Search results for: intonation functions

Authors: Tatiana Stanchuliak

Abstract:

It is common knowledge that intonation is not laid over a ready text. On the contrary, intonation forms and accompanies the text on the level of its birth in the speaker’s mind. As a result, intonation plays one of the fundamental roles in the process of transferring a thought into external speech. Intonation structure can highlight the semantic significance of textual elements and become a ranging mark in understanding the information structure of the text. Intonation functions by means of prosodic characteristics, one of which is intonation salience, whose function in texts results in making some textual elements more prominent than others. This function of intonation, therefore, performs as organizing. It helps to form the frame of key elements of the text. The study under consideration made an attempt to look into the inner nature of salience and create a sort of a text intonation model. This general goal brought to some more specific intermediate results. First, there were established degrees of salience on the level of the smallest semantic element - intonation group, as well as prosodic means of creating salience, were examined. Second, the most frequent combinations of prosodic means made it possible to distinguish patterns of salience, which then became constituent elements of a text intonation model. Third, the analysis of the predicate structure allowed to divide the whole text into smaller parts, or units, which performed a specific function in the developing of the general communicative intention. It appeared that such units can be found in any text and they have common characteristics of their intonation arrangement. These findings are certainly very important both for the theory of intonation and their practical application.

Keywords: accentuation , inner speech, intention, intonation, intonation functions, models, patterns, predicate, salience, semantics, sentence stress, text

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Authors: Tayebeh Norouzi

Abstract:

This study aims to investigate the acquisition process of intonation. It examines the intonation structure of Tokyo Japanese and its realization by Iranian learners of Japanese. Seven Iranian learners of Japanese, differing in fluency, and two Japanese speakers participated in the experiment. Two sentences were used to test the phonological and phonetic characteristics of lexical pitch-accent as well as the intonation patterns produced by the speakers. Both sentences consisted of similar words with the same number of syllables and lexical pitch-accents but different syntactic structure. Speakers were asked to read each sentence three times at normal speed, and the data were analyzed by Praat. The results show that lexical pitch-accent, Accentual Phrase (AP) and AP boundary tone realization vary depending on sentence type. For sentences of type XdeYwo, the lexical pitch-accent is realized properly. However, there is a rise in AP boundary tone regardless of speakers’ level of fluency. In contrast, in sentences of type XnoYwo, the lexical pitch-accent and AP boundary tone vary depending on the speakers’ fluency level. Advanced speakers are better at grouping words into phrases and produce more native-like intonation patterns, though they are not able to realize downstep properly. The non-native speakers tried to realize proper intonation patterns by making changes in lexical accent and boundary tone.

Keywords: intonation, Iranian learners, Japanese prosody, lexical accent, second language acquisition.

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Authors: Mohammad Hadi Mahmoodi, Soroor Zekrati

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The aim of this study was to investigate the relationship between hemispheric dominance and intonation learning of Iranian EFL students. In order to gain this goal, 52 female students from three levels of beginner, elementary and intermediate in Paradise Institute, and 18 male university students at Bu-Ali Sina University constituted the sample. In order to assist students learn the correct way of applying intonation to their everyday speech, the study proposed an interactive approach and provided students with visual aid through which they were able to see the intonation pattern on computer screen using 'Speech Analyzer' software. This software was also used to record subjects’ voice and compare them with the original intonation pattern. Edinburg Handedness Questionnaire (EHD), which ranges from –100 for strong left-handedness to +100 for strong right-handedness was used to indicate the hemispheric dominance of each student. The result of an independent sample t-test indicated that girls learned intonation pattern better than boys, and that right brained students significantly outperformed the left brained ones. Using one-way ANOVA, a significant difference between three proficiency levels was also found. The posthoc Scheffer test showed that the exact difference was between intermediate and elementary, and intermediate and beginner levels, but no significant difference was observed between elementary and beginner levels. The findings of the study might provide researchers with some helpful implications and useful directions for future investigation into the domain of the relationship between mind and second language learning.

