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On Functions With Nondecreasing Increments

Thesis Info

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Author

Mehmood, Faraz

Program

PhD

Institute

University of Karachi

City

Karachi

Province

Sindh

Country

Pakistan

Thesis Completing Year

2019

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/11543/1/Faraz%20Mehmood%20maths%202019

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676726795971

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In last few decades, the concept of functions with non-decreasing increments got attention of many mathematicians due to its worth and importance. We would like to establish a connection/relation among functions with non-decreasing increments and arithmetic integral mean, Wright convex functions, convex functions, r?convex functions, Jensen m?convex functions, m?convex functions, m ? r?convex functions, k?monotonic functions, absolutely monotonic functions, completely monotonic functions, Laplace Transform and exponentially convex functions, by using thenite di erence operator as di erent cases ofm h f. We also consider function with nondecreasing increments of third order and get the generalisations of the Levinson''stype inequality and Jensen-Mercer''s-type inequality by using Jensen-Boas inequality. We will deduce some general identities of Popoviciu type for discrete case for sums for function and sequence in two dimension using higher order r divided difference, positivity of these expressions are characterised for higher order r?convex functions. We will also obtain some general identities of Popoviciu type for integral R R P(y; z)f(y; z)dy dz of higher order di erentiable function and positivity of these expressions are characterised for higher order r?convex and completely monotonic functions. We would discuss some applications in terms of generalised Cauchy-type means and exponential convexity as well. We would get the generalisation of discrete identity and inequality ofCeby sev-type and discuss generalisation of integral identities and inequalities ofCeby sev-type and K. Fan-type for higher order r?convex functions with two variables. Further, we will obtain double weighted integrals Montgomery''s identities and using these identities we deduce generalisation of Ostrowski- and Gr uss-type inequalities for double weighted integrals for di erentiable functions of higher order with two variables.
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پروانے جل اٹھے

پروانے جل اٹھے

5جولائی 1977ء یوم سیاہ

جب چئیر مین ذوالفقار علی بھٹو کے پروانے جل اٹھے تھے۔ ہے کوئی جو اپنے لیڈر کے لیے خود پر تیل چھڑک کر آ گ لگا لے ۔تقریباََ سات کارکنوں نے اپنی جانیں ذوالفقار علی بھٹو شہید پر قربان کیں چئیر مینذوالفقار علی بھٹو نے سپریم کورٹ میں کھڑے ہو کر کہ کوئی کسی کے لیے اپنی انگلی کو نہیں جلا سکتا میرے جان نثاروں نے اپنی جانیں قربان کر دیں ۔

جب پروانے جل اٹھے تھے۔ یہ قبریں پنجاب اور ایک تصویر آزاد کشمیر کے ان پروانوں کی ہیں جو اپنے قائد کو پھا نسی چڑھتا ہوا نہیں دیکھ سکتے تھے اور خود پر تیل چھڑک کر خود سوزی کر کے اپنے قائد چئیر مین ذوالفقار علی بھٹو شہید پر قربان ہو گئے ۔پہلی قبر گوجرانوالہ کے وحید قریشی کی دوسری تیسری فیصل آ باد کے رشید عاجز اور یعقوب پرویز کھوکھر ،چوتھی تصویر آزاد کشمیر کے خود سوزی کر نے والے عزیز ملک کی ہے ۔

 

Spirituality and Psychological Well-Being Among Muslims and Christians Adolescents and Young Adults

The present study examining the relationship between spirituality and psychological well-being among Muslims and Christians adolescents and young adults. Daily Spiritual Experience Scale and Warwick Edinburgh Mental Well-being Scale were used to examine the study variables. The present study was carried out on the sample of (N = 254) i.e. Muslims (n = 123) and Christians (n = 131). The sub sample of Muslims and Christians were further divided into males (n = 48) and females (n = 75). Similarly Christian males (n = 60) and Christian females (n = 71). The instruments used to measure the variables possessed satisfactory reliability i.e. Spirituality (α =.80) for Muslims and (α =.92) for Christians and psychological well-being (α =.82) for Muslims and (α =.84) for Christians. Results of the study revealed that spirituality not only had significant positive correlation with psychological well-being but also found to be significant positive predictor of psychological well-being among Muslims as well as Christians. Additional findings of the study further revealed that significant differences exist in the terms of gender and age. Limitations, suggestion and implications were also discussed at the end of the study.

