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Algebras of Block Toeplitz Matrices and Matrix Valued Truncated Toeplitz Operators and Related Problems

Thesis Info

Access Option

External Link

Author

Khan, Muhammad Ahsan

Program

PhD

Institute

Government College University

Institute Type

Public

Affiliation

Abdul Salam School of Mathematical Sciences

City

Kotli

Province

Punjab

Country

Pakistan

Thesis Completing Year

2020

Thesis Completion Status

Completed

Subject

Mathemaics

Language

English

Link

http://prr.hec.gov.pk/jspui/bitstream/123456789/13384/1/Muhammad%20Ahsan%20Khan%20maths%202020%20gcu%20lhr%20prr.pdf

Added

2021-02-17 19:49:13

Modified

2024-03-24 20:25:49

ARI ID

1676725451585

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The present thesis entitled ”Algebras of block Toeplitz matrices and matrix valued truncated Toeplitz operators and related problems” includes a part of research work carried out by the author during the last four years at the Abdus Salam School of Mathematical sciences Government college university Lahore Pakistan and Simion Stoilow Insitute of the Mathematics of the Romania Academy Bucharest Romania. The thesis comprises five chapters and each chapter is subdivided into various sections. The maximal algebras of scalar Toeplitz matrices are known to be formed by generalized circulants. A similar simple description cannot be obtained for block Toeplitz matrices. The identification of algebras consisting of block Toeplitz matrices is a harder problem, that has received little attention up to now. We consider the case when the block entries of the matrices belong to a commutative algebra A. We introduce and investigate certain families of maximal commutative algebras of block Toeplitz matrices. After obtaining some general results, we classify all the maximal algebras for certain particular cases of A. In the last decade, a large amount of research has concentrated on a generalization of Toeplitz matrices, namely truncated Toeplitz operators. For matrix valued truncated Toeplitz operators, which are compressions of multiplication operators to model subspaces of the vector valued Hardy space H2(E) , likewise block Toeplitz matrices case the problem of characterizing the maximal algebras becomes now very hard and there seem to be no chances to treat it in a general way. However, under certain assumptions, we obtain criteria for commutation relations for matrix valued truncated Toeplitz operators. The results show an analogy to the case of Toeplitz matrices which will be the basis for the extension of Sedlock classes of truncated Toeplitz operators to matrix valued case.
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یہ عشق میں نہ سوچ تو کہ کیا نہیں ملا

یہ عشق میں نہ سوچ، تُجھے کیا نہیں ملا
ہے کر لیا، تو خاک میں اپنی جبیں ملا

ہم راہ دیکھتے ہی رہے جس کی عمر بھر
آیا وہ شہر میں بھی تو ہم سے نہیں ملا

بچپن میں دل کی بستی میں رہتے تھے کتنے لوگ
دیکھا شباب میں تو فقط اک مکیں ملا

اگلے جہاں کے عہد پہ ہم کو دیا ہے ٹال
کم بخت ہم کو وہ تو بلا کا ذہیں ملا

کہتے رہے تھے یار جسے ہم تمام عمر
اک دن عدو کی بزم میں وہ نازنیں ملا

گر یاں دیا نہ تُو نے تو نہ لوں گا حشر میں
یارب اسے اگر ہے ملانا، یہیں ملا

رویہ صارف (Consumer Behavior) اور اسلامی اخلاقیات

Consumer Behavior means behavior of consumers about buying goods and services in the market. Consumer behavior entails all activities associated with the purchase, use and disposal of goods and services, including the consumer emotional, mental and behavioral responses that precede or follow these activities. The following article discusses consumer behavior in the light of Islamic ethics, and explains different aspects of consumer behavior according to Islamic moral values and also given suggestions to consumers and marketers to adopt Islamic buying and selling behavior, which is beneficial not only for individual but also to human as a whole. The consumer behavior in the modern capitalist economy plays very important role, not only in consumption but also in the production and services.

Chromosomal Abnormalities and Epstein Barr Virus in Acute Lymphoblastic Leukemia in Children

Background: Acute lymphoblastic leukemia (ALL) is a disease typically characterized by the accumulation of immature abnormal lymphoid progenitor cells (lymphoblasts) in the bone marrow, which have abnormal proliferation and differentiation. A number of acquired chromosomal abnormalities arising from translocations, deletions, duplications and inversions have been identified in 80% of childhood and 79% of adulthood ALL. 1. Objectives of Study: To determine the frequency of chromosomal abnormalities in children suffering from ALL, To evaluate structural and numerical chromosomal abnormalities in patients with ALL, To find out the frequency of Epstein Barr Virus in ALL cases. Subjects and Methods: An observational study was conducted at the Liaquat University of Medical and Health Sciences, Jamshoro and Isra University Hospital, Hyderabad. 100 diagnosed childhood ALL cases were selected through non-probability purposive sampling according to inclusion and exclusion criteria. The Blood samples were collected in bottles containing Ethylene diamine tetra acetic acid (EDTA) as an anticoagulant and were processed on automatic hematoanalyzer, Sysmex KX 21. Fixed cell suspensions prepared from diagnostic bone marrow. For routine cytogenetic analysis and FISH, samples were obtained from the diagnosed cases of acute lymphoblastic leukemia. Methods for detecting EBV infection were based on RT-PCR. The data was analyzed on SPSS version 21.0 (IBM, Corporation, USA) and Microsoft excel. The continuous variables were presented as mean ± SD and analyzed using student’s t-test. Categorical variables were analyzed by Chi-square test and results were presented as frequencies and percentages. Data was presented in tables, graphs and charts. P-value of significance was taken at ≤0.05. Results: Numerical and structural chromosomal abnormalities were noted in 69% and 60% of cases respectively (p=0.001). Chromosomal ploidy showed Diploidy and Aneuploidy in 29% and 69% of cases respectively (p=0.0001). Hyperploidy, hypoploidy and pseudoploidy were noted in 51%, 6% and 12% of cases respectively (0.001). Chromosomal structural abnormalities noted were; t (12; 21)(p13; q22) t(9;22)(q34;q11), t(8;14)(q24;q32), t(5;14)(q31;q32), t(17;19)(q22;q13), t (7;11) (q35; q13), t (1;7) (p32; q35), t (7;19) (q35; p13), t(1;19)(p13;q23), t(8;22)(q24;q11) and unknown 13%. Ph + chromosome (t (9; 22) (q34; q11) was noted in 6% of cases and EBV in 19% of total study population Conclusion: ALL cases are characterized by leukocytosis and anemia. Epstein Barr Virus was found in 19% childhood ALL in present study. Present study shows good prognostic cytogenetic abnormalities like hyperdiploidy and t (12; 21)(p13; q22) in Pakistani children with ALL and frequency of poor prognostic cytogenetic aberrations like hypoploidy and t (9; 22) (q34; q11.2) is comparable to previous studies.