Keywords: intonation, hemispheric dominance, visual aid, language learning, second language learning

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Authors: D. Beziakina, E. Bulgakova

Abstract:

This study continues the research on Tuvinian intonation and presents a general cross-dialect analysis of intonation of Tuvinian declarative utterances, specifically the character of the tone movement in order to test the hypothesis about the prevalence of level tone in some Tuvinian dialects. The results of the analysis of basic pitch characteristics of Tuvinian speech (in general and in comparison with two other Turkic languages - Uzbek and Azerbaijani) are also given in this paper. The goal of our work was to obtain the ranges of pitch parameter values typical for Tuvinian speech. Such language-specific values can be used in speaker identification systems in order to get more accurate results of ethnic speech analysis. We also present the results of a cross-dialect analysis of declarative intonation in the poorly studied Tuvinian language.

Keywords: speech analysis, statistical analysis, speaker recognition, identification of person

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Authors: Esha Banerjee, Atul Kumar Ojha, Girish Nath Jha

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This study aims to develop an intonation labeled database for Hindi. Although no single standard for prosody labeling exists in Hindi, researchers in the past have employed perceptual and statistical methods in literature to draw inferences about the behavior of prosody patterns in Hindi. Based on such existing research and largely agreed upon intonational theories in Hindi, this study attempts to develop a manually annotated prosodic corpus of Hindi speech data, which can be used for training speech models for natural-sounding speech in the future. 100 sentences ( 500 words) each for declarative and interrogative types have been labeled using Praat.

Keywords: speech dataset, Hindi, intonation, labeled corpus

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Authors: Qi Wang

Abstract:

Chinese has become a powerful worldwide language and millions of learners are studying it all over the words. Chinese is a tone language with unique meaningful characters, which makes foreign learners master it with more difficulties. On the other hand, as each foreign language, the learners of Chinese first will learn the basic Chinese Sound Structure (the initials and finals, tones, Neutral Tone and Tone Sandhi). It’s quite common that in the following studies, teachers made a lot of efforts on drilling and error correcting, in order to help students to pronounce correctly, but ignored the training of suprasegmental features (e.g. stress, intonation). This paper analysed the oral data based on our graduation students (two-year program) from 2006-2013, presents the intonation pattern of our graduates to speak Chinese as second language -high and plain with heavy accents, without lexical stress, appropriate stop endings and intonation, which led to the misunderstanding in different real contexts of communications and the international official Chinese test, e.g. HSK (Chinese Proficiency Test), HSKK (HSK Speaking Test). This paper also demonstrated how the Chinese to use the suprasegmental features strategically in different functions and moods (declarative, interrogative, imperative, exclamatory and rhetorical intonations) in order to train the learners to achieve better Communicative Performance.

Keywords: second language learning, suprasegmental, communication, HSK (Chinese Proficiency Test)

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Authors: Daria Beziakina, Elena Bulgakova

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The paper deals with cross-gender and cross-linguistic comparison of pitch characteristics for Tuvinian with two other Turkic languages - Uzbek and Azerbaijani, based on the results of statistical analysis of pitch parameter values and intonation patterns used by male and female speakers. The main goal of our work is to obtain the ranges of pitch parameter values typical for Tuvinian speakers for the purpose of automatic language identification. We also propose a cross-gender analysis of declarative intonation in the poorly studied Tuvinian language. The ranges of pitch parameter values were obtained by means of specially developed software that deals with the distribution of pitch values and allows us to obtain statistical language-specific pitch intervals.

Keywords: speech analysis, statistical analysis, speaker recognition, identification of person

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Authors: Fuad Al Sarari

Abstract:

The purpose of the present paper is to study subclasses of analytic functions which generalize the classes of Janowski functions introduced by Polatoglu. We study certain convolution conditions. This leads to a study of the sufficient condition and the neighborhood results related to the functions in the class S (T; H; F ): and a study of new subclasses of analytic functions that are defined using notions of the generalized Janowski classes and -symmetrical functions. In the quotient of analytical representations of starlikeness and convexity with respect to symmetric points, certain inherent properties are pointed out.

