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Keywords = Isma‘ilism

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23 pages, 4107 KiB  
Article
Assessing Recharge Zones for Groundwater Potential in Dera Ismail Khan (Pakistan): A GIS-Based Analytical Hierarchy Process Approach
by Anwaar Tabassum, Asif Sajjad, Ghayas Haider Sajid, Mahtab Ahmad, Mazhar Iqbal and Aqib Hassan Ali Khan
Water 2025, 17(11), 1586; https://doi.org/10.3390/w17111586 - 23 May 2025
Viewed by 982
Abstract
Groundwater constitutes the primary source of liquid freshwater on Earth and is essential for ecosystems, agriculture, and human consumption. However, rising demand, urbanization, and climate change have intensified groundwater depletion, particularly in semi-arid regions. Therefore, assessing groundwater recharge zones is essential for sustainable [...] Read more.
Groundwater constitutes the primary source of liquid freshwater on Earth and is essential for ecosystems, agriculture, and human consumption. However, rising demand, urbanization, and climate change have intensified groundwater depletion, particularly in semi-arid regions. Therefore, assessing groundwater recharge zones is essential for sustainable water resource management in vulnerable areas such as Dera Ismail Khan, Pakistan. This study aims to delineate groundwater potential zones (GWPZs), using an integrated approach combining the Geographic Information System (GIS), remote sensing (RS), and the analytical hierarchy process (AHP). Twelve factors were identified in a study conducted using GIS-based AHP to determine the groundwater recharge zones in the region. These include land use/land cover (LULC), rainfall, drainage density, soil type, slope, road density, water table depth, and remote sensing indices such as Normalized Difference Vegetation Index (NDVI), Normalized Difference Built-up Index (NDBI), Moisture Stress Index (MSI), Worldview Water Index (WVWI), and Land Surface Temperature (LST). The results show that 17.52% and 2.03% of the area have “good” and “very good” potential for groundwater recharge, respectively, while 48.63% of the area has “moderate” potential. Furthermore, gentle slopes (0–2.471°), high drainage density, shallow water depths (20–94 m), and densely vegetated areas (with a high NDVI) are considered important influencing factors for groundwater recharge. Conversely, areas with steep slopes, high temperatures, and dense built-up areas showed “poor” potential for recharge. This approach demonstrates the effectiveness of integrating advanced remote sensing indices with the AHP model in a semi-arid context, validated through high-accuracy field data (Kappa = 0.93). This methodology offers a cost-effective decision support tool for sustainable groundwater planning in similar environments. Full article
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13 pages, 1519 KiB  
Review
Reviewing Water Wars and Water Weaponisation Literatures: Is There an Unnoticed Link?
by Paula Duarte Lopes and Margarida Gama
Water 2025, 17(6), 897; https://doi.org/10.3390/w17060897 - 20 Mar 2025
Viewed by 1225
Abstract
The prediction made by former Vice President of the World Bank, Ismail Serageldin, that the wars of the 21st century will be about water, remains on the international political agenda. Yet, there is enough evidence corroborating that water wars have not occurred in [...] Read more.
The prediction made by former Vice President of the World Bank, Ismail Serageldin, that the wars of the 21st century will be about water, remains on the international political agenda. Yet, there is enough evidence corroborating that water wars have not occurred in the past and that there are sufficient mechanisms in place to prevent them in the future. Simultaneously, domestic water violent conflicts have been taking place, usually as immediate reactions to localised disputes. More importantly, water weaponisation has been gaining visibility during violent conflicts, violating international humanitarian law without any consequences so far. This paper reviews the water wars and water weaponisation literatures, arguing that there is an under-researched link between these two literatures and practices. This review suggests that the water weaponisation discourse and practice may facilitate the context for the water wars prophecy to become true. Full article
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3 pages, 145 KiB  
Book Review
Book Review: Kadare (2024). A Dictator Calls. Translated by John Hodgson. London: Vintage Digital. ISBN: 9781529920574
by Yutaka Okuhata
Literature 2025, 5(1), 1; https://doi.org/10.3390/literature5010001 - 30 Dec 2024
Viewed by 801
Abstract
On 1 July 2024, Ismail Kadare, one of the most prestigious and successful writers in Albania, died at the age of eighty-eight [...] Full article
12 pages, 270 KiB  
Article
The Test of Sports and Folk Narratives with the Notion of Haram: Citing the Example of the Branch of Wrestling
by Ünsal Yılmaz Yeşildal, Doğukan Batur Alp Gülşen and Cihat Burak Korkmaz
Religions 2024, 15(11), 1311; https://doi.org/10.3390/rel15111311 - 26 Oct 2024
Cited by 1 | Viewed by 1266
Abstract
Culture consists of material and spiritual values and tools that a nation has accumulated in the historical process. In addition to the most basic contexts such as language and religion, contexts such as sporting activities, art, public medicine, and the public calendar are [...] Read more.
