Chapter 3 - Basic operations of level-set functions

The basic set operations between fuzzy sets are defined using the min and max functions; however, later on, new operators were introduced that used other functions, which, nevertheless, had similar properties to functions min and max. The resulting fuzzy set theories are more suitable for the description and processing of specific data sets. Crisp and fuzzy multisets have found numerous applications but still the basic operations are based on functions min and max. It is straightforward to replace these functions in the fuzzy part of fuzzy multisets; however, it is not as easy but is feasible to do the same with the multisets and the “crisp” part of fuzzy multisets. The new mathematical structures are called triangular multisets and triangular fuzzy multisets, respectively. The aim is to facilitate the processing of certain data sets so they can be used in multi-criteria decision making and computing.

In this article, an “algebra” (sometimes also called a “universal algebra”) is a nonempty set X together with a set of “basic operations”; each operation is a finitary total function on X , i.e., a function ƒ : X n → X for some n ∈ {1, 2, …}. While our operations always have finite arity, we allow infinitely many basic operations in our algebras. Clone theory investigates the structure of all algebras on a given set. It turns out that several questions originating in algebra can be solved (or at least attacked) with set-theoretic methods. This article collects a few examples of this approach, based on recent results of Pinsker, Shelah and the author. Background . A large part of universal algebra investigates the relations between the basic operations of an algebra; here we take a different point of view—all term operations are considered to live an an equal level; thus we will, for example, not distinguish between a Boolean algebra and the corresponding Boolean ring, because each ring operation can be written as composition of operations of the Boolean algebra, and conversely. This point of view is motivated by the observation that many properties of algebras depend only on the clone of its term functions or polynomials, without any need to distinguish basic operations; for example, an equivalence relation is a congruence relation iff it is closed under all unary polynomials.

For large-scale hydraulic and hydroelectric project, it has complex underground tunnel group. Traditional design for tunnel was usually based on the method of CAD in 2D which could not embody the geological features where the tunnel was passed, and also could not describe the space relations of the tunnel networks. With the development of graphics and GIS, 3D Design for hydraulic tunnel is to be possible. This paper proposes one method for modeling hydraulic tunnel in 3D and analyzing its space distance in 3D based on distance analysis of GIS. The space distance is important for construction safety protection. The design of tunnel can be divided into two parts: the section and the axis. The design of section is based on its function of tunnel, such as non-pressure tunnel and pressure tunnel. The design of axis is based on space distribution. The main procedure of modeling the tunnel: determining the axis, and then choosing the corresponding section. The section needs to be performed the translated and rotated matrix to the proper position along the axis. The next step is to extrude the section along the axis, so the entity model of one tunnel is created. Two entities are the outer one and the inner one. The tunnel model can be acquired through the outer tunnel subtracting the inner one by Boolean operations. The tunnel model is one entity by complex geometric transformations and Boolean operations. Performing the space distance analysis directly is quite difficult, and it needs find an indirect mean. The axis of tunnel has relatively simple expression, so, the link between the tunnel model and its corresponding axis and section is established. Here, the C++ as the program design language, the map class is used to implement the link. Thus, it can find the axis and section equation from the tunnel. In virtue of such method, analyzing the space distance of complex solid model becomes easy. The calculating model is created and discussed how to calculate the minimum distance between tunnels. At first, the minimum distance between axes is calculated, which formed a minimum distance Vector. Second, the parts between the axes and sidewall of tunnel are subtracted. Finally, this method is applied to one large-scale hydraulic and hydroelectric project.

As the semiconductor technology is scaling down to nanometer regime, accurate layout analysis requires operations with simulated contours of VLSI layouts which are non-rectilinear.The basic operations on non-rectilinear shapes include union, connected components and boundary. Whatever precision one would choose for the representation of vertices of shapes and the exact representation of, say, A∩B would require an even higher precision. Consequently rounding of intermediate results of a chain of basic operations seems to be inevitable. The presence of rounding errors is unacceptable because the implementation of algorithms for all operations which involve topology, such as connected components, find boundary, etc, becomes impossible or extremely complexWe present a complete solution to the following problem: Let a VLSI design or a simulated through a lithography process image of a VLSI design be given by a number of two-dimensional point sets. Each point set consists of polygons with arbitrary (not necessarily 90°) angles and with vertices on an integer grid. Perform any amount of sequential basic operations so that:• The resulting point sets are mathematically exact, that is no rounding errors are allowed. In particular the connectivity of the point sets remains intact, the boundaries remain undistorted and the statements like = B) ∪ ∩ B), (A B) is disjoint with B and ∩ ⊂ A always hold.• The amount of time and memory per one basic geometric operation on elementary polygons (trapezoids) is constant. For example it would not be an acceptable solution to represent the vertices of the new shapes which appear as a result of the operations of intersection, difference, etc, by unlimited length rational numbers.

