Boolean multiplication

Figure 3.4.4-1 summarizes conventional fault tree symbols. The many symbols are daunting, but remember that the computer only performs AND (Boolean multiplication) and OR (Boolean addition) operations. All else are combinations of these. 

Boolean multiplication is distributive with respect to Boolean union that is,... 

This modification for Boolean matrix multiplication permits use of the Boolean union operation (logical OR operation or logical sum) instead of regular multiplication and union operations. The Boolean union operation can be executed much faster on a digital computer. Experience has shown that performing the Boolean union of rows instead of the standard Boolean multiplication of matrices can reduce the computation time by as much as a factor of four. 

Table 2.1-1 compares the ordinary algebra of continuous variables with the Boolean algebra of 1 s and Os. This table uses the symbols and -h for the operations of intersection (AND) and union (OR) which mathematicians represent by n and u respectively. The symbols and -f which are the symbols of multiplication and addition, are used because of the similarity of their use to AND and OR in logic. 

Searching is fast and easy. Along with a simple keyword search, IPN offers alternative searches by patent number, boolean text, and advanced text that allows for multiple field searching. Browsing provides an organized approach to searching for patents. Through a review of specific classifications, you can identify topics and patents of interest. 

Boolean matrix multiplication is carried out according to the usual matrix rules, expressed in terms of the logical product and logical sum as defined above. 

One method of partitioning the system equations is to compute the maximal loops using powers of the adjacency matrix as discussed in Section II. Certain modifications to the methods of Section II are needed in order to reduce the computation time. The first modification is to obtain the product of the matrices using Boolean unions of rows instead of the multiplication technique previously demonstrated to obtain a power of an adjacency matrix. To show how the Boolean union of rows can replace the standard matrix multiplication, consider the definition of Boolean matrix multiplication, Eq. (2), which can be expanded to... 

The rule structure allows simple Boolean functions to be performed. Multiple numbered IF clauses are logically ANDed together. Multiple clauses which are part of the same numbered IF are logically ORed. The logical NOT does not exist, but can be simulated using predicates with the opposite meaning in the IF clause, (i.e. BIGGER is equivalent to NOT SMALLER). Table IV lists the currently available predicates for IF clauses. 

In order to bridge the gap between the discretized micro- and macro-worlds, averaging of the variables is necessary. Macroscopic variables in the N-S equation, are the density p and the momentum I, which are functions of the lattice space vector r and time t. The local density p is the summation of the average number of particles travelling along each of six (hexagonal) directions, with velocity c. Multiplication of the density p by the velocity vector u equals linear momentum (I = pu). Boolean algebra is applied for the expressions of the discretized variables density and momentum, respectively, as follows ... 

Another Boolean value to indicate if attacks on multiple virtual machines were performed in sequence or simultaneously. 

T. Akutsu, S. Miyano, and S. Kuhara, Algorithms for identifying Boolean networks and related biological networks based on matrix multiplication and fingerprint fnnction. / Comput Biol 7(3 ) 331-343 (2000). 

Multiple indices may be searched simultaneously using Entrez s Boolean operators. For example, a query in PubChem Compound of "Br[Element] AND l[CovalentUnitCount]" will find all chemical structures containing the element bromine and that are not part of a mixture. Please note that Entrez Boolean operators are capitalized (e.g., "AND," "OR," and "NOT"). 

Selection of D enables the user to define a search (Figure 4) in terms of all attributes of the wider biological record, now perceived as including the structure (Option C) and values which are specific to particular tests (Option T). The search may include multiple terms, earlier sets and Boolean logic. 

FTA and its Boolean nature is limited to considering components with only two states, either good or failed [Bossche, 1988, paragraph 2]. In reality, components may have multiple degraded states that have different impacts on system behaviour, and further variations depending on flight phase or time variables. It is very difficult if not impossible to consider this type of complexity in FTA, and other tools (such as Markov) may be useful. 

The Boolean logic of a fault tree requires that each basic event is independent. While we have tackled repeated events, and how they can be reduced using Boolean logic, there is also the case where a common cause can affect multiple seemingly independent components simultaneously. The common causes being referred to are either an external influence that may cause multiple otherwise independent parts to fail simultaneously (such as particular risks, or manufacturing defects for common batch components), or an implicit dependency in the failures where the failure of... 

A fluidic amplifier controls the pressure and the flow rate in a microchip through the interaction of multiple fluid streams or jets. Fluidic amplifiers can be used for both analog (proportional) and digital (bistable) control. They can also be used to perform various Boolean functions in a microfluidic chip for the purpose of control logic. 

Semantics are given to the graph structure by associating nodes with function decompositions. In the following, a binary decomposition that can be described in terms of equation (1), will be called a Kronecker decomposition. Rmctions diow xi) and dhigh xi) only depend on the boolean variable and (or) on constant values. Operations -I- and represent addition and multiplication in the domain of function /. 

Theoretical results support the conjecture, that there exist no single graph type that can represent boolean and arithmetic functions with the same efficiency. Therefore, it is important to provide a framework for the integration of multiple graph types. 

Fault Tree Analysis (FTA) is a well known and widely used safety tool, implementing a deductive, top down approach. It starts with a top level hazard, which has to be known in advance and "works the way down" through all causal factors of this hazard, combined with Boolean Logic (mainly AND and OR gates). It can consider hardware, software and human errors and identifies both single and multiple points of failure. Both a quantitative and qualitative analysis is possible. 

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