Unrestricted Order Research Articles

Bright solitons of the model with arbitrary refractive index and unrestricted dispersion

Objective:The generalized Schrödinger equation with arbitrary refractive index and unrestricted order is considered. The objective of this paper is to demonstrate the bright solitons of this mathematical model can be found by the same formula taking into account the order of nonlinear differential equation. Method:The algorithm of constructing optical solitons of the generalized nonlinear Schrödinger equation with unrestricted dispersion using the simplest equation method is presented and applied for finding of optical solitons. Results:The optical solitons of the generalized nonlinear Schrödinger equation of the twelfth order are found at three powers on nonlinearity. The application of the method for finding solution of the generalized Korteweg–de Vries equation with arbitrary power of nonlinearity and unrestricted dispersion is discussed.

Optical Solitons of the Generalized Nonlinear Schrödinger Equation with Kerr Nonlinearity and Dispersion of Unrestricted Order

The family of the generalized Schrödinger equations with Kerr nonlinearity of unrestricted order is considered. The solutions of equations are looked for using traveling wave reductions. The Painlevé test is applied for finding arbitrary constants in the expansion of the general solution into the Laurent series. It is shown that the equation does not pass the Painlevé test but has two arbitrary constants in local expansion. This fact allows us to look for solitary wave solutions for equations of unrestricted order. The main result of this paper is the theorem of existence of optical solitons for equations of unrestricted order that is proved by direct calculation. The optical solitons for partial differential equations of the twelfth order are given in detail.

فقہی قاعدہ تَرْكُ الِاسْتِفْصَالِ كعُمُومِ الْمَقَال کا تحقیقی وتجزیاتی مطالعہ

This discussion is about an important rule of Islamic jurisprudence. This rule arises when the Prophet (peace be on him) forsakes the demand for details on an event or a question which is likely to be an event or a question in many cases instead of confining it to a particular situation. An unrestricted order is imposed and this order is common to all cases of this condition. This rule was first deduced by Imām al-Shāfi‘ī, and then it was adopted in many cases but not all of those cases are entirely out of the question: tark al-istifṣāl ka ‘umūm al-maqāl (abandoning the demand for details like a normal conversation). Most of the jurists have unrestrictedly followed this rule, some have restricted it to certain conditions and some have rejected it. This rule apparently contradicts another rule that is also attributed to Imām al-Shafi‘ī, which is: Qaḍāya al-aḥwāl idhā taṭarraqa ilayhā ’l-iḥtimāl kasāḥā thawb al-ijmāl wa saqaṭ bihā ’l-istidlāl (when a case is probable, this probability creates brevity and the argument based on it turns void). Therefore, one of the purposes of this discussion is to dispel the illusion of contradiction between these two rules.

Unordered rule discovery using Ant Colony Optimization

In this article, a novel unordered classification rule list discovery algorithm is presented based on Ant Colony Optimization (ACO). The proposed classifier is compared empirically with two other ACO-based classification techniques on 26 data sets, selected from miscellaneous domains, based on several performance measures. As opposed to its ancestors, our technique has the flexibility of generating a list of IF-THEN rules with unrestricted order. It makes the generated classification model more comprehensible and easily interpretable. The results indicate that the performance of the proposed method is statistically significantly better as compared with previous versions of AntMiner based on predictive accuracy and comprehensibility of the classification model.

A Bound on the MSE of Oversampled Dithered Quantization With Feedback

We analyze the behavior of the mean squared error (MSE) achievable by oversampled, uniform scalar quantization using feedback, pre- and post-filters of unrestricted order, when encoding wide-sense stationary discrete-time random sources having (possibly) unbounded support. Our results are based upon the use of subtractively dithered uniform scalar quantizers. We consider the number of quantization levels, N, to be given and fixed, which lends itself to fixed-rate encoding, and focus on the cases in which N is insufficient to avoid overload. In order to guarantee the stability of the closed-loop, we consider the use of a clipper before the scalar quantizer. Our results are valid for zero-mean sources having independent innovations whose moments satisfy some mild requirements, which are met by infinite-support distributions such as Gaussian and Laplacian. We show that, for fixed N, the MSE can be made to decay with the oversampling ratio lambda as O(e-c 0 lambda 1/3) when lambda tends to infinity, where c 0 [0.5(N-1)]2/3. We note that the latter bound is asymptotic in lambda but not in N, and that it includes clipping errors.

