Development of symbolic algorithms for certain algebraic processes

This study investigates the problem of computing the exact greatest common divisor of two polynomials relative to an orthogonal basis, defined over the rational number field. The main objective of the study is to design and implement an effective and efficient symbolic algorithm for the general class of dense polynomials, given the rational number defining terms of their basis. From a general algorithm using the comrade matrix approach, the nonmodular and modular techniques are prescribed. If the coefficients of the generalized polynomials are multiprecision integers, multiprecision arithmetic will be required in the construction of the comrade matrix and the corresponding systems coefficient matrix. In addition, the application of the nonmodular elimination technique on this coefficient matrix extensively applies multiprecision rational number operations. The modular technique is employed to minimize the complexity involved in such computations. A divisor test algorithm that enables the detection of an unlucky reduction is a crucial device for an effective implementation of the modular technique. With the bound of the true solution not known a priori, the test is devised and carefully incorporated into the modular algorithm. The results illustrate that the modular algorithm illustrate its best performance for the class of relatively prime polynomials. The empirical computing time results show that the modular algorithm is markedly superior to the nonmodular algorithms in the case of sufficiently dense Legendre basis polynomials with a small GCD solution. In the case of dense Legendre basis polynomials with a big GCD solution, the modular algorithm is significantly superior to the nonmodular algorithms in higher degree polynomials. For more definitive conclusions, the computing time functions of the algorithms that are presented in this report have been worked out. Further investigations have also been suggested

Marginal abatement cost curves (MACCs): important approaches to obtain (firm and sector) greenhouse gases (GHGs) reduction

The study aims to identify appropriate methods that can help organisations to reduce energy use and emissions by using an effective concept of sustainability. In different countries, estimates of marginal abatement costs for reducing GHG emissions have been widely used. Around the world, many researchers have focused on MACCs and reported different results. This may due to different assumptions used which in turn lead to uncertainty and inaccuracy. Under these circumstances, much attention has been paid to the need for the role of MACC in providing reliable information to decision makers and various stakeholders. By reviewing the literature, this paper has analysed MACCs in terms of the role of different approaches to MACCs, representations of MACCs, MACC applications, pricing carbon, verification, and sectors analysis for energy and emissions projections. This paper concludes that MACCs should depend on actual data to provide more reliable information that may assist (firms and sectors) stockholders to determine what appropriate method for reducing emission

Computing Covers Using Prefix Tables

An \emph{indeterminate string} $x = x[1..n]$ on an alphabet $\Sigma$ is a sequence of nonempty subsets of $\Sigma$; $x$ is said to be \emph{regular} if every subset is of size one. A proper substring $u$ of regular $x$ is said to be a \emph{cover} of $x$ iff for every $i \in 1..n$, an occurrence of $u$ in $x$ includes $x[i]$. The \emph{cover array} $\gamma = \gamma[1..n]$ of $x$ is an integer array such that $\gamma[i]$ is the longest cover of $x[1..i]$. Fifteen years ago a complex, though nevertheless linear-time, algorithm was proposed to compute the cover array of regular $x$ based on prior computation of the border array of $x$. In this paper we first describe a linear-time algorithm to compute the cover array of regular string $x$ based on the prefix table of $x$. We then extend this result to indeterminate strings.Comment: 14 pages, 1 figur

Inferring an Indeterminate String from a Prefix Graph

An \itbf{indeterminate string} (or, more simply, just a \itbf{string}) \s{x} = \s{x}[1..n] on an alphabet $\Sigma$ is a sequence of nonempty subsets of $\Sigma$. We say that \s{x}[i_1] and \s{x}[i_2] \itbf{match} (written \s{x}[i_1] \match \s{x}[i_2]) if and only if \s{x}[i_1] \cap \s{x}[i_2] \ne \emptyset. A \itbf{feasible array} is an array \s{y} = \s{y}[1..n] of integers such that \s{y}[1] = n and for every $i \in 2..n$, \s{y}[i] \in 0..n\- i\+ 1. A \itbf{prefix table} of a string \s{x} is an array \s{\pi} = \s{\pi}[1..n] of integers such that, for every $i \in 1..n$, \s{\pi}[i] = j if and only if \s{x}[i..i\+ j\- 1] is the longest substring at position $i$ of \s{x} that matches a prefix of \s{x}. It is known from \cite{CRSW13} that every feasible array is a prefix table of some indetermintate string. A \itbf{prefix graph} \mathcal{P} = \mathcal{P}_{\s{y}} is a labelled simple graph whose structure is determined by a feasible array \s{y}. In this paper we show, given a feasible array \s{y}, how to use \mathcal{P}_{\s{y}} to construct a lexicographically least indeterminate string on a minimum alphabet whose prefix table \s{\pi} = \s{y}.Comment: 13 pages, 1 figur

Gravitational Properties of the Proca Field

We study various properties of a Proca field coupled to gravity through minimal and quadrupole interactions, described by a two-parameter family of Lagrangians. St\"uckelberg decomposition of the effective theory spells out its model-dependent ultraviolet cutoff, parametrically larger than the Proca mass. We present pp-wave solutions that the model admits, consider linear fluctuations on such backgrounds, and thereby constrain the parameter space of the theory by requiring null-energy condition and the absence of negative time delays in high-energy scattering. We briefly discuss the positivity constraints$-$derived from unitarity and analyticity of scattering amplitudes$-$that become ineffective in this regard.Comment: 23 pages, revised positivity-bound analysis, references adde

Pengaruh Negatif Era Teknologi Informasi dan Komunikasi pada Remaja (Perspektif Pendidikan Islam)

The development of information and communication technologies can no longer be avoided. Due to the presence of information and communication technology, one's life will be easier. Someone can send a message, send and find information quickly and easily. But behind the convenience to be aware of any negative impact of information and communication technology to teenagers. In this era of information and  communication  technology,  every  business  and  activities  as  well  as  a deliberate act to achieve a goal must have a foundation footing good and strong. Islamic education as a form of human effort, must have a foundation to which all activities  and  all  the  formulation  of  educational  goals  of  Islam  is  connected. Islamic religious education in overcoming the negative influence of the era of information and communication technologies in adolescents are educators especially for  the  elderly or  people  who  provide supplies  knowledge  of Islam against teenagers so that they can fortify themselves in the act, think, and act in accordance with the provisions outlined by Allah SWT. for the safety of his life