Design and Experimental Research of Variable Formula Fertilization Control System Based on Prescription Diagram

In order to solve the problems such as the inability to automatically mix a variety of solid fertilizers and the unreasonable fertilizer amount, improve fertilizer utilization, and reduce production costs, this study designs a variable formula fertilization control system based on a prescription diagram, including pressure sensor, speed sensor, servo motor, fertilizer discharge actuator, Programmable Logic Controller (PLC controller), vehicle control terminal, etc. Based on pre-loaded soil prescription diagram and combining fertilizer pressure and ground wheel speed detection information, the system obtained a formula fertilization control strategy through calculation to realize the function of fast and automatic formula of nitrogen, phosphorus, and potassium fertilizers and precise variable fertilization. The experimental study on the performance of the variable formula fertilization control system showed the following: the measurement error range of the pressure sensor was 0.005~0.03%; the relationship between the motor speed and the amount of nitrogen, phosphorus, and potassium fertilizer discharged was calibrated. Three gears were established for the motor speed: low (10 r/min), medium (30 r/min), and high (50 r/min); the measurement accuracy of the speed sensor was above 98%. The test verified that the control accuracy of the variable formula fertilization system reached more than 95%, which met the requirements of fast automatic formula and precise variable fertilization and had good practicability and economy.

Populations facing a nonlinear environmental gradient: Steady states and pulsating fronts

We consider a population structured by a space variable and a phenotypical trait, submitted to dispersion, mutations, growth and nonlocal competition. This population is facing an environmental gradient: to survive at location [Formula: see text], an individual must have a trait close to some optimal trait [Formula: see text]. Our main focus is to understand the effect of a nonlinear environmental gradient. We thus consider a nonlocal parabolic equation for the distribution of the population, with [Formula: see text], [Formula: see text]. We construct steady states solutions and, when [Formula: see text] is periodic, pulsating fronts. This requires the combination of rigorous perturbation techniques based on a careful application of the implicit function theorem in rather intricate function spaces. To deal with the phenotypic trait variable [Formula: see text] we take advantage of a Hilbert basis of [Formula: see text] made of eigenfunctions of an underlying Schrödinger operator, whereas to deal with the space variable [Formula: see text] we use the Fourier series expansions. Our mathematical analysis reveals, in particular, how both the steady states solutions and the fronts (speed and profile) are distorted by the nonlinear environmental gradient, which are important biological insights.

A Remark on the Change of Variable Theorem for the Riemann Integral

In 1961, Kestelman first proved the change in the variable theorem for the Riemann integral in its modern form. In 1970, Preiss and Uher supplemented his result with the inverse statement. Later, in a number of papers (Sarkhel, Výborný, Puoso, Tandra, and Torchinsky), the alternative proofs of these theorems were given within the same formulations. In this note, we show that one of the restrictions (namely, the boundedness of the function f on its entire domain) can be omitted while the change of variable formula still holds.

Virtual and arrow Temperley–Lieb algebras, Markov traces, and virtual link invariants

Let [Formula: see text] be the algebra of Laurent polynomials in the variable [Formula: see text] and let [Formula: see text] be the algebra of Laurent polynomials in the variable [Formula: see text] and standard polynomials in the variables [Formula: see text] For [Formula: see text] we denote by [Formula: see text] the virtual braid group on [Formula: see text] strands. We define two towers of algebras [Formula: see text] and [Formula: see text] in terms of diagrams. For each [Formula: see text] we determine presentations for both, [Formula: see text] and [Formula: see text]. We determine sequences of homomorphisms [Formula: see text] and [Formula: see text], we determine Markov traces [Formula: see text] and [Formula: see text], and we show that the invariants for virtual links obtained from these Markov traces are the [Formula: see text]-polynomial for the first trace and the arrow polynomial for the second trace. We show that, for each [Formula: see text] the standard Temperley–Lieb algebra [Formula: see text] embeds into both, [Formula: see text] and [Formula: see text], and that the restrictions to [Formula: see text] of the two Markov traces coincide.

