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Class Groups
Published in Richard A. Mollin, Algebraic Number Theory Second, 2011
Remark 3.16 Note that in Exercises 3.43-3.44, and 3.52-3.53, we are essentially dealing with the solutions of Pell’s equationx2-Dy2=±1. Euler misattributed a method of solving this equation to John Pell (1611-1685), whence its name. However, another English mathematician, William Brouncker (1601–1665) actually found the method. Lagrange was the first to prove that the positive Pell equation always has infinitely many solutions-see Biography 3.3 on page 93. The above exercises show that the Pell equation is actually about the fundamental unit of a quadratic field. Often, in an elementary number theory course, continued fractions are employed to solve the equation-see [53,§5.3, pp. 232-239] for instance.
Studying some networks using topological descriptors and multi-criterion decision making
Published in Molecular Physics, 2023
Guoping Zhang, Arfa Mushtaq, Adnan Aslam, Saima Parveen, Salma Kanwal
Choosing a leading network is our main perspective. We have four networks These networks are of the same class and . tetrahedron is the fundamental unit of these networks. Essentially, every silicate contains tetrahedron where the corner and the centre vertices represent oxygen and silicon ions, respectively. has a network structure of tetrahedron which is at the centre and is at each vertex. So this chemical compound has many physical properties which include density, melting point, boiling point, entropy and thermal conductivity, etc. In order to choose the leading network with respect to topological indices as criteria, the decision matrix is
Confidently extracting hierarchical taxonomy information from unstructured maintenance records of industrial equipment
Published in International Journal of Production Research, 2023
Abhijeet S. Bhardwaj, Dharmaraj Veeramani, Shiyu Zhou
Before presenting the details of our method, we provide below a brief overview of essential NLP terminology. Word: the fundamental unit of text (‘repair’).Token: single word or a group of words concatenated together to indicate a single entity. Token is uni-gram (‘filter’), bi-gram (‘drive-motor’) or tri-gram (‘motor-cooling-system’)Vocabulary: set of all words referred as set Document: A document (d) is a collection of words (e.g. maintenance record). We denote the set of all documents by Corpus: collection of multiple documents
An efficient QCA-based full adder design with power dissipation analysis
Published in International Journal of Electronics Letters, 2023
Ismail Gassoumi, Lamjed Touil, Abdellatif Mtibaa
QCA technology is a new paradigm which implements complex very large-scale integration (VLSI) circuits with very less power consumption and achieves high speed at nanoscale level compared to clasical technology. The fundamental unit of this technology is is a tiny square cell. This cell contains four holes with two injected electrons. Electrons can tunnel between dots. The two possible configurations of a cell illustrated in Figure 1b (Sasamal, Singh, Mohan et al., 2020a; Zahmatkesh et al., 2019). The cell with polarisation −1 represents logic ‘0’ and the cell with polarisation +1 represents logic ‘1’. If a set of cells is put beside each other, the polarisation of the first cell will be moved to the adjacent cell and so on towards the output cell. This chain of cells represents the QCA wire as depicted in Figure 1c. In QCA technology, the majority voter and inverter gates are the primitive building blocks to synthesise different circuits as shown in Figure 1d and e (Cho et al., 2007). AND and OR gates can be designed using the majority gate by fixing one of the three input cells to logic 0 and logic 1 respectively (Cho et al., 2007). For signal refreshing and control of the direction of signal propagation in the QCA design, a clock mechanism should be used as shown in Figure 1a (Sasamal, Singh, Mohan et al., 2020a). The QCA-based circuits can be divided into four sequential zones, each zone passing through four states (switch, hold, release, and relax) to ensure the circuit remains close to the ground state (Sasamal, Singh, Mohan et al., 2020a).