Keywords: convolution conditions, subordination, Janowski functions, starlike functions, convex functions

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Authors: Nkereke Essien

Abstract:

The study seeks to analyze tone in the Educated Nigerian accent of English (ENAE) using the three tones: Low (L), High (H) and Low-High (LH). The aim is to find out whether there are any differences or similarities in the performance of the experimental group and the control. To achieve this, twenty educated Nigerian speakers of English who are educated in the language were selected by a Stratified Random Sampling (SRS) technique from two federal universities in Nigeria. They were given a passage to read and their intonation patterns were compared with that of a native speaker (control). The data were analyzed using Pierrehumbert’s (1980) intonation system of analysis. Three different approaches were employed in the analysis of the intonation Phrase (IP) as used by Pierrehumbert: perceptual, statistical and acoustic. We first analyzed our data from the passage and utterances using Willcoxon Matched Pairs Signs Ranks Test to establish the differences between the performance of the experimental group and the control. Then, the one-way Analysis of variance (ANOVA) statistical and Tukey-Krammar Post Hoc Tests were used to test for any significant difference in the performances of the twenty subjects. The acoustic data were presented to corroborate both the perceptual and statistical findings. Finally, the tonal patterns of the selected subjects in the three categories - A, B, C, were compared with those of the control. Our findings revealed that the tonal pattern of the Educated Nigerian Accent of English (ENAE) is significantly different from the tonal pattern of the Standard British Accent of English (SBAE) as represented by the control. A high preference for unidirectional tones, especially, the high tones was observed in the performance of the experimental group. Also, high tones do not necessarily correspond to stressed syllables and low tones to unstressed syllables.

Keywords: accent, intonation phrase (IP), tonal patterns, tone

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Authors: Nabil Shair

Abstract:

Arab music maqām (Arab modal framework) is founded, among other main characteristics, on microtonal intervals. Notwithstanding the importance and multifaceted nature of intonation in Arab music, there is a paucity of studies examining this subject based on scientific and quantitative approaches. The present-day theory concerning the Arab tone system is largely based on the pioneering treatise of Mīkhā’īl Mashāqah (1840), which proposes the theoretical division of the octave into 24 equal quarter tones. This kind of equal-tempered division is incompatible with the performance practice of Arab music, as many professional Arab musicians conceptualize additional pitches beyond the standard 24 notes per octave. In this paper, we refute the standard theory presenting the scale of well-tempered quarter tones by implementing a quantitative analysis of the performed intonation of two prominent tetrachords in Arab music, namely bayyātī and ḥijāz. This analysis was conducted with the help of advanced computer programs, such as Sonic Visualiser and Tony, by which we were able to obtain precise frequency data (Hz) of each tone every 0.01 second. As a result, the value (in cents) of all three intervals of each tetrachord was measured and accordingly compared to the theoretical intervals. As a result, a specific distribution of a range of deviation from the equal-tempered division of the octave was detected, especially the detection of a diminished first interval of bayyātí and diminished second interval of ḥijāz. These types of intonation entail a considerable amount of flexibility, mainly influenced by surrounding tones, direction and function of the measured tone, ornaments, text, personal style of the performer and interaction with the audience. This paper seeks to contribute to the existing literature dealing with intonation in Arab music, as it is a vital part of the performance practice of this musical tradition. In addition, the insights offered by this paper and its novel methodology might also contribute to the broadening of the existing pedagogic methods used to teach Arab music.

Keywords: Arab music, intonation, performance practice, music theory, oral music, octave division, tetrachords, music of the middle east, music history, musical intervals

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Authors: Mairo Musa Galadima, Phoebe Mshelia

Abstract:

In Nigeria, the national policy of education stipulates that the kindergarten primary schools and the legislature are to use the three popular Nigerian Languages namely: Hausa, Igbo and Yoruba. However, the English language seems to be preferred and this calls for this paper. Attempts were made to draw out the challenges faced by learners in understanding English speech sounds and using them to communicate effectively in English; using 5(five) selected secondary school in Bauchi. It was discover that challenges abound in the wrong use of stress and intonation, transfer of phonetic features from their first language. Others are inadequate qualified teachers and relevant materials including text-books. It is recommended that teachers of English should lay more emphasis on the teaching of supra-segmental features and should be encouraged to go for further studies, seminars and refresher courses.

