Boolean Sum Research Articles

A Boolean sum interpolation for multivariate functions of bounded variation

This paper deals with the approximation error of trigonometric interpolation for multivariate functions of bounded variation in the sense of Hardy-Krause. We propose interpolation operators related to both the tensor product and sparse grids on the multivariate torus. For these interpolation processes, we investigate the corresponding error estimates in the Lp norm for the class of functions under consideration. In addition, we compare the accuracy with the cardinality of these grids in both approaches.

Morphology of the connected components of the boolean sum of two digraphs (≤ 5)-hypomorphic up to complementation

Morphology of the connected components of the boolean sum of two digraphs (≤ 5)-hypomorphic up to complementation

Bivariate Bernstein Chlodovsky Operators Preserving Exponential Functions and Their Convergence Properties

This paper is devoted to an extension of the bivariate generalized Bernstein-Chlodovsky operators preserving the exponential function exp ( 2 , 2 ) where exp ( α , β ) = e − α x − β y , α , β ∈ R 0 + and x , y ≥ 0 . For these operators, we first examine the weighted approximation properties for continuous functions in the weighted space, and in the latter case, we also obtain the convergence rate for these operators using a weighted modulus of continuity. Then, we investigate the order of approximation regarding local approximation results via Peetre’s K-functional. We introduce the GBS (Generalized Boolean Sum) operators of generalized Bernstein-Chlodovsky operators, and we estimate the degree of approximation in terms of the Lipschitz class of Bögel continuous functions and the mixed modulus of smoothness. Finally, we provide some numerical and graphical examples with different values to demonstrate the rate of convergence of the constructed operators.

Random Multi-access Scheme Based on Group Testing Theory for Massive Machine-type Communication

Considering emerging massive machine-type communication with very few devices simultaneously transmitting short messages, random multiple access has currently become a research hot spot since the traditional multiple access protocol struggles to work properly. Inspired by the superimposed code, a channel assignment scheme based on non-adaptive group testing theory is proposed to allow multiple users to share a single channel at the same time. A superimposed code with threshold value d, which can recover the original code through the Boolean sum of up to d codes, is constructed by studying the problem of group testing. When the number of active users with a message to be transmitted is no more than the threshold value, active users encode their messages by the superimposed code and multiple active users transmit their messages together by sharing a single channel. The receiver can decode the mixed messages by the disjunction of the superimposed code. We focus on the design of the superimposed code by group testing theory, formulate this problem as the construction of a transversal design, and use the Latin square to form a useful transversal design. Theoretical proof shows that our design works well under limited circumstances where the number of active users does not exceed the threshold value. Additionally, our design can recover the original codes by the mixture of codes, which are more easily than the algorithms based on compressive sensing. Simulation analysis presents that our design performs better than the traditional competitive access protocols in terms of successful probability, throughput, and packet loss rate.

Kernel-based construction operators for Boolean sum and ruled geometry

Boolean sum and ruling are two well-known construction operators for both parametric surfaces and trivariates. In many cases, the input freeform curves in ▪ or surfaces in ▪ are complex, and as a result, these construction operators might fail to build the parametric geometry so that it has a positive Jacobian throughout the domain.In this work, we focus on cases in which those constructors fail to build parametric geometries with a positive Jacobian throughout while the freeform input has a kernel point. We show that in the limit, for high enough degree raising or enough refinement, our construction scheme must succeed if a kernel exists. In practice, our experiments, on quadratic, cubic and quartic Bézier and B-spline curves and surfaces show that for a reasonable degree raising and/or refinement, the vast majority of construction examples are successful.

On bivariate Kantorovich exponential sampling series

In this paper, we introduce and analyze the approximation properties of bivariate generalization for the family of Kantorovich type exponential sampling series. We derive the basic convergence result and Voronovskaya type theorem for the proposed sampling series. Using logarithmic modulus of smoothness, we establish the quantitative estimate of order of convergence for the Kantorovich type exponential sampling series. Furthermore, we study the convergence results for the generalized Boolean sum (GBS) operator associated with bivariate Kantorovich exponential sampling series. At the end, we provide a few examples of kernels to which the presented theory can be applied along with the graphical representation and error estimates.

Some new inequalities and numerical results of bivariate Bernstein-type operator including Bézier basis and its GBS operator

We investigate some new inequalities by estimating the rate of convergence by means of the complete modulus of continuity and a class of Lipschitz functions for the bivariate Bernstein-type operator including Bézier basis and present an example including numerical results comparing its rate of convergence. Moreover, we introduce its GBS (Generalized Boolean Sum) operator and obtain its rate of convergence with the help of the mixed modulus of continuity and Lipschitz class of Bögel continuous functions with exemplifying numerical results. Our research will demonstrate that the GBS operator possesses at least better numerical results than the bivariate Bernstein-type operator. All mentioned results point out the novelty of this study.

