Functions In Polynomial Forms Research Articles

Алгоритм построения полиномиального решения задачи программного управления для динамической системы в частных производных

The completely controlled dynamic system in partial derivatives is considered. The problem of constructing state and control functions in an analytical form is solved. The basic method is the cascade decomposition method, which is algorithmically implemented in three stages: forward cascade decomposition, central stage and reverse. The method is based on the properties of the matrix coefficient at the derivative of the control function. Decomposition means a p-step transition from the original system to a reduced system that is quite similar in form to the original one, but with respect to functions from subspaces. The given conditions are reduced in the process of decomposition. When passing to the p-th step system, additional conditions appear on the partial derivatives of the components of the state function. The number of extra conditions at each point is equal to the number of decomposition steps. The matrix coefficient at the derivative of the control function of the reduced system of the last step is surjective. It is this property that determines the presence of the property of complete controllability of the system under consideration. The first stage of decomposition - the stage of the direct move ends with the detection of the number of decomposition steps and the identification of the property of complete controllability. The task of the central stage of decomposition is to construct the state function of the reduced system of the last step in an analytical form. The state function of the reduced system is the basis function that determines the form of the state function of the original system. Necessary and sufficient conditions for the existence of a basis function in polynomial form are established. The minimum degree of the polynomial is also set, which is determined by the number of decomposition steps. Formulas for constructing vector functions - coefficients of the basis function polynomial are given. Formulas for constructing the control function of the reduced system are given in polynomial form. During the last stage of decomposition, the state function of the original system is successively restored in polynomial form. This polynomial function satisfies the given conditions at the start and end points. The final stage is the construction of the control function of the original system in polynomial form as well. While the last stage of decomposition the state function of the original system is successively restored in polynomial form. This polynomial function satisfies the given conditions at the start and end points. The final stage is the construction of the control function of the original system also in polynomial form. A step-by-step algorithm for solving the program control problem for a dynamic system in partial derivatives has been developed. Formulas for constructing state and control functions in polynomial form are given. An example of a three-dimensional dynamical partial differential system with a surjective matrix coefficient in the first-step splitting system is given. The implementation of the proposed algorithm is demonstrated. The state and control functions are constructed in the form of a polynomial of minimum degree.

Optimization of the operational parameters of a picking-type pneumatic planter using response surface methodology

Singulation of seeds has been investigated extensively by researchers all over the world and a large number of precision seeding systems with design variations have been developed for different crops. A picking type metering mechanism was developed at CAET, AAU, Godhra, Gujarat, India. The performance of the picking type seed-metering device of a pneumatic planter was investigated under laboratory conditions to optimize the operating parameters for lady's finger seed. The picking of single seed the three operational parameters i.e. hole diameters for the nozzle: 1.0, 1.5, 2.5 and 3.0 mm; forward speed: 0.37, 0.56, 0.83, 1.11 and 1.30 m/s and vacuum pressure: 19.33, 39.32, 43.98, 58.64 and 68.63 kPa were selected for the study. The metering system of the planter was set to place the seed to seed spacing at 300 mm. The response surface methodology (RSM) technique was used to optimize the operational parameters of a precision planter. For optimizing the forward speed, vacuum pressure and nozzle size for developed machine was evaluated by examining the miss index, multiple index, quality of feed index and precision. The data obtained in the experiments were used to develop functions in polynomial form using multiple regression technique. The optimum value was found to be around 0.96 m/s, 36.25 kPa and 2.0 mm of forward speed, vacuum pressure and the holes diameter of nozzle, respectively. The most important variable that governs planting phenomenon is the combination of hole diameter of nozzle and vacuum pressure accounts 89.18 per cent.

More constructions of semi-bent and plateaued functions in polynomial forms

Plateaued functions and their subclass semi-bent functions have useful applications in cryptography and communications. In this paper we give new constructions of quadratic semi-bent functions in polynomial forms on the finite field $$\mathbb {F}_{2^n}$$ for both odd and even n. We also present some characterizations of e-plateaued functions with few trace terms when n is even.