Keywords: kindergarten, stress, phonetic and intonation, Nigeria

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Authors: Emmanuel Uba, Oluwasegun Omidiora, Eugenia Abiodun-Eniayekan

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The role of phonology on reading comprehension is long established in the literature. However, the vast majority of studies on the relationship between phonology and reading or comprehension among adults involve investigating the role of intonation, stress, and segmental knowledge on understanding texts. Not much attention is paid to junctural observation and its effect on the interpretation of texts. This study, therefore, presents a preliminary case-study investigation of the effect of juncture on comprehension of texts among adult Nigerian learners of English. Eighty adult learners of English in Nigeria were presented with fifteen seemingly ambiguous sentences to interpret. The sentences were structured in a way that pausing at different points would produce different interpretations. The results reveal that wrong application of pause is capable of affecting comprehension even when other phonological factors such as stress and intonation are observed properly.

Keywords: comprehension, juncture, phonology, reading

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Authors: Mairo Musa Galadima, Phoebe Mshelia

Abstract:

In Nigeria, the national policy of education stipulates that the kindergarten-primary schools and the legislature are to use the three popular Nigerian Languages namely: Hausa, Igbo, and Yoruba. However, the English language seems to be preferred and this calls for this paper. Attempts were made to draw out the challenges faced by learners in understanding English speech sounds and using them to communicate effectively in English; using 5 (five) selected secondary school in Bauchi. It was discovered that challenges abound in the wrong use of stress and intonation, transfer of phonetic features from their first language. Others are inadequately qualified teachers and relevant materials including textbooks. It is recommended that teachers of English should lay more emphasis on the teaching of supra-segmental features and should be encouraged to go for further studies, seminars and refresher courses.

Keywords: stress and intonation, phonetic and challenges, teaching and learning English, secondary schools

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Authors: Anca-Diana Bibiri, Mihaela Mocanu, Adrian Turculeț

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In a language the inventory of greetings is dynamic with frequent input and output, although this is hardly noticed by the speakers. In this register, there are a number of constant, conservative elements that survive different language models (among them, the classic formulae: bună ziua! (good afternoon!), bună seara! (good evening!), noapte bună! (good night!), la revedere! (goodbye!) and a number of items that fail to pass the test of time, according to language use at a time (ciao!, pa!, bai!). The source of innovation depends both of internal factors (contraction, conversion, combination of classic formulae of greetings), and of external ones (borrowings and calques). Their use imposes their frequencies at once, namely the elimination of the use of others. This paper presents a sociolinguistic approach of contemporary Romanian greetings, based on prosodic surveys in two research projects: AMPRom, and SoRoEs. Romanian language presents a rich inventory of questions (especially partial interrogatives questions/WH-Q) which are used as greetings, alone or, more commonly accompanying a proper greeting. The representative of the typical formulae is Ce mai faci? (How are you?), which, unlike its English counterpart How do you do?, has not become a stereotype, but retains an obvious emotional impact, while serving as a mark of sociolinguistic group. The analyzed corpus consists of structures containing greetings recorded in the main Romanian cultural (urban) centers. From the methodological point of view, the acoustic analysis of the recorded data is performed using software tools (GoldWave, Praat), identifying intonation patterns related to three sociolinguistics variables: age, sex and level of education. The intonation patterns of the analyzed statements are at the interface between partial questions and typical greetings.

Keywords: acoustic analysis, greetings, Romanian language, sociolinguistics

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Authors: Muna Tabuni

Abstract:

The paper contains an investigation of the behavior of the Zeros of Bargmann functions for one and two-mode systems. A brief introduction to Harmonic oscillator formalism for one and two-mode is given. The Bargmann analytic representation for one and two-mode has been studied. The zeros of Bargmann analytic function for one-mode are considered. The Q Husimi functions are introduced. The Bargmann functions and the Husimi functions have the same zeros. The Bargmann functions f(z) have exactly q zeros. The evolution time of the zeros are discussed. The zeros of Bargmann analytic functions for two-mode are introduced. Various examples have been given.