Convergence theorems in Orlicz and Bögel continuous functions spaces by means of Kantorovich discrete type sampling operators

In this study, we prove the convergence theorems on the space of compactly supported functions and in the general setting of Orlicz spaces for a general class of Kantorovich type discrete operators defined by Carlo Bardaro and Ilaria Mantellini. We also define the generalized Boolean sum (GBS) operator for the class of bivariate Kantorovich type discrete operators and examine the approximation properties of GBS operators in the space of Bögel functions.

Rate of convergence of parametrically generalized bivariate Baskakov-stancu operators

In the present work, we consider Stancu variant of a bivariate parametrically generalised Baskakov operator. We discuss the rate of convergence of these operators by means of partial moduli of continuity and modulus of continuity of second order. Also, we prove Vornovskaja type assymptotic theorem. Furthermore, the Generalized Boolean Sum (GBS) operators associated to Stancu variant of a- Baskakov operators are constructed and we study their order of convergence using mixed modulus of smoothness for B?ogel continuous and B?ogel differentiable functions. Some surface plotting illustrating the approximation for different values of Stancu variables and the error of approximation by the proposed operators are also given using MATLAB programming.

Iterates of q-Bernstein operators on triangular domain with all curved sides

Abstract In this article, Phillips-type Bernstein operators ( ℬ m , q t F ) ( t , s ) ({{\mathcal{ {\mathcal B} }}}_{m,q}^{t}F)\left(t,s) and ( ℬ n , q s F ) ( t , s ) ({{\mathcal{ {\mathcal B} }}}_{n,q}^{s}F)\left(t,s) , their products, and Boolean sum based on q-integer have been studied on a triangle with all curved sides. Furthermore, convergence of iterates of these operators have been analyzed using the weakly Picard operators technique and the contraction principle.

Cheney–Sharma Type Operators on a Triangle with Straight Sides

We consider two types of Cheney–Sharma operators for functions defined on a triangle with all straight sides. We construct their product and Boolean sum, we study their interpolation properties and the orders of accuracy and we give different expressions of the corresponding remainders, highlighting the symmetry between the methods. We also give some illustrative numerical examples.

On the rate of convergence of modified \(\alpha\)-Bernstein operators based on q-integers

In the present paper we define a q-analogue of the modified a-Bernstein operators introduced by Kajla and Acar (Ann. Funct. Anal. 10 (4) 2019, 570-582). We study uniform convergence theorem and the Voronovskaja type asymptotic theorem. We determine the estimate of error in the approximation by these operators by virtue of second order modulus of continuity via the approach of Steklov means and the technique of Peetre's \(K\)-functional. Next, we investigate the Gruss- Voronovskaya type theorem. Further, we define a bivariate tensor product of these operatos and derive the convergence estimates by utilizing the partial and total moduli of continuity. The approximation degree by means of Peetre's K- functional , the Voronovskaja and Gruss Voronovskaja type theorems are also investigated. Lastly, we construct the associated GBS (Generalized Boolean Sum) operator and examine its convergence behavior by virtue of the mixed modulus of smoothness.

A Review on Some Linear Positive Operators Defined on Triangles

We consider results regarding Bernstein and Cheney–Sharma-type operators that interpolate functions defined on triangles with straight and curved sides and we introduce a new Cheney–Sharma-type operator for the triangle with one curved side, highlighting the symmetry between the methods. We present some properties of the operators, their products and Boolean sums and some results regarding the remainders of the corresponding approximation formulas, using modulus of continuity and Peano’s theorem. Additionally, we consider some numerical examples to show the approximation properties of the given operators.

Approximation by Modified Bivariate Bernstein-Durrmeyer and GBS Bivariate Bernstein-Durrmeyer Operators on a Triangular Region

In this paper, the approximation properties and the rate of convergence of modified bivariate Bernstein-Durrmeyer Operators on a triangular region are examined. Furthermore, definitions and some properties of modulus of continuity for functions of two variables are given. Voronovskaya and Gr\"{u}ss Voronovskaja type theorems are used to determine the order of approximation. The GBS (Generalized Boolean Sum) operator of Bivariate Bernstein-Durrmeyer type on a triangular region is studied. Lastly, some numerical examples are given and related graphs are plotted for comparison.

Some inequalities and numerical results estimating error of approximation for tensor product kind bivariate quantum beta-type operators and pertaining to GBS variant

In this study, we investigate some inequalities estimating the error of approximation for new defined tensor product kind quantum beta-type operators on rectangular regions, and we give an inequality in weighted mean. Moreover, we introduce generalized Boolean sum (GBS) operators pertaining to tensor product kind bivariate quantum beta-type operators, and we also give inequalities estimating the error of approximation for GBS variant concerning the mixed modulus of continuity for Bögel continuous and Bögel differentiable functions, respectively. Finally, we present some applications illustrating numerical results of approximation errors with the help of Maple software.

