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The paper aims to build the relationship between an entire function of restricted hyper-order with its linear c-shift operator.

Standard methodology for papers in difference and shift operators and value distribution theory have been used.

The relation between an entire function of restricted hyper-order with its linear c-shift operator was found under the periphery of sharing a set of two small functions IM (ignoring multiplicities) when exponent of convergence of zeros is strictly less than its order. This research work is an improvement and extension of two previous papers.

This is an original research work.

Our purpose of this study is to construct an algorithm for finding a zero of the sum of two maximally monotone mappings in Hilbert spaces and discus its convergence. The assumption that one of the mappings is α-inverse strongly monotone is dispensed with. In addition, we give some applications to the minimization problem. Our method of proof is of independent interest. Finally, a numerical example which supports our main result is presented. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

This paper aims to develop a credit-risk model in which firms face rollover risk, and the markets for defaulted assets are segmented due to entry costs. The paper shows that reducing the entry costs in this economy may decrease the total surplus of the economy. This outcome can arise because when market barriers are lifted, the gap between the liquidation prices across the markets will shrink, but then the market that would experience a price drop may face more bankruptcies because the rollover risk will increase in that market. The paper describes under which condition such an intervention policy improves or hurts the total surplus.

An appropriate equivalent model is the key to the effective analysis of the system and structure in which permanent magnet takes part. At present, there are several equivalent models for calculating the interacting magnetic force between permanent magnets including magnetizing current, magnetic charge and magnetic dipole–dipole model. How to choose the most appropriate and efficient model still needs further discussion.

This paper chooses cuboid, cylindrical and spherical permanent magnets as calculating objects to investigate the detailed calculation procedures based on three equivalent models, magnetizing current, magnetic charge and magnetic dipole–dipole model. By comparing the accuracies of those models with experiment measurement, the applicability of three equivalent models for describing permanent magnets with different shapes is analyzed.

Similar calculation accuracies of the equivalent magnetizing current model and magnetic charge model are verified by comparison between simulation and experiment results. However, the magnetic dipole–dipole model can only accurately calculate for spherical magnet instead of other nonellipsoid magnets, because dipole model cannot describe the specific characteristics of magnet's shape, only sphere can be treated as the topological form of a dipole, namely a filled dot.

This work provides reference basis for choosing a proper model to calculate magnetic force in the design of electromechanical structures with permanent magnets. The applicability of different equivalent models describing permanent magnets with different shapes is discussed and the equivalence between the models is also analyzed.

The main purpose of this study resides essentially in the development of a new tool to quantify the biomass in the bioreactor operating under steady state conditions.

Modeling is the most relevant tool for understanding the functioning of some complex processes such as biological wastewater treatment. A steady state model equation of activated sludge model 1 (ASM1) was developed, especially for autotrophic biomass (XBA) and for oxygen uptake rate (OUR). Furthermore, a respirometric measurement, under steady state and endogenous conditions, was used as a new tool for quantifying the viable biomass concentration in the bioreactor.

The developed steady state equations simplified the sensitivity analysis and allowed the autotrophic biomass (XBA) quantification. Indeed, the XBA concentration was approximately 212 mg COD/L and 454 mgCOD/L for SRT, equal to 20 and 40 d, respectively. Under the steady state condition, monitoring of endogenous OUR permitted biomass quantification in the bioreactor. Comparing XBA obtained by the steady state equation and respirometric tool indicated a percentage deviation of about 3 to 13%. Modeling bioreactor using GPS-X showed an excellent agreement between simulation and experimental measurements concerning the XBA evolution.

These results confirmed the importance of respirometric measurements as a simple and available tool for quantifying biomass.

The present work focuses on the primality and the Cartesian product of graphs.

Given a graph G, a subset M of V (G) is a module of G if, for a, b ∈ M and x ∈ V (G) \ M, xa ∈ E(G) if and only if xb ∈ E(G). A graph G with at least three vertices is prime if the empty set, the single-vertex sets and V (G) are the only modules of G.