Keywords: Bargmann functions, two-mode, zeros, harmonic oscillator

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Authors: Melike Aydoğan, Dürdane Öztürk

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Let H(D) be the linear space of all analytic functions defined on the open unit disc. A log-harmonic mappings is a solution of the nonlinear elliptic partial differential equation where w(z) ∈ H(D) is second dilatation such that |w(z)| < 1 for all z ∈ D. The aim of this paper is to define some inequalities of starlike logharmonic functions of order α(0 ≤ α ≤ 1).

Keywords: starlike log-harmonic functions, univalent functions, distortion theorem

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Authors: Alicia C. Sanchez

Abstract:

This paper investigates the implementation of RAFU functions (radical functions) in robotics and automation. Specifically, the main goal is to show how these functions may be useful in lane-keeping control and the lateral control of autonomous machines, vehicles, robots or the like. From the knowledge of several points of a certain route, the RAFU functions are used to achieve the lateral control purpose and maintain the lane-keeping errors within the fixed limits. The stability that these functions provide, their ease of approaching any continuous trajectory and the control of the possible error made on the approximation may be useful in practice.

Keywords: automatic navigation control, lateral control, lane-keeping control, RAFU approximation

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Authors: Rashidah Omar, Suzeini Abdul Halim, Aini Janteng

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The coefficients estimate problem for Taylor-Maclaurin series is still an open problem especially for a function in the subclass of bi-univalent functions. A function f ϵ A is said to be bi-univalent in the open unit disk D if both f and f^-1 are univalent in D. The symbol A denotes the class of all analytic functions f in D and it is normalized by the conditions f(0) = f’(0) – 1=0. The class of bi-univalent is denoted by  The subordination concept is used in determining second and third Taylor-Maclaurin coefficients. The upper bound for second and third coefficients is estimated for functions in the subclasses of bi-univalent functions which are subordinated to the function φ. An analytic function f is subordinate to an analytic function g if there is an analytic function w defined on D with w(0) = 0 and |w(z)| < 1 satisfying f(z) = g[w(z)]. In this paper, two subclasses of bi-univalent functions associated with Hohlov operator are introduced. The bound for second and third coefficients of functions in these subclasses is determined using subordination. The findings would generalize the previous related works of several earlier authors.

Keywords: analytic functions, bi-univalent functions, Hohlov operator, subordination

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Authors: İbrahim Aktaş, Árpád Baricz

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In this paper, the radii of star likeness of the Jackson and Hahn-Exton q-Bessel functions are considered, and for each of them three different normalizations is applied. By applying Euler-Rayleigh inequalities for the first positive zeros of these functions tight lower, and upper bounds for the radii of starlikeness of these functions are obtained. The Laguerre-Pólya class of real entire functions plays an important role in this study. In particular, we obtain some new bounds for the first positive zero of the derivative of the classical Bessel function of the first kind.

Keywords: bessel function, lommel function, radius of starlikeness and convexity, Struve function

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Authors: Tulin Coskun

Abstract:

We investigate the approximation of analytic functions of several variables in polydisc by the sequences of linear k-positive operators in Gadjiev sence. The approximation of analytic functions of complex variable by linear k-positive operators was tackled, and k-positive operators and formulated theorems of Korovkin's type for these operators in the space of analytic functions on the unit disc were introduced in the past. Recently, very general results on convergence of the sequences of linear k-positive operators on a simply connected bounded domain within the space of analytic functions were proved. In this presentation, we extend some of these results to the approximation of analytic functions of several complex variables by sequences of linear k-positive operators.