Approximation by Tensor-Product Kind Bivariate Operator of a New Generalization of Bernstein-Type Rational Functions and Its GBS Operator

We introduce a tensor-product kind bivariate operator of a new generalization of Bernstein-type rational functions and its GBS (generalized Boolean sum) operator, and we investigate their approximation properties by obtaining their rates of convergence. Moreover, we present some graphical comparisons visualizing the convergence of tensor-product kind bivariate operator and its GBS operator.

The study of the properties of structural models of elements of educational environment

The widespread use of mathematical modeling is often constrained by the insufficient level of technological maturity and the lack of theoretical foundations to justify the effectiveness of different classes of models. One of the ways to improve educational project management systems is to solve problems of modeling the structure and trajectory of changes in the parameters of the system in education management, as a complex poorly structured organizational and technical system. To solve the problem of analysis of structural schemes of complex technical systems, which include education systems, it is proposed to use the analytical methodof determining closed cycles in complex control systems. The method is based on the use of specific properties of adjacencymatrices. It is shown that the degrees of the adjacency matrix follow the general structure of the oriented graph with certain regularities of map-ping the arcs of the graph. This allows you to create a matrix of reach of the studied topological structure with the selection of con-tours in an oriented graph. A method for identifying cycles in graphs based on the formation of the Boolean sum of degrees ofthe reach matrix with its subsequent transposition and superposition has been developed, which allows obtaininga mapping of a contour in a graph in the form of a square submatrix filled with units.

Understanding anomalies in animal behaviour: data on cow activity in relation to health and welfare

We collected data on the behaviour of dairy cows in barns, clinical signs of diseases as well as events that may stress or agitate the cows. A Real-Time Locating System gives the position of individual cows every second. The position of the cow is determined by triangulation based on radio waves emitted by a tag fixed on each cow neck collar and captured by antennas in the barn. The cow’s activity is inferred from its position: ‘eating’ if the cow is positioned at the feeding table, ‘resting’ if the cow is in a resting area (typically cubicles), else ‘in alleys’. We aggregated this information to get the time spent in each activity per hour. We also calculated the activity level of the cow for each hour of the day by attributing a weight to the time spent in each activity. For each cow and day, we collected information on health events or other events that may affect behaviour. There were 11 types of events. Six events were linked to health: lameness; mastitis; LPS (i.e. administration of lipopolysaccharide (LPS) in the mammary gland, an experimental treatment to induce udder inflammation); subacute ruminal acidosis; other diseases (such as colic, diarrhoea, ketosis, milk fever or other infectious diseases); and accidents (such as retained placenta or vaginal laceration). Two events were linked to reproduction: oestrus and calving. Three events were stress events: animal mixing, disturbance (i.e. mild intervention on animals such as late feeding, alarm test) and marginal management changes (ration changes, fill bed). In addition, a Boolean sums up whether this hour was considered as normal or not. Data contain four datasets. It consists of univariate time series. Each time series corresponds to the hourly activity level of a cow. Datasets 1 and 2 are from the INRAE Herbipôle experimental farm and include data from experiments; datasets 3 and 4 are from commercial farms. They contain data on respectively 28, 28, 30 and 300 cows monitored for 6 months, 2 months, 40 days and one year. The data can be used to study the links between health, reproduction events and stress on the one hand and cow behaviour on the other hand. More specifically, it can be used to build and test tools for an earlier detection of health and disturbances, with a view to inform caretakers so that corrective actions can be rapidly put in place.

A strong converse inequality for the iterated Boolean sums of the Bernstein operator

"We establish a two-term strong converse estimate of the rate of approximation by the iterated Boolean sums of the Bernstein operator. The characterization is stated in terms of appropriate moduli of smoothness or K-functionals."

A new kind of Bi-variate $ \lambda $ -Bernstein-Kantorovich type operator with shifted knots and its associated GBS form

<p style='text-indent:20px;'>In this paper, we introduce a bi-variate case of a new kind of <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-Bernstein-Kantorovich type operator with shifted knots defined by Rahman et al. [<xref ref-type="bibr" rid="b31">31</xref>]. The rate of convergence of the bi-variate operators is obtained in terms of the complete and partial moduli of continuity. Next, we give an error estimate in the approximation of a function in the Lipschitz class and establish a Voronovskaja type theorem. Also, we define the associated GBS(Generalized Boolean Sum) operators and study the degree of approximation of Bögel continuous and Bögel differentiable functions by these operators with the aid of the mixed modulus of smoothness. Finally, we show the rate of convergence of the bi-variate operators and their GBS case for certain functions by illustrative graphics and tables using MATLAB algorithms.</p>