvendors is higher than for a risk-neutral retailer. We also find that the RE equilibrium sale price for a risk-averse newsvendor is decreasing in salvage price in some situations.

Analytical Solution for Elastic and Elastoplastic Bending of a Curved Beam Composed of Inhomogeneous Materials

We present an analytical solution for elastic and elastoplastic bending problem of a curved beam composed of inhomogeneous materials. Suppose the material is isotropic, ideally elastoplastic and it obeys Tresca’s yield criterion and the corresponding associated flow rule. And the elastic modulus and yield limit of the material vary radially according to general power functions. The expressions of stresses and displacements of a curved beam in both purely elastic stress state and partially plastic stress state are derived. The influence of material inhomogeneity on the elastoplastic behavior of a curved beam is demonstrated in numerical examples. Analytical solutions presented here can serve as benchmark results for evaluating numerical solutions.

The Higher Order Crack Tip Fields for Arbitrary FGM Composite Functions

As the material properties function at the crack tip greatly affects the structure of crack tip field, especially in the study of comparatively thin FGM composites whose properties functions are equipped with the feature of high gradient variation, it is not enough to adopt the simple linear function to depict the property testpoints of the material. A new method, which is more precise to deal with the experimental data and to depict FGM composite properties functions (e.g. elastic modulus E), is proposed in this paper. Beside the linear function, quadratic term, cubic terms and so on are added. Then, the solution is first given for the general power function, and the more precise analytical solution of higher order crack tip fields will be obtained by means of superposition method. Actually, any other complicated or arbitrary material functions can be treated similarly by using Taylor series expansion at the crack tip. Finally, the obtained higher order crack tip fields or the eigen-functions which can depict the arbitrary material function would provide the theoretical basis for the fracture experiments and numerical simulation, and this method is of great universal significance.

Modifications to the Conduit Flow Process Mode 2 for MODFLOW‐2005

As a result of rock dissolution processes, karst aquifers exhibit highly conductive features such as caves and conduits. Within these structures, groundwater flow can become turbulent and therefore be described by nonlinear gradient functions. Some numerical groundwater flow models explicitly account for pipe hydraulics by coupling the continuum model with a pipe network that represents the conduit system. In contrast, the Conduit Flow Process Mode 2 (CFPM2) for MODFLOW-2005 approximates turbulent flow by reducing the hydraulic conductivity within the existing linear head gradient of the MODFLOW continuum model. This approach reduces the practical as well as numerical efforts for simulating turbulence. The original formulation was for large pore aquifers where the onset of turbulence is at low Reynolds numbers (1 to 100) and not for conduits or pipes. In addition, the existing code requires multiple time steps for convergence due to iterative adjustment of the hydraulic conductivity. Modifications to the existing CFPM2 were made by implementing a generalized power function with a user-defined exponent. This allows for matching turbulence in porous media or pipes and eliminates the time steps required for iterative adjustment of hydraulic conductivity. The modified CFPM2 successfully replicated simple benchmark test problems.

Delving Deeper: Derivatives of Generalized Power Functions

After several years of teaching multiple sections of first-semester calculus, it was easy for me to think that I had nothing new to learn. But every year and every class bring a new group of students with their unique gifts and insights. In a recent class, after covering the derivative rules for power functions and exponential functions, I asked the class about the derivative of a function like y = xsinx, which is neither a power function (the power is not constant) nor an exponential function (the base is not constant).