Culture consists of material and spiritual values and tools that a nation has accumulated in the historical process. In addition to the most basic contexts such as language and religion, contexts such as sporting activities, art, public medicine, and the public calendar are also important environments that reveal their own cultural elements. Among these contexts, religion is very effective in shaping the daily life of the individual and, thus, society through the rules it enjoins. Religion does not dominate only the world of belief of the individual. Through the world of belief, it also directs their relations with the social institutions they are involved in. Sport is one of the most important activities and social institutions that stand out with various functions in daily life, with wrestling being one of the branches of sports that have emerged as a result of the imitation of the struggle of human beings with nature and other creatures with which they share nature. In particular, those involved in the nomadic way of life had to hunt in order not to starve and fight in order to survive. Wrestling, which emerged as a result of these obligations, held an important place among all Turks in the world for a period of time, especially in the transition periods of life, such as birth, marriage, and death. One of the conditions set forth by women as a condition of marriage was that their suitor defeated them in wrestling. Examples of this condition are also observed in literary texts belonging to different periods when Turks were not yet acquainted with Islam and the concepts of halal and haram, which entered their lives together with Islam. According to the provisions of the Holy Qur’an, right/unprohibited thoughts and actions are associated with the words good and halal, while wrong/prohibited thoughts and actions are associated with the words sin and haram. In this study, the social and cultural phases of wrestling as a sports branch among Turks in the historical process will be evaluated on the basis of the history of religions and religious references, in addition to the literary texts belonging to historical periods when Turks were members of different religions, in the context of two events that have been experienced and reported in the news. The study was carried out using the method of document analysis, a method of qualitative research, and the data obtained by this method were evaluated using content analysis. The narratives of Alıp Manaş, Alpamış, Alpamıs, Alıpmenşen, and Bamsı Beyrek, which are evaluated in this context, belong to the periods when the Turks had not been introduced to Islam or had only recently been introduced to it. Alıp Manaş was collated from different Turkic tribes such as the Altais, Alpamış from the Uzbeks, Alpamıs the Kazakhs/Karakalpaks, Alıpmenşen the Bashkirs/Tatars, and Bamsı Beyrek the Oghuz Turks. The narratives of Kirmanshah, Köse Kenan-Dânâ Hanım, Bey Böyrek, Shah Ismail, and Yaralı Mahmut, which are evaluated in the study, belong to the periods when the Turks became Muslim en masse, and are related only among the Oghuz Turks. These narratives are included in the study because they are similar to Alıp Manaş, Alpamış, Alpamıs, Alıpmenşen, and Bamsı Beyrek and they belong to the period when Islam was largely established among the Turkish masses in Anatolia. The effect of the new religion on wrestling, which is a branch of sport, will be revealed through these narratives belonging to different tribes and religious periods. Once more, an event that occurred in recent history, and was the subject of the news, was subjected to document analysis, and content analysis was carried out through the text of the news and evaluated in the context of the study. This study aims to explain the effect of religious rules on sports branches with theological, folkloric, and sociological references based on ancient literary texts belonging to the Turks and two incidents which were experienced. Full article
(This article belongs to the Special Issue Sport and Religion: Continuities, Connections, Concerns)
33 pages, 46215 KiB  
Article
The Necessity and Goodness of Animals in Sijistānī’s Kashf Al-Maḥjūb
by Peter Adamson and Hanif Amin Beidokhti
Philosophies 2024, 9(3), 72; https://doi.org/10.3390/philosophies9030072 - 20 May 2024
Viewed by 1798
Abstract
The Neoplatonic notion of “emanation” implies a required progression through hierarchical stages, originating from the highest principle (the One or God) and cascading down through a series of principles. While this process is deemed necessary, it is also inherently good, even “choiceworthy”, aligning [...] Read more.