Characterization of Shisham Dalbergia Sissoo Against Dieback Disease in Various Ecological Zones of Punjab

Shisham (Dalbergia sissoo) is an important tree of Pakistan and is widely grown in different areas of the country mainly for furniture, timber and fuel. Dieback is a serious threat to this tree and has caused huge damage not only in Pakistan but also in India, Nepal and Bangladesh. A nursery experiment of sexually (seedlings) and asexually (cuttings) propagated D. sissoo was conducted during 2009 and 2010. Both seedlings and cuttings were inoculated with the most commonly found fungi in the dieback affected trees i.e Fusarium solani, Botryodiplodia theobromae, Curvularia lunata and Ganoderma lucidum. Overall highest disease incidence was observed in plants inoculated with F. solani (31.39%). In seedlings F. solani caused 46.18% disease while in cuttings it was only 16.61%. No disease was recorded in controlled conditions. A significant (P<0.05) correlation of seedlings and cuttings was observed with climatic variables. Good association of seedlings (r= 0.734) and cuttings (r=0.629) was observed with maximum temperature. Disease predictive models of seedlings and cuttings were developed with climatic variables; Y = -58.3 +7.58x 1 +0.0054x 2 -1.14x 3 +2.47x 4 -1.09x 5 R 2 = 0.62 (2 years data) Where Y= Disease in cuttings, x 1 = Rainfall, x 2 = Relative humidity, x 3 = Minimum temperature, x 4 = Maximum temperature and x 5 = Wind velocity R 2 =0.48 (2 years data) Y = -134 +15x 1 +0.158 x 2 -2.32 x 3 +5.27 x 4 -3.70x 5 Where Y= Disease in seedlings, x 1 = Rainfall, x 2 = Relative humidity, x 3 = Minimum temperature, x 4 = Maximum temperature and x 5 = Wind velocity A comprehensive survey of shisham was carried out in four agro-ecological zones (Sandy deserts, Northern irrigated plains, Barani areas and Sulaiman piedmont) of Punjab province. For survey eleven districts were selected being the most productive and shisham frequenting districts from the above said zones. Trees were divided into age classes and on the basis of disease severity into healthy, partially affected and fully affected trees. Age class one (1-20 years old trees) was observed as the healthiest while age class three (above 40 years) was found to be the most affected in all zones and districts. Maximum number of healthy trees was observed in age class one of Barani areas (92.16%) Maximum number of partially affected trees was seen in Sulaiman piedmont (22.89%) and less number was in Barani areas. Maximum number of dead or fully affected trees was found in class three of Northern irrigated palins (23.60%) and only 4.68% dead trees were recorded in age class one of Barani areas. In selected districts maximum number of healthy trees was in class one of Rawalpindi district (95%) and maximum partially affected trees were in age class three in Dera Ghazi Khan (22.89%). Maximum number of dead trees (33%) was found in age class three of Toba Tek Singh. Rawalpindi was the least affected district with only 5 % dead trees in class one. Water table between 15-20 feet of depth was considered as the most suitable depth where less number of dead trees were recorded in all age classes. Significant (P<0.05) correlation was observed between age class, relative humidity, minimum temperature, maximum temperature and wind velocity. Correlation was non significant with rainfall and water table. Good association was observed between tree age and dieback disease in Healthy (r=0.626), partially affected (r=0.539) and fully affected trees (r=0.613). A disease predictive model based on two years disease survey data was developed: FA = 2.51 + 4.15 x 1 + 5.25 x 2 + 0.00866 x 3 - 0.113 x 4 - 0.0611 x 5 - 0.0115 x 6 - 0.0092 x 7 + 0.541 x8 R 2 = 0.89 Where FA= Fully affected, x 1 = Age, x 2 = Year, x 3 = Water table, x 4 = Rainfall, x 5 = Relative humidity, x 6 = Minimum temperature, x 7 = Maximum temperature, x 8 = Wind velocity