Motivated by works obtained on “the Cartesian product of graphs” and “the primality,” this paper creates a link between the two notions.

In fact, we study the primality of the Cartesian product of two connected graphs minus k vertices, where k ∈ {0, 1, 2}.

The purpose of the study is to obtain and analyze vibro-acoustic characteristics.

A unified analysis model for the rotary composite laminated plate and conical–cylindrical double cavities coupled system is established. The related parameters of the unified model are determined by isoparametric transformation. The modified Fourier series are applied to construct the admissible displacement function and the sound pressure tolerance function of the coupled systems. The energy functional of the structure domain and acoustic field domain is established, respectively, and the structure–acoustic coupling potential energy is introduced to obtain the energy functional. Rayleigh–Ritz method was used to solve the energy functional.

The displacement and sound pressure response of the coupled systems are acquired by introducing the internal point sound source excitation, and the influence of relevant parameters of the coupled systems is researched. Through research, it is found that the impedance wall can reduce the amplitude of the sound pressure response and suppress the resonance of the coupled systems. Besides, the composite laminated plate has a good noise reduction effect.

This study can provide the theoretical guidance for vibration and noise reduction.

The purpose of this paper is to investigate and review the impact of the use of statistical quality control (SQC) development and analytical and numerical methods on average run length for econometric applications.

This study used several academic databases to survey and analyze the literature on SQC tools, their characteristics and applications. The surveys covered both parametric and nonparametric SQC.

This survey paper reviews the literature both control charts and methodology to evaluate an average run length (ARL) which the SQC charts can be applied to any data. Because of the nonparametric control chart is an alternative effective to standard control charts. The mixed nonparametric control chart can overcome the assumption of normality and independence. In addition, there are several analytical and numerical methods for determining the ARL, those of methods; Markov Chain, Martingales, Numerical Integral Equation and Explicit formulas which use less time consuming but accuracy. New ideas of mixed parametric and nonparametric control charts are effective alternatives for econometric applications.

In terms of mixed nonparametric control charts, this can be applied to all data which no limitation in using of the proposed control chart. In particular, the data consist of volatility and fluctuation usually occurred in econometric solutions. Furthermore, to find the ARL as a performance measure, an explicit formula for the ARL of time series data can be derived using the integral equation and its accuracy can be verified using the numerical integral equation.

In this paper we talk about complex matrix quaternions (biquaternions) and we deal with some abstract methods in mathematical complex matrix analysis.

We introduce and investigate the complex space HC consisting of all 2 × 2 complex matrices of the form ξ=z1+iw1z2+iw2−z‾2−iw‾2z‾1+iw‾1, (z1,w1,z2,w2)∈C4.

We develop on HC a new matrix holomorphic structure for which we provide the fundamental operational calculus properties.

We give sufficient and necessary conditions in terms of Cauchy–Riemann type quaternionic differential equations for holomorphicity of a function of one complex matrix variable ξ∈HC. In particular, we show that we have a lot of holomorphic functions of one matrix quaternion variable.

The purpose of this paper is to investigate an alternative to established hysteresis models.

Different mathematical representations of the magnetic hysteresis are compared and some differences are briefly discussed. After this, the application of the T(x) function is presented and an inductor model is developed. Implementation details of the used transient circuit simulator code are further discussed. From real measurement results, parameters for the model are extracted. The results of the final simulation are finally discussed and compared to measurements.

The T(x) function possesses a fast mathematical formulation with very good accuracy. It is shown that this formulation is very well suited for an implementation in transient circuit simulator codes. Simulation results using the developed model are in very good agreement with measurements.

For the purpose of this paper, only soft magnetic materials were considered. However, literature suggests, that the T(x) function can be extended to hard magnetic materials. Investigations on this topic are considered as future work.

While the mathematical background of the T(x) function is very well presented in the referenced papers, the application in a model of a real device is not very well discussed yet. The presented paper is directly applicable to typical problems in the field of power electronics.