The Neoplatonic notion of “emanation” implies a required progression through hierarchical stages, originating from the highest principle (the One or God) and cascading down through a series of principles. While this process is deemed necessary, it is also inherently good, even “choiceworthy”, aligning with the identification of the first principle with the Good. Plotinus, a prominent Neoplatonist, emphasizes the beauty and goodness of the sensible world, governed by divine providence. This perspective, transmitted through Arabic adaptations of Plotinus, influences Islamic philosophers too. This paper delves into the thought of the Ismāʿīlī philosopher Abū Yaʿqūb al-Sijistānī (d. after. 349/971), exploring the interplay of necessity and goodness in his cosmology, with a focus on non-human animals. Sijistānī’s Persian Uncovering the Veiled provides a unique perspective on animals, presenting them as both necessary unfoldings of the universal intellect and inherently good beings with intrinsic value. The paper concludes with an appendix featuring an improved edition and English translation of relevant passages. Full article
(This article belongs to the Special Issue Ancient and Medieval Theories of Soul)
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17 pages, 6993 KiB  
Article
Isma‘ili Continuity and Social Change: Chronotopes and Practicing Taqiyya within the Sulaymani Community of Saudi Arabia
by Aidah Aljuran and Jarred Brewster
Languages 2023, 8(4), 275; https://doi.org/10.3390/languages8040275 - 22 Nov 2023
Viewed by 2677
Abstract
This paper examines discourses around the religious and social practice of taqiyya among members of the Sulaymani Isma‘ili community in Saudi Arabia. Isma‘ilism, in the context of 1200 years of anti-Shi‘a discrimination, cultivated the practice known as taqiyya (Arabic, ‘circumscription’) as a tool [...] Read more.
This paper examines discourses around the religious and social practice of taqiyya among members of the Sulaymani Isma‘ili community in Saudi Arabia. Isma‘ilism, in the context of 1200 years of anti-Shi‘a discrimination, cultivated the practice known as taqiyya (Arabic, ‘circumscription’) as a tool for self-preservation, which was then further rationalized and reinforced by the sect’s esoteric theology. Taqiyya consists of concealing religious identity, public avoidance of certain rituals, and, in some instances, claiming to be a member of the unmarked Sunni majority. Sweeping changes in Saudi society in the last several years have meant a growing ambivalence about taqiyya and its continued utility. This is significant since taqiyya for many of our interlocutors in this study is not merely a survival tactic. Instead, it is better understood as an embodied disposition cultivated against the backdrop of household privacy. This disposition intimately links everyday comportment to the esoteric cosmology of Isma‘ilism, which is the distinctive and iconic feature of the faith. Our interlocutors’ narratives demonstrated how the invocation of different spatial and temporal frameworks provides a basic heuristic by which to interpret these individuals’ accounts of taqiyya. For some of these individuals, taqiyya is an essential and timeless practice, while for others, the meaning has been reshaped by the recent socio-political reforms. Full article
(This article belongs to the Special Issue Language Use in the Middle East and North Africa)
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23 pages, 1692 KiB  
Article
Molecular Survey of Rickettsia raoultii in Ticks Infesting Livestock from Pakistan with Notes on Pathogen Distribution in Palearctic and Oriental Regions
by Shehla Shehla, Mashal M. Almutairi, Abdulaziz Alouffi, Tetsuya Tanaka, Shun-Chung Chang, Chien-Chin Chen and Abid Ali
Vet. Sci. 2023, 10(11), 636; https://doi.org/10.3390/vetsci10110636 - 29 Oct 2023
Cited by 4 | Viewed by 2814
Abstract
Ticks are hematophagous ectoparasites that transmit different pathogens such as Rickettsia spp. to domestic and wild animals as well as humans. Genetic characterizations of Rickettsia spp. from different regions of Pakistan are mostly based on one or two genetic markers and are confined [...] Read more.