Keywords: analytic functions, approximation of analytic functions, Linear k-positive operators, Korovkin type theorems

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Authors: Nutthapat Kaewrattanapat, Wannarat Bunchongkien

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The purposes of this research are to study and develop the algorithm of Thai spoonerism words by semi-automatic computer programs, that is to say, in part of data input, syllables are already separated and in part of spoonerism, the developed algorithm is utilized, which can establish rules and mechanisms in Thai spoonerism words for bi-syllables by utilizing analysis in elements of the syllables, namely cluster consonant, vowel, intonation mark and final consonant. From the study, it is found that bi-syllable Thai spoonerism has 1 case of spoonerism mechanism, namely transposition in value of vowel, intonation mark and consonant of both 2 syllables but keeping consonant value and cluster word (if any). From the study, the rules and mechanisms in Thai spoonerism word were applied to develop as Thai spoonerism word software, utilizing PHP program. the software was brought to conduct a performance test on software execution; it is found that the program performs bi-syllable Thai spoonerism correctly or 99% of all words used in the test and found faults on the program at 1% as the words obtained from spoonerism may not be spelling in conformity with Thai grammar and the answer in Thai spoonerism could be more than 1 answer.

Keywords: algorithm, spoonerism, computational linguistics, Thai spoonerism

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Authors: Chahid K. Ghaddar

Abstract:

The spreadsheet engine is exploited via a non-conventional mechanism to enable novel worksheet solver functions for computational calculus. The solver functions bypass inherent restrictions on built-in math and user defined functions by taking variable formulas as a new type of argument while retaining purity and recursion properties. The enabling mechanism permits integration of numerical algorithms into worksheet functions for solving virtually any computational problem that can be modelled by formulas and variables. Several examples are presented for computing integrals, derivatives, and systems of deferential-algebraic equations. Incorporation of the worksheet solver functions with the ubiquitous spreadsheet extend the utility of the latter as a powerful tool for computational mathematics.

Keywords: calculus, differential algebraic equations, solvers, spreadsheet

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Authors: Luiz F. J. Maia

Abstract:

This paper presents an approach to fuzzy control, with the use of new pertinence functions, applied in the case of an inverted pendulum. Appropriate definitions of pertinence functions to fuzzy sets make possible the implementation of the controller with only one control rule, resulting in a smooth control surface. The fuzzy control system can be implemented with analog devices, affording a true real-time performance.

Keywords: control surface, fuzzy control, Inverted pendulum, pertinence functions

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Authors: Mohammad Mehdi Mazarei, Ali Asghar Behroozpoor

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In this paper, we define an optimization problem corresponding to smooth and nonsmooth functions which its optimal solution is the first derivative of these functions in a domain. For this purpose, a linear programming problem corresponding to optimization problem is obtained. The optimal solution of this linear programming problem is the approximate generalized first derivative. In fact, we approximate generalized first derivative of nonsmooth functions as tailor series. We show the efficiency of our approach by some smooth and nonsmooth functions in some examples.

Keywords: general derivative, linear programming, optimization problem, smooth and nonsmooth functions

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Authors: J. Juan Peña, J. Morales, J. García-Ravelo, J. García-Martínez

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It is known that by introducing alternative forms of exponential and logarithmic functions, the Tsallis entropy Sᵩ is itself a generalization of Shannon entropy S. In this work, from a deformation through a scaling function applied to the differential operator, it is possible to generate a q-deformed calculus as well as a q-deformed arithmetic, which not only allows generalizing the exponential and logarithmic functions but also any other standard function. The updated q-deformed differential operator leads to an updated integral operator under which the functions are integrated together with a weight function. For each differentiable function, it is possible to identify its q-deformed partner, which is useful to generalize other algebraic relations proper of the original functions. As an application of this proposal, in this work, a generalization of exponential and logarithmic functions is studied in such a way that their relationship with the thermodynamic functions, particularly the entropy, allows us to have a q-deformed expression of these. As a result, from a particular scaling function applied to the differential operator, a q-deformed arithmetic is obtained, leading to the generalization of the Tsallis entropy.

Keywords: q-calculus, q-deformed arithmetic, entropy, exponential functions, thermodynamic functions

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Authors: Khosrow Maleknejad, Yaser Rostami

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In this paper, semi-orthogonal B-spline scaling functions and wavelets and their dual functions are presented to approximate the solutions of integro-differential equations.The B-spline scaling functions and wavelets, their properties and the operational matrices of derivative for this function are presented to reduce the solution of integro-differential equations to the solution of algebraic equations. Here we compute B-spline scaling functions of degree 4 and their dual, then we will show that by using them we have better approximation results for the solution of integro-differential equations in comparison with less degrees of scaling functions.