Soliton molecules in the (2+1)-dimensional Nizhnik–Novikov–Veselov equation

The [Formula: see text]-soliton solution of the (2+1)-dimensional Nizhnik–Novikov–Veselov equation is constructed. The line soliton molecule, the breather and the lump soliton are presented successively for [Formula: see text]. The three-soliton molecule structure, interaction of one-soliton molecule and a line soliton, the soliton molecules consisting of a line soliton and the breather/lump soliton of the solution [Formula: see text] are constructed for [Formula: see text]. Moreover, the four-soliton molecule structure, interaction of the soliton molecule and a line soliton, the soliton molecule consisting of the line soliton molecule and a lump soliton, the elastic interaction between the line soliton molecule and a lump soliton, the soliton molecules consisting of the line soliton molecule and the breather, two breather solitons, the breather soliton and a lump of the variable [Formula: see text] for this equation are also derived for [Formula: see text] by applying the velocity resonance, the module resonance of wave number and the long-wave limit ideas. To illustrate these phenomena, the analysis explicit solutions are all given and their dynamics features are all displayed through figures.

Efficient Estimation of a Scale Parameter of Bivariate Lomax Distribution by Ranked Set Sampling

Ranked set sampling (RSS) is an efficient technique for estimating parameters and is applicable whenever ranking on a set of sampling units can be done easily by a judgment method or based on an auxiliary variable. In this paper, we assume [Formula: see text]to have bivariate Lomax distribution where a study variable [Formula: see text]is difficult and/or expensive to measure and is correlated with an auxiliary variable [Formula: see text] which is readily measurable. The auxiliary variable is used to rank the sampling units. In this article, we propose an estimator for the scale parameter of bivariate Lomax distribution using some of the modified RSS schemes. Efficiency comparison of the proposed estimators is performed numerically as well as graphically. A simulation study is also performed to demonstrate the performance of the proposed estimators. Finally, we implement the results to real-life datasets. AMS classification codes: 62D05, 62F07, 62G30

Coherent states attached to the quantum disc algebra and their associated polynomials

The one-variable [Formula: see text]-coherent states attached to the [Formula: see text]-disc algebra are constructed and used to obtain the [Formula: see text]-Bargmann–Fock realization of its Fock representation. Then, this realization is used to obtain the [Formula: see text]-continuous Hermite polynomials as well as continuous and discrete [Formula: see text]-Hermite polynomials by using a pair of Hermitian canonical conjugate operators and two pairs of the non-Hermitian conjugate operators, respectively. Besides, we introduce a two-variable family of [Formula: see text]-coherent states attached to the Fock representation space of the [Formula: see text]-disc algebra and its opposite algebra and obtain their simultaneous [Formula: see text]-Bargmann–Fock realization. For an appropriate non-Hermitian operator, the latter realization is served to obtain the well-known little [Formula: see text]-Jacobi polynomials used in constructing the [Formula: see text]-disc polynomials.

A Phasor Analysis Method for Charge-Controlled Memory Elements

Memory elements, including memristor, memcapacitor, meminductor and second-order memristor, have been widely exploited recently to realize circuit systems for a broad scope of applications. This paper introduces a phasor analysis method for memory elements to help with the understanding of the complex nonlinear phenomena in circuits with memory elements. With the proposed method, all different memory elements could be described in a unified form and the series-connected circuit with memristor, memcapacitor, meminductor and second-order memristor could be simply modeled as one variable [Formula: see text]. Thus, the phasor vectors provided a way to conveniently calculate the [Formula: see text]–[Formula: see text] relation of different memory elements and to clearly understand the similarities and differences between all memory elements. Then some interesting phenomena were introduced when combining different memory elements. Moreover, a specific [Formula: see text] with certain [Formula: see text]–[Formula: see text] relations could be easily obtained with the method. And through the inverse calculation, the specific [Formula: see text] could be decomposed to a certain combination of memory elements. Meanwhile, the parameters of [Formula: see text] in the phasor domain were analyzed. Furthermore, the frequency characteristic for a [Formula: see text] circuit could be easily analyzed with the method and a particular series resonance was introduced.

Zero-Hopf Periodic Orbits for a Rössler Differential System

We study the zero-Hopf bifurcation of the Rössler differential system [Formula: see text] where the dot denotes the derivative with respect to the independent variable [Formula: see text] and [Formula: see text], [Formula: see text], [Formula: see text] are real parameters.

Periodic Solutions of Continuous Third-Order Differential Equations with Piecewise Polynomial Nonlinearities

We consider third-order autonomous continuous piecewise differential equations in the variable [Formula: see text]. For such differential equations with nonlinearities of the form [Formula: see text], we investigate their periodic solutions using the averaging theory. We remark that since the differential system is only continuous we cannot apply to it the classical averaging theory, that needs that the differential system be at least of class [Formula: see text].