Ticks are hematophagous ectoparasites that transmit different pathogens such as Rickettsia spp. to domestic and wild animals as well as humans. Genetic characterizations of Rickettsia spp. from different regions of Pakistan are mostly based on one or two genetic markers and are confined to small sampling areas and limited host ranges. Therefore, this study aimed to molecularly screen and genetically characterize Rickettsia spp. in various tick species infesting camels, sheep, and goats. All the collected tick specimens were morphologically identified, and randomly selected tick species (148) were screened molecularly for the detection of Rickettsia spp. by amplifying three rickettsial DNA fragments, namely, the citrate-synthase gene (gltA), outer-membrane protein A (ompA), and outer-membrane protein B (ompB). After examining 261 hosts, 161 (61.7%) hosts were found infested by 564 ticks, including 287 (50.9%) nymphs, 171 (30.3%) females, and 106 (18.8%) males in five districts (Kohat, Dera Ismail Khan, Lower Dir, Bajaur, and Mansehra). The highest occurrence was noted for Hyalomma dromedarii (number = 72, 12.8%), followed by Haemaphysalis sulcata (n = 70, 12.4%), Rhipicephalus turanicus (n = 64, 11.3%), Rhipicephalus microplus (n = 55, 9.7%), Haemaphysalis cornupunctata (n = 49, 8.7%), Hyalomma turanicum (n = 48, 8.5%), Hyalomma isaaci (n = 45, 8.0%), Haemaphysalis montgomeryi (n = 44, 7.8%), Hyalomma anatolicum (n = 42, 7.5%), Haemaphysalis bispinosa (n = 38, 6.7%), and Rhipicephalus haemaphysaloides (n = 37, 6.6%). A subset of 148 ticks were tested, in which eight (5.4%) ticks, including four Hy. turanicum, two Ha. cornupunctata, one Ha. montgomeryi, and one Ha. bispinosa, were found positive for Rickettsia sp. The gltA, ompA, and ompB sequences revealed 100% identity and were phylogenetically clustered with Rickettsia raoultii reported in China, Russia, USA, Turkey, Denmark, Austria, Italy, and France. Additionally, various reports on R. raoultii from Palearctic and Oriental regions were summarized in this study. To the best of our knowledge, this is the first report regarding genetic characterization and phylogenetic analysis of R. raoultii from Pakistan. Further studies to investigate the association between Rickettsia spp. and ticks should be encouraged to apprise effective management of zoonotic consequences. Full article
(This article belongs to the Special Issue Tick-Borne Diseases and Their Control)
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23 pages, 4826 KiB  
Article
Feasibility Study of a Reverse Osmosis Desalination Unit Powered by Photovoltaic Panels for a Sustainable Water Supply in Algeria
by Zahia Tigrine, Hanene Aburideh, Djamila Zioui, Sarra Hout, Naima Sahraoui, Yasmine Benchoubane, Amina Izem, Djilali Tassalit, Fatma Zohra Yahiaoui, Mohamed Khateb, Nadjib Drouiche and Seif El Islam Lebouachera
Sustainability 2023, 15(19), 14189; https://doi.org/10.3390/su151914189 - 26 Sep 2023
Cited by 3 | Viewed by 5788
Abstract
In recent years, reverse osmosis water desalination has developed rapidly and has become the most competitive and widely used technology in the world. The number of desalination plants is increasing rapidly as freshwater needs increase. Various membrane technologies have been developed and improved, [...] Read more.
In recent years, reverse osmosis water desalination has developed rapidly and has become the most competitive and widely used technology in the world. The number of desalination plants is increasing rapidly as freshwater needs increase. Various membrane technologies have been developed and improved, including nanofiltration (NF) and reverse osmosis (RO), whose desalination costs have been relatively reduced. Therefore, this work proposes an experimental study for a small desalination unit based on RO generated by renewable energy, which is mainly suitable for arid regions or desert areas that do not have electricity and water and can be applied for emergency treatment to meet strong freshwater resource needs. In this study, to meet the drinking water demand, a reverse osmosis desalination system is designed and evaluated in order to improve and optimize its operation. This system has a daily capacity of 2 m3. We used brackish groundwater, which has been characterized as reference water, to produce synthetic water for different salinities until seawater. The analysis is based on data obtained from experiments carried out in the standalone RO pilot designed for the production of fresh water. For this purpose, we conducted relevant experiments to examine the influence of applied pressure, salt concentration and temperature on the RO membrane performance. The effects of different factors that affect the energy consumption in the RO desalination process were analyzed, and those with significant influence were explored. The effectiveness of RO desalination coupled with a photovoltaic (PV) energy system is shown. We found the recovery rate for system operation to be 32%. An optimization study is presented for the operation of an autonomous RO desalination system powered by photovoltaic panels. The energy produced by the PV system was used to feed two pumps forthe production of drinking waterwithanRO membrane, under the conditions of the town of Bou-Ismail. As results, a 3 kWp PV system was installed based on the energy demand. The design data have shown that a 3 kWp PV system can power a 1.8 W RO load given the Bou-Ismail climate. Energy consumption in the case study under Bou-Ismail weather conditions were analyzed. The desalination of brackish water at a TDS value of 5 g/L requires an energy of about 1.5 kWh/m3. Using seawater at a TDS value of 35 g/L, this value increases to 5.6 kWh/m3. The results showed that the optimal recovery rate for system operation was determined to be 32% for a feedwater salinity of 35 g/L, and 80% for a feedwater salinity of 1 g/L. Full article
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2 pages, 180 KiB  
Editorial
Preface: The 3rd International Day on Computer Science and Applied Mathematics (ICSAM’23)
by Abdeslam Jakimi and Mohamed Oualla
Comput. Sci. Math. Forum 2023, 6(1), 11; https://doi.org/10.3390/cmsf2023006011 - 31 Aug 2023
Viewed by 1459
Abstract
The Scientific Association of Academic Researchers (ASCA) and Software Engineering & Information Systems Engineering Team (GL-ISI) organized, in collaboration with Moulay Ismail University, the 3rd International Day on Computer Science and Applied Mathematics (ICSAM’23) on May 13, 2023, in the Faculty of Sciences [...] Read more.