Keywords: ıntegro-differential equations, quartic B-spline wavelet, operational matrices, dual functions

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Authors: Fernando Maass, Pablo Martin, Jorge Olivares

Abstract:

The Bessel function J₀(x) is very important in Electrodynamics and Physics, as well as its zeros. In this work, a method to obtain high accuracy approximation is presented through an application to that function. In most of the applications of this function, the values of the zeros are very important. In this work, analytic approximations for this function have been obtained valid for all positive values of the variable x, which have high accuracy for the function as well as for the zeros. The approximation is determined by the simultaneous used of the power series and asymptotic expansion. The structure of the approximation is a combination of two rational functions with elementary functions as trigonometric and fractional powers. Here us in Pade method, rational functions are used, but now there combined with elementary functions us fractional powers hyperbolic or trigonometric functions, and others. The reason of this is that now power series of the exact function are used, but together with the asymptotic expansion, which usually includes fractional powers trigonometric functions and other type of elementary functions. The approximation must be a bridge between both expansions, and this can not be accomplished using only with rational functions. In the simplest approximation using 4 parameters the maximum absolute error is less than 0.006 at x ∼ 4.9. In this case also the maximum relative error for the zeros is less than 0.003 which is for the second zero, but that value decreases rapidly for the other zeros. The same kind of behaviour happens for the relative error of the maximum and minimum of the functions. Approximations with higher accuracy and more parameters will be also shown. All the approximations are valid for any positive value of x, and they can be calculated easily.

Keywords: analytic approximations, asymptotic approximations, Bessel functions, quasirational approximations

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Authors: Priestly Malambo, Sonja Van Putten, Hanlie Botha, Gerrit Stols

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The relevance of what is content is taught in tertiary teacher training has long been in question. This study attempts to understand how advanced mathematics courses equip student teachers to teach functions at secondary school level. This paper reports on an investigation that was conducted in an African university, where preservice teachers were purposefully selected for participation in individual semi-structured interviews after completing a test on functions as taught at secondary school. They were asked to justify their reasoning in the test and to explain functions in a way that might bring about understanding of the topic in someone who did not know how functions work. These were final year preservice mathematics teachers who had studied advanced mathematics courses for three years. More than 50% of the students were not able to explain concepts or to justify their reasoning about secondary school functions in a coherent way. The results of this study suggest that the study of advanced mathematics does not automatically enable students to teach secondary school functions, and that, although these students were able to do advanced mathematics, they were unable to explain the working of functions in a way that would allow them to teach this topic successfully.

Keywords: secondary school, mathematical reasoning, student-teachers, functions

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Authors: Yilmaz Simsek

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In this paper, we study the Bernstein-type basis functions with their generating functions. We give various properties of these polynomials with the aid of their generating functions. These polynomials and generating functions have many valuable applications in mathematics, in probability, in statistics and also in mathematical physics. By using the Bernstein-Galerkin and the Bernstein-Petrov-Galerkin methods, we give some applications of the Bernstein-type polynomials for solving high even-order differential equations with their numerical computations. We also give Bezier-type curves related to the Bernstein-type basis functions. We investigate fundamental properties of these curves. These curves have many applications in mathematics, in computer geometric design and other related areas. Moreover, we simulate these polynomials with their plots for some selected numerical values.

Keywords: generating functions, Bernstein basis functions, Bernstein polynomials, Bezier curves, differential equations

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Authors: Pablo Martin, Jorge Olivares, Fernando Maass

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The use of fractional derivatives to different problems in Engineering and Physics has been increasing in the last decade. For this reason, we have here considered partial derivatives when the integral is a spherical Bessel function of the first kind in both regular and modified ones simple initial conditions have been also considered. In this way, the solution has been found as a combination of hypergeometric functions. The case of a general rational value for α of the fractional derivative α has been solved in a general way for alpha between zero and two. The modified spherical Bessel functions of the first kind have been also considered and how to go from the regular case to the modified one will be also shown.

Keywords: caputo fractional derivatives, hypergeometric functions, linear differential equations, spherical Bessel functions

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