Qualitative temporal and spatial reasoning is in many cases based on binary relations such as before, after, starts, contains, contact, part of, and others derived from these by relational operators. The calculus of relation algebras is an equational formalism; it tells us which relations must exist, given several basic operations, such as Boolean operations on relations, relational composition and converse. Each equation in the calculus corresponds to a theorem, and, for a situation where there are only finitely many relations, one can construct a composition table which can serve as a look up table for the relations involved. Since the calculus handles relations, no knowledge about the concrete geometrical objects is necessary. In this sense, relational calculus is "pointless". Relation algebras were introduced into temporal reasoning by Allen (1983, Communications of the ACM 26(1), 832--843) and into spatial reasoning by Egenhofer and Sharma (1992, Fifth International Symposium on Spatial Data Handling, Charleston, SC). The calculus of relation algebras is also well suited to handle binary constraints as demonstrated e.g. by Ladkin and Maddux (1994, Journal of the ACM 41(3), 435--469). In the present paper I will give an introduction to relation algebras, and an overview of their role in qualitative temporal and spatial reasoning.

A extended finite element (X-FEM) method and corresponding Fortran code are developed in the modeling and simulation for nonlinear geometry and fracture mechanics problems. X-FEM can model a domain without explicitly meshing the crack surface. This method can treat an arbitrary crack independent of the mesh and crack growth. The X-FEM formulas with nonlinear geometry are deduced. In order to model the crack discontinuity, a Heaviside step function and a two-dimensional asymptotic crack-tip displacement field are added to the traditional finite element approximation for the local enrichment based on the theory of partition of unity. The crack is described by two level set functions. The X-FEM computational algorithm is presented in the framework of Lagrangian description in order to model the arbitrary discontinuities in large deformations. The stress intensifying the factors of a crack are calculated by using the multi-point displacement extrapolation method and least square fitting. Finally, a numerical example is presented to demonstrate the accuracy and efficiency of the X-FEM and the FORTRAN code in large deformation crack problems. It is found that X-FEM is superior to the traditional FEM in the modeling and simulation of crack being and crack growth program.

We describe continuous linear operators acting in a countable inductive limit of weighted Frchet spaces of entire functions of several complex variables and commuting in these spaces with systems of partial differentiation and translation operators. Under the made assumptions, the commutants of the systems of differentiation and translation operators coincide. They consist of convolution operators defined by an arbitrary continuous linear functional on . At that, we do not assume that the set of the polynomials is dense in . In the space topological dual to , we introduce the natural multiplication. Under this multiplication, the algebra is isomorphic to the aforementioned commutant with the usual multiplication, which is the composition of the operators. This isomorphism is also topological if is equipped by the weak topology, while the commutant is equipped by the weak operator topology. This implies that the set of the polynomials of the differentiation operators is dense in the commutant with topology of pointwise convergence. We also study the possibility of representing an operator in the commutant as an infinite order differential operator with constant coefficients. We prove the immediate continuity of linear operators commuting with all differentiation operators in a weighted (LF)-space of entire functions isomorphic via Fourier-Laplace transform to the space of infinitely differentiable functions compactly supported in a real multi-dimensional space.

Deactivation of commercially relevant cobalt catalysts for Low Temperature Fischer-Tropsch (LTFT) synthesis is discussed with a focus on the two main long-term deactivation mechanisms proposed: Carbon deposits covering the catalytic surface and re-oxidation of the cobalt metal. There is a great variety in commercial, demonstration or pilot LTFT operations in terms of reactor systems employed, catalyst formulations and process conditions. Lack of sufficient data makes it difficult to correlate the deactivation mechanism with the actual process and catalyst design. It is well known that long term catalyst deactivation is sensitive to the conditions the actual catalyst experiences in the reactor. Therefore, great care should be taken during start-up, shutdown and upsets to monitor and control process variables such as reactant concentrations, pressure and temperature which greatly affect deactivation mechanism and rate. Nevertheless, evidence so far shows that carbon deposition is the main long-term deactivation mechanism for most LTFT operations. It is intriguing that some reports indicate a low deactivation rate for multi-channel micro-reactors. In situ rejuvenation and regeneration of Co catalysts are economically necessary for extending their life to several years. The review covers information from open sources, but with a particular focus on patent literature.