The Scientific Association of Academic Researchers (ASCA) and Software Engineering & Information Systems Engineering Team (GL-ISI) organized, in collaboration with Moulay Ismail University, the 3rd International Day on Computer Science and Applied Mathematics (ICSAM’23) on May 13, 2023, in the Faculty of Sciences and Technology of Errachidia. Full article
(This article belongs to the Proceedings of The 3rd International Day on Computer Science and Applied Mathematics)
11 pages, 1099 KiB  
Article
Zinc and Potassium Fertilizer Synergizes Plant Nutrient Availability and Affects Growth, Yield, and Quality of Wheat Genotypes
by Aneela Bashir, Qudrat Ullah Khan, Ahmad Alem, Awatif A. Hendi, Umber Zaman, Shahid Ullah Khan, Khalil ur Rehman, Asghar Ali Khan, Ihsan Ullah, Yasir Anwar and Ehab A. Abdelrahman
Plants 2023, 12(12), 2241; https://doi.org/10.3390/plants12122241 - 7 Jun 2023
Cited by 11 | Viewed by 2924
Abstract
The growth and productivity of wheat crops depend on the availability of essential nutrients such as zinc (Zn) and potassium (K2O), which play critical roles in the plant’s physiological and biochemical processes. This study aimed to investigate the synergizing effect of [...] Read more.
The growth and productivity of wheat crops depend on the availability of essential nutrients such as zinc (Zn) and potassium (K2O), which play critical roles in the plant’s physiological and biochemical processes. This study aimed to investigate the synergizing effect of zinc and potassium fertilizers on uptake of both the nutrients, growth, yield, and quality of the Hashim-08 cultivar and local landrace, during the 2019–2020 growing season in Dera Ismail Khan, Pakistan. The experiment was designed using a split plot pattern in a randomized complete pattern, with main plots for the wheat cultivars and subplots for the fertilizer treatments. Results indicated that both cultivars responded positively to the fertilizer treatments, with the local landrace exhibiting maximum plant height and biological yield, and improved Hashim-08, showing increased agronomic parameters, including the number of tillers and grains and spike length. Application of Zn and K2O fertilizers significantly enhanced agronomic parameters, such as the number of grains per plant, spike length, thousand-grain weight, grain yield, harvest index, Zn uptake of grain, dry gluten content, and grain moisture content, while crude protein and grain potassium remained relatively unchanged. The soil’s Zn and K content dynamics were found to vary among treatments. In conclusion, the combined application of Zn and K2O fertilizers proved beneficial in improving the growth, yield, and quality of wheat crops, with the local landrace exhibiting lower grain yield but greater Zn uptake through fertilizer application. The study’s findings highlight that the local landrace showed good response to the growth and qualitative parameter when compared with the Hashim-08 cultivar. Additionally, the combined application of Zn and K showed a positive relation in terms of nutrient uptake and soil Zn and K content. Full article
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12 pages, 314 KiB  
Article
Orthogonal Polynomials with Singularly Perturbed Freud Weights
by Chao Min and Liwei Wang
Entropy 2023, 25(5), 829; https://doi.org/10.3390/e25050829 - 22 May 2023
Cited by 2 | Viewed by 1490
Abstract
In this paper, we are concerned with polynomials that are orthogonal with respect to the singularly perturbed Freud weight functions. By using Chen and Ismail’s ladder operator approach, we derive the difference equations and differential-difference equations satisfied by the recurrence coefficients. We also [...] Read more.