On the Maximum Number of Bent Components of Vectorial Functions

In this paper, we show that the maximum number of bent component functions of a vectorial function $F:GF(2)^{n}\to GF(2)^{n}$ is $2^{n}-2^{n/2}$ . We also show that it is very easy to construct such functions. However, it is a much more challenging task to find such functions in polynomial form $F\in GF(2^{n})[x]$ , where $F$ has only a few terms. The only known power functions having such a large number of bent components are $x^{d}$ , where $d=2^{n/2}+1$ . In this paper, we show that the binomials $F^{i}(x)=x^{2^{i}}(x+x^{2^{n/2}})$ also have such a large number of bent components, and these binomials are inequivalent to the monomials $x^{2^{n/2}+1}$ if $0 . In addition, the functions $F^{i}$ have differential properties much better than $x^{2^{n/2}+1}$ . We also determine the complete Walsh spectrum of our functions when $n/2$ is odd and $\gcd (i,n/2)=1$ .

Further results on semi-bent functions in polynomial form

Plateaued functions have been introduced by Zheng and Zhang in 1999 as good candidates for designing cryptographic functions since they possess many desirable cryptographic characteristics. Plateaued functions bringtogether various nonlinear characteristics and include twoimportant classes of Boolean functions defined in even dimension: the well-known bent functions ($0$-plateaued functions) and the semi-bent functions ($2$-plateaued functions). Bent functions have been extensively investigated since 1976. Very recently, the study of semi-bent functions has attracted a lot of attention in symmetric cryptography. Many intensive progresses in the design of such functions have been made especially in recent years.The paper is devoted to the construction of semi-bent functions on the finite field $\mathbb{F}_{2^n}$ ($n=2m$) in the line of a recent work of S. Mesnager [IEEE Transactions on Information Theory, Vol 57, No 11, 2011]. We extend Mesnager's results and present a new construction of infinite classes of binary semi-bent functions in polynomial trace. The extension is achieved by inserting mappings $h$ on $\mathbb{F}_{2^n}$ which can be expressed as $h(0) = 0$ and $h(uy) = h_1(u)h_2(y)$ with $u$ ranging over the circle $U$ of unity of $\mathbb{F}_{2^n}$, $y \in \mathbb{F}_{2^m}^{*}$ and $uy \in \mathbb{F}_{2^n}^{*}$, where $h_1$ is a isomorphism on $U$ and $h_2$ is an arbitrary mapping on $\mathbb{F}_{2^m}^{*}$. We then characterize the semi-bentness property of the extended family in terms of classical binary exponential sums and binary polynomials.

New Constructions of Quadratic Bent Functions in Polynomial Form

New quadratic bent functions in polynomial form are constructed in this paper. The constructions give new Boolean bent, generalized Boolean bent and p-ary bent functions. Based on Z 4 -valued quadratic forms, a simple method provides several new constructions of generalized Boolean bent functions. From these generalized Boolean bent functions a method is presented to transform them into Boolean bent and semi-bent functions. Moreover, many new p-ary bent functions can also be obtained by applying similar methods.

New constructions of semi-bent functions in polynomial forms

Semi-bent functions are a kind of useful functions in cryptography. In this paper we give new constructions of n-variable quadratic semi-bent functions in polynomial forms for both odd and even n, and we also present many n-variable semi-bent functions with few trace terms when n is even.

Improved Polynomial Fuzzy Modeling and Controller with Stability Analysis for Nonlinear Dynamical Systems

This study presents an improved model and controller for nonlinear plants using polynomial fuzzy model-based (FMB) systems. To minimize mismatch between the polynomial fuzzy model and nonlinear plant, the suitable parameters of membership functions are determined in a systematic way. Defining an appropriate fitness function and utilizing Taylor series expansion, a genetic algorithm (GA) is used to form the shape of membership functions in polynomial forms, which are afterwards used in fuzzy modeling. To validate the model, a controller based on proposed polynomial fuzzy systems is designed and then applied to both original nonlinear plant and fuzzy model for comparison. Additionally, stability analysis for the proposed polynomial FMB control system is investigated employing Lyapunov theory and a sum of squares (SOS) approach. Moreover, the form of the membership functions is considered in stability analysis. The SOS-based stability conditions are attained using SOSTOOLS. Simulation results are also given to demonstrate the effectiveness of the proposed method.