Berry-phase treatment of the homogeneous electric field perturbation in insulators

A perturbation theory of the static response of insulating crystals to homogeneous electric fields, that combines the modern theory of polarization (MTP) with the variation-perturbation framework is developed, at unrestricted order of perturbation. First, we address conceptual issues related to the definition of such a perturbative approach. In particular, in our definition of an electric-field-dependent energy functional for periodic systems, the position operator appearing in the perturbation term is replaced by a Berry-phase expression, along the lines of the MTP. Moreover, due to the unbound nature of the perturbation, a regularization of the Berry-phase expression for the polarization is needed in order to define a numerically-stable variational procedure. Regularization is achieved by means of discretization, which can be performed either before or after the perturbation expansion. We compare the two possibilities and apply them to a model tight-binding Hamiltonian. Lowest-order as well as generic formulas are presented for the derivatives of the total energy, the normalization condition, the eigenequation, and the Lagrange parameters.

On-line rankings of graphs

A (vertex) k-ranking of a graph G=( V, E) is a proper vertex coloring ϕ: V→{1,…, k} such that each path with endvertices of the same color i contains an internal vertex of color ⩾ i+1. In the on-line coloring algorithms, the vertices v 1,…, v n arrive one by one in an unrestricted order, and only the edges inside the set { v 1,…, v i } are known when the color of v i has to be chosen. We characterize those graphs for which a 3-ranking can be found on-line. We also prove that the greedy (First-Fit) on-line algorithm, assigning the smallest feasible color to the next vertex at each step, generates a (3 log 2 n) -ranking for the path with n⩾2 vertices, independently of the order in which the vertices are received.

Set mappings of unrestricted order

A set mapping on a set S is a function f mapping S into the powerset of S such that x ∉ f(x) for each x in S. The set map f has order θ if θ is the least cardinal such that |f(x)| < θ for each x in S. A subset H of S is free for f if x ∉ f(y) for all x, y in H. In this paper we use classical results about set mappings of large order to investigate conditions which ensure a large free set for set mappings of unrestricted order.

Text-Music Relationships in Arapaho Songs

THE RELATIONSHIP between song texts and music in primitive cultures is one of the less known fields within comparative musicology. Few musical styles have been studied with regard to the linguistic aspects of the songs. It is the purpose of this paper to survey this problem in the music of the Arapaho.' We first list briefly the main characteristics of the Arapaho musical style. It is, as far as is known, entirely monophonic and primarily vocal, the songs being accompanied by drums and rattles. Rather large intervals and large melodic range predominate. Melodic movement is almost exclusively downward, and the singing is characterized by a great deal of vocal tension and rhythmic pulsation on the longer notes. The form of the songs may be called strophic; one long, descending melodic line which flattens out at the end is repeated several times; it, in turn, consists of several phrases each of which also descends, but each of which begins somewhat lower than the previous one, making a somewhat terrace-shaped contour. According to their texts, the Arapaho songs can be divided into (a) those which have only meaningless syllables, and (b) those which have meaningful texts. The latter group is in the minority, if the approximately one hundred twenty songs which are known can be considered representative. To be sure, it is known that some songs may be sung both with a meaningful text and entirely with meaningless syllables. We discuss first the larger category, that of songs with only meaningless syllables. This category is again divided, according to the types of syllables used, into two groups. The first includes most of the song types; it is definitely the older and includes the songs of all functions except Peyote. The songs presently under consideration are characterized by meaningless syllables which begin with h, w, or y, and which end in a vowel or with w or y. Typical syllables are hey, hay, ya, how, wu, and similar ones. An attempt was made to determine if any set patterns are used in determining the position of the syllables, or their order, within a song. Little positive information could be gained, however. Generally speaking, the syllables appear to be placed in a song in rather unrestricted order. On the other hand, however, it was found that singers used almost

Societies and Academies

Societies and Academies