In this paper, we are concerned with polynomials that are orthogonal with respect to the singularly perturbed Freud weight functions. By using Chen and Ismail’s ladder operator approach, we derive the difference equations and differential-difference equations satisfied by the recurrence coefficients. We also obtain the differential-difference equations and the second-order differential equations for the orthogonal polynomials, with the coefficients all expressed in terms of the recurrence coefficients. Full article
(This article belongs to the Special Issue Random Matrices: Theory and Applications)
1 pages, 167 KiB  
Reply
Reply to Arshad, R.; Jalil, M. Comment on “Nizam Chew, L.C.; Ismail, E.S. S-box Construction Based on Linear Fractional Transformation and Permutation Function. Symmetry 2020, 12, 826”
by Liyana Chew Nizam Chew and Eddie Shahril Ismail
Symmetry 2023, 15(5), 1006; https://doi.org/10.3390/sym15051006 - 29 Apr 2023
Viewed by 1095
Abstract
In this reply paper, we provide a response to the comment paper of a comment on “S-box Construction Based on Linear Fractional Transformation and Permutation Function” written by Razi Arshad and Mudassar Jalil [...] Full article
2 pages, 193 KiB  
Comment
Comment on Nizam Chew, L.C.; Ismail, E.S. S-box Construction Based on Linear Fractional Transformation and Permutation Function. Symmetry 2020, 12, 826
by Razi Arshad and Mudassar Jalil
Symmetry 2023, 15(5), 1005; https://doi.org/10.3390/sym15051005 - 29 Apr 2023
Cited by 2 | Viewed by 1080
Abstract
The aim of this comment paper is to identify a technical error in the published article of Nizam Chew et al. [...] Full article
16 pages, 308 KiB  
Article
Properties of q-Symmetric Starlike Functions of Janowski Type
by Afis Saliu, Isra Al-Shbeil, Jianhua Gong, Sarfraz Nawaz Malik and Najla Aloraini
Symmetry 2022, 14(9), 1907; https://doi.org/10.3390/sym14091907 - 12 Sep 2022
Cited by 25 | Viewed by 1996
Abstract
The word “symmetry” is a Greek word that originated from “symmetria”. It means an agreement in dimensions, due proportion, and arrangement; however, in complex analysis, it means objects remaining invariant under some transformation. This idea has now been recently used in geometric function [...] Read more.
The word “symmetry” is a Greek word that originated from “symmetria”. It means an agreement in dimensions, due proportion, and arrangement; however, in complex analysis, it means objects remaining invariant under some transformation. This idea has now been recently used in geometric function theory to modify the earlier classical q-derivative introduced by Ismail et al. due to its better convergence properties. Consequently, we introduce a new class of analytic functions by using the notion of q-symmetric derivative. The investigation in this paper obtains a number of the latest important results in q-theory, including coefficient inequalities and convolution characterization of q-symmetric starlike functions related to Janowski mappings. Full article
(This article belongs to the Special Issue Symmetry in Geometric Functions and Mathematical Analysis II)
13 pages, 280 KiB  
Article
Approximation of Real Functions by a Generalization of Ismail–May Operator
by Adrian Holhoş
Mathematics 2022, 10(10), 1650; https://doi.org/10.3390/math10101650 - 12 May 2022
Cited by 3 | Viewed by 1790
Abstract
In this paper, we generalize a sequence of positive linear operators introduced by Ismail and May and we study some of their approximation properties for different classes of continuous functions. First, we estimate the error of approximation in terms of the usual modulus [...] Read more.
In this paper, we generalize a sequence of positive linear operators introduced by Ismail and May and we study some of their approximation properties for different classes of continuous functions. First, we estimate the error of approximation in terms of the usual modulus of continuity and the second-order modulus of Ditzian and Totik. Then, we characterize the bounded functions that can be approximated uniformly by these new operators. In the last section, we obtain the most important results of the paper. We give the complete asymptotic expansion for the operators and we deduce a Voronovskaya-type theorem, results that hold true for smooth functions with exponential growth. Full article
(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)
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