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

On a small class of Boolean sums

Neciporuk [3], Lamagna/Savage [1] and Tarjan [6] determined the monotone network complexity of a set of Boolean sums if each two sums have at most one variable in common. By this result they could define explicitely a set of n Boolean sums which depend on n variables and whose monotone complexity is of order n 3/2. In the main theorem of this paper we prove a more general lower bound on the monotone network complexity of Boolean sums. Our lower bound is for many Boolean sums the first nontrivial lower bound. On the other side we can prove that the best lower bound which the main theorem yields is the n 3/2-bound cited above. For the proof we use the technical trick of assuming that certain functions are given for free.

The [formula omitted]-hypomorphy of digraphs up to complementation

Volumetric Boolean sum

Degree of simultaneous approximation of bivariate functions by Gordon operators

Expressing Coons-Gordon surfaces as nurbs

Enhancement of Gordon-Coons interpolations by “bubble functions”

Neciporuk, Lamagna/Savage and Tarjan determined the monotone network complexity of a set of Boolean sums if any two sums have at most one variable in common. Wegener then solved the case that any two sums have at most k variables in common. We extend his methods and results and consider the case that any set of h +1 distinct sums have at most k variables in common. We use our general results to explicitly construct a set of n Boolean sums over n variables whose monotone complexity is of order n 5/3. The best previously known bound was of order n 3/2. Related results were obtained independently by Pippenger.

The article presents examples of solving the tasks of analysis of safety of complex technical systems by using logical-probabilistic methods (LPM). LPM are characterised by sufficient visualisation and simplicity of formalisation of the hazardous state of the object in the form of the shortest paths of hazardous state or hazardous state scenario. Parametric representation of logical functions (Boolean sum and products of initiating events and initiating conditions) is carried out using the methods of orthogonalisation. Determination of individual and total contribution of events in probability of realisation of a dangerous condition can allow you to plan the activities to ensure the safety state of the object. In this article the examples are being solved with the use of ARBITR software.

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.

For measurable functions f of two real variables there are considered the Boolean sums\(\tilde L_{m,n} f\) of parametric extensions of certain univariate Durmeyer-type operators\(\tilde L_m \) and\(\tilde L_n \). The weighted mixed moduli of continuity of\(\tilde L_{m,n} f\) are estimated and the degrees of approximation of f by\(\tilde L_{m,n} f\) in some weighted norms are investigated.

In the present paper we consider Jackson-type operators obtained via a Boolean sum, as studied by Cao, Gonska et al. We also present an alternative to their classical method of discretization using appropriate quadrature formulas. Our goal is to obtain for the discretized operators the same degree of approximation as Cao and Gonska (in particular, DeVore-Gopengauz inequalities), at the same time making sure that the operators discretized by means of our method will inherit more properties from the initial operators than by using quadrature formulas. As an example, we will consider convolution-type operators that preserve convexity up to a certain order, and we will show that the operators discretized with our technique also preserve monotonicity and convexity.

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 spline-based differential quadrature

The purpose of the paper is to introduce a new analogue of Phillips-type Bernstein operators B m , q u f u , v and B n , q v f u , v , their products P m n , q f u , v and Q n m , q f u , v , their Boolean sums S m n , q f u , v and T n m , q f u , v on triangle T h , which interpolate a given function on the edges, respectively, at the vertices of triangle using quantum analogue. Based on Peano’s theorem and using modulus of continuity, the remainders of the approximation formula of corresponding operators are evaluated. Graphical representations are added to demonstrate consistency to theoretical findings. It has been shown that parameter q provides flexibility for approximation and reduces to its classical case for q = 1 .