An effective simulation mechanism for construction operations

Time is a critical factor in processing a very large volume of data a.k.a ‘Big Data’. Many existing data mining algorithms (supervised and unsupervised) become futile because of the ubiquitous use of horizontal processing i.e. row-by-row processing of stored data. Processing time for big data is further exacerbated by its high dimensionality (# of features) and high cardinality (# of records). To address this processing-time issue, we proposed a vertical approach with predicate trees (pTree). Our approach structures data into columns of bit slices, which range from few to hundreds and are processed vertically i.e. column by column. We tested and compared our vertical approach to traditional (horizontal) approach using three basic Boolean operations namely addition, subtraction and multiplication with 10 data sizes. The length of data size ranged from half a billion bits to 5 billion bits. The results are analyzed w.r.t processing speed time and speed gain for both the approaches. The result shows that our vertical approach outperformed the traditional approach for all Boolean operations (add, subtract and multiply) across all data sizes and results in speed-gain between 24% to 96%. We concluded from our results that our approach being in data-mining ready format is best suited to apply to operations involving complex computations in big data application to achieve significant speed gain.

Stochastic realization theory provides a natural theoretical background for recent identification methods, called subspace methods, which have shown superior performance for multivariable state-space model-building. The basic steps of subspace algorithms are geometric operations on certain vector spaces generated by observed input-output time series which can be interpreted as “sample versions” of the abstract geometric operations of stochastic realization theory. The construction of the state space of a stochastic process is one such basic operation.

We propose a new O(n)-space implementation of the GKO-Cauchy algorithm for the solution of linear systems with Cauchy-like matrix. Despite its slightly higher computational cost, this new algorithm makes a more efficient use of the processor cache memory. Thus, for matrices of size larger than about 500-1000, it outperforms the existing algorithms. We present an applicative case of Cauchy-like matrices with non-reconstructible main diagonal. In this special instance, the O(n) space algorithms can be adapted nicely to provide an efficient implementation of basic linear algebra operations in terms of the low displacement-rank generators.

A computer system is proposed for bi or tri-dimensional graphic display of various structural shapes (molecular bodies) and property shape representations (one electron properties, electrostatic potential…). Calculation spaces, linked to molecular coordinates, are jointly handled within a unique representation space structured as a 3 D stage or grid Box. The Box fixed coordinates ensure Boolean and geometric operations through interactive graphic sub-systems. Direct images, subimages and composite images can be generated, as well as processed data and/or images. The polymodelling system presented — POLYMOD — leads to rapid and original animation operations on modelizations based on lengthy calculations.

Solid modeling is the technique for representing and manipulating a complete description of physical objects. Valid representations of solid are developed base on a mathematical description of rigid object. A valid solid representation scheme provides sufficient information for geometric operations on objects, e.g. Boolean operations, collision detection. The description here introduces the Fundamental concepts in solid modeling are introduced. Characteristics of a solid representation scheme are discussed. Two popular solid representation schemes, the constructive solid geometry and the boundary representation schemes, are described. The fundamental algorithms for solid modeling, point membership classification, and Boolean operations, are also discussed. Validity of the boundary representation scheme, the Euler operators, as well as the Winged‐edge data structure for the implementation of boundary representation scheme are also presented.

With the development of artificial intelligence, the separation of memory and processor in the traditional von-Neumann architecture has led to the bottleneck of data transmission hindering the development of energy efficient computing. The computing-in-memory (CIM) paradigm is expected to solve the problems of memory wall and power wall. In this work, we propose a time-domain (TD) computing scheme based on the spin transfer torque magnetic random access memory (STT-MRAM). Basic Boolean logic operations, such as AND/OR/Full-adder (FA), are implemented through converting the bit-line voltage to time delay and time-to-digital converter (TDC). The proposal is simulated using the 28 nm CMOS process and 40 nm MTJ compact model. Monte-Carlo simulations show that 94.2% to 100% computation accuracy can be obtained and the delay of AND/OR and FA is 2.5 ns and 3.5 ns. The energy consumption of AND/OR and FA achieve 59.43 fJ and 97.56 fJ, respectively.