Bent and Hyper-Bent Functions in Polynomial Form and Their Link With Some Exponential Sums and Dickson Polynomials

Bent functions are maximally nonlinear Boolean functions with an even number of variables. They were introduced by Rothaus in 1976. For their own sake as interesting combinatorial objects, but also because of their relations to coding theory (Reed-Muller codes) and applications in cryptography (design of stream ciphers), they have attracted a lot of research, specially in the last 15 years. The class of bent functions contains a subclass of functions, introduced by Youssef and Gong in 2001, the so-called hyper-bent functions, whose properties are still stronger and whose elements are still rarer than bent functions. Bent and hyper-bent functions are not classified. A complete classification of these functions is elusive and looks hopeless. So, it is important to design constructions in order to know as many of (hyper)-bent functions as possible. This paper is devoted to the constructions of bent and hyper-bent Boolean functions in polynomial forms. We survey and present an overview of the constructions discovered recently. We extensively investigate the link between the bentness property of such functions and some exponential sums (involving Dickson polynomials) and give some conjectures that lead to constructions of new hyper-bent functions.

A new class of bent and hyper-bent Boolean functions in polynomial forms

Bent functions are maximally nonlinear Boolean functions and exist only for functions with even number of inputs. This paper is a contribution to the construction of bent functions over $${\mathbb{F}_{2^{n}}}$$ (n = 2m) having the form $${f(x) = tr_{o(s_1)} (a x^ {s_1}) + tr_{o(s_2)} (b x^{s_2})}$$ where o(s i ) denotes the cardinality of the cyclotomic class of 2 modulo 2 n ? 1 which contains s i and whose coefficients a and b are, respectively in $${F_{2^{o(s_1)}}}$$ and $${F_{2^{o(s_2)}}}$$ . Many constructions of monomial bent functions are presented in the literature but very few are known even in the binomial case. We prove that the exponents s 1 = 2 m ? 1 and $${s_2={\frac {2^n-1}3}}$$ , where $${a\in\mathbb{F}_{2^{n}}}$$ (a ? 0) and $${b\in\mathbb{F}_{4}}$$ provide a construction of bent functions over $${\mathbb{F}_{2^{n}}}$$ with optimum algebraic degree. For m odd, we give an explicit characterization of the bentness of these functions, in terms of the Kloosterman sums. We generalize the result for functions whose exponent s 1 is of the form r(2 m ? 1) where r is co-prime with 2 m + 1. The corresponding bent functions are also hyper-bent. For m even, we give a necessary condition of bentness in terms of these Kloosterman sums.

On Quadratic Bent Functions in Polynomial Forms

In this correspondence, we construct some new quadratic bent functions in polynomial forms by using the theory of quadratic forms over finite fields. The results improve some previous work. Moreover, we solve a problem left by Yu and Gong in 2006.

Optimisation of the seed spacing uniformity performance of a vacuum-type precision seeder using response surface methodology

The objective of this study was to optimise the seed spacing uniformity performance of a precision seeder using response surface methodology (RSM) and to verify the optimum levels of the variables. The variables considered in the study consisted of the vacuum on the seed plate, the diameter of seed holes and the peripheral speed of the seed plate. Cotton seeds were used for sowing and experiments conducted in this study were based on the central composite design (CCD), one of the designs in RSM. Data obtained in the laboratory were divided into three different groups in order to obtain values of the multiple index, quality of feed index and miss index. An additional performance criterion was also proposed and used as an indicator of the sowing performance. This was the root-mean-square deviation from the theoretical seed spacing. The data obtained in the laboratory were then used to develop functions in polynomial form that allowed the calculation of the optimum level of each independent variable considered in the study. The optimum levels of vacuum pressure and the diameter of holes for precision seeding of cotton seeds were found to be around 5.5 kPa and 3 mm, respectively. No optimum value was obtained for the peripheral speed of the seed plate. It was found that the lower the peripheral speed of the plate, the higher is the performance.

Constructions of quadratic bent functions in polynomial forms

In this correspondence, the constructions and enumerations of all bent functions represented by a polynomial form of f(x)=/spl Sigma//sub i=1//sup n/2-1/c/sub i/Tr(x/sup 1+2(i)/)+c/sub n/2/Tr/sub 1//sup n/ /sup /2/(x/sup 1+2(n/2)/), c/sub i//spl isin//sub 2/ F/sub 2/ are presented for special cases of n. Using an iterative approach, the construction of bent functions of n variables with degree n/2 is also provided using the constructed quadratic bent functions.

Construction space harmonic functions in polynomial form and spherical functions by complex-functional

In this paper, applying the theory of complex-functional, not only the space harmonic functions in polynomial form, but also the spherical functions are obtained.