Efficient Black-Box Identity Testing for Free Group Algebras

Hrubes and Wigderson [Pavel Hrubes and Avi Wigderson, 2014] initiated the study of noncommutative arithmetic circuits with division computing a noncommutative rational function in the free skew field, and raised the question of rational identity testing. For noncommutative formulas with inverses the problem can be solved in deterministic polynomial time in the white-box model [Ankit Garg et al., 2016; Ivanyos et al., 2018]. It can be solved in randomized polynomial time in the black-box model [Harm Derksen and Visu Makam, 2017], where the running time is polynomial in the size of the formula. The complexity of identity testing of noncommutative rational functions, in general, remains open for noncommutative circuits with inverses. We solve the problem for a natural special case. We consider expressions in the free group algebra F(X,X^{-1}) where X={x_1, x_2, ..., x_n}. Our main results are the following. 1) Given a degree d expression f in F(X,X^{-1}) as a black-box, we obtain a randomized poly(n,d) algorithm to check whether f is an identically zero expression or not. The technical contribution is an Amitsur-Levitzki type theorem [A. S. Amitsur and J. Levitzki, 1950] for F(X, X^{-1}). This also yields a deterministic identity testing algorithm (and even an expression reconstruction algorithm) that is polynomial time in the sparsity of the input expression. 2) Given an expression f in F(X,X^{-1}) of degree D and sparsity s, as black-box, we can check whether f is identically zero or not in randomized poly(n,log s, log D) time. This yields a randomized polynomial-time algorithm when D and s are exponential in n

Pattern of Electrocardiographic Changes in Pre-operative check-up in Patients Undergoing ENT Surgery

Background: The 12-lead ECG is the primary clinical tool for non-invasive assessment of cardiac electrical function. Before any surgery, ECG is commonly used in pre-operative check-up. Therefore pre-operative check-up of large number of people can be a valuable tool for accurate estimates of the prevalence of ECG abnormalities of general population. Objectives: To observe the extent of ECG changes in a group of population undergoing ENT surgery and to describe the prevalence of changes in relation to age and sex. Methods: 704 consecutive adult stable patients were included between age of 20 – 60 years without previously diagnosed case of any other systemic diseases who came to this hospital for elective surgical procedures and were investigated with new ECG for pre-operative check-up in private Hospital between January 201 to December 2010. Results: Total number of ECG recordings were 704; out of them 479 were male & 225 were female. Among 704 ECG recording 486(69.03%) were normal. The number of old myocardial infarction (MI) was in 36 cases (5.11%). Cardiac Ischaemia observed by ST depressions/T inversion was present in 15 cases (2.13%). Conclusion: The observation of the study is concludes that nearly 70% of all asymptomatic people (between 20 – 60 years) possessing normal ECG but Ischaemic changes (like Q wave/ST depression & T inversion/ LBBB) was present in about 8% cases. ECG changes of IHD (Ischaemic Heart Disease) increases with age and male groups at higher risk of developing IHD. Key Words: ECG; Pre-operative Check-up; Myocardial Infarction (MI). DOI: http://dx.doi.org/10.3329/bsmmuj.v4i2.8630 BSMMU J 2011; 4(2):56-5

A cross-sectional study on morbidity pattern of elderly population residing in a rural area of Tripura

Background: The elderly is one of the most vulnerable and high-risk groups in terms of health and their health seeking behaviour is crucial in any society. A major component of the burden of illness for the elderly derives from prevalent chronic disease. The objective of study aims to find out morbidity pattern of elderly population aged 60 years and above.Methods: A community based cross-sectional study was done in rural areas of Madhupur, Sepahijala district, Tripura from August 2015-January 2016. A total of 260 (elderly aged 60 years and above) study participants were selected by simple random sampling.Results: Majority (52.7%) were between 60-70 years of age, least (1.5%) was in 90-100 years age group. Most of the study population (84.6%) were Hindu and female were more than male (51.9 % vs 48.1%). Majority (38.8%) of them were suffering from two (2) morbidities and 8.1% of study population had 4 and more morbidities. Non-specific generalized weakness was the most common (62.7%) morbidity, followed by gastrointestinal problems (56%) found in geriatric population. Musculoskeletal problems (low back pain, joint pain, osteoarthritis) were 45% followed by anaemia (42%) and impaired vision (36%). Increasing age group and non-smoke tobacco habit among the elderly population was associated with number of morbidities (≥ 3 morbidities/ person) per person (p <0.05).Conclusions: The study showed high prevalence of morbidities among elderly population. Non-specific generalized weakness was one of the most important problems in this age group. We have to find out the underlying cause of this non-specific generalized weakness by further clinical examination and laboratory investigations in future research

On Explicit Branching Programs for the Rectangular Determinant and Permanent Polynomials

We study the arithmetic circuit complexity of some well-known family of polynomials through the lens of parameterized complexity. Our main focus is on the construction of explicit algebraic branching programs (ABP) for determinant and permanent polynomials of the rectangular symbolic matrix in both commutative and noncommutative settings. The main results are: - We show an explicit O^*(binom{n}{downarrow k/2})-size ABP construction for noncommutative permanent polynomial of k x n symbolic matrix. We obtain this via an explicit ABP construction of size O^*(binom{n}{downarrow k/2}) for S_{n,k}^*, noncommutative symmetrized version of the elementary symmetric polynomial S_{n,k}. - We obtain an explicit O^*(2^k)-size ABP construction for the commutative rectangular determinant polynomial of the k x n symbolic matrix. - In contrast, we show that evaluating the rectangular noncommutative determinant over rational matrices is #W[1]-hard

Fast Exact Algorithms Using Hadamard Product of Polynomials

Let C be an arithmetic circuit of poly(n) size given as input that computes a polynomial f in F[X], where X={x_1,x_2,...,x_n} and F is any field where the field arithmetic can be performed efficiently. We obtain new algorithms for the following two problems first studied by Koutis and Williams [Ioannis Koutis, 2008; Ryan Williams, 2009; Ioannis Koutis and Ryan Williams, 2016]. - (k,n)-MLC: Compute the sum of the coefficients of all degree-k multilinear monomials in the polynomial f. - k-MMD: Test if there is a nonzero degree-k multilinear monomial in the polynomial f. Our algorithms are based on the fact that the Hadamard product f o S_{n,k}, is the degree-k multilinear part of f, where S_{n,k} is the k^{th} elementary symmetric polynomial. - For (k,n)-MLC problem, we give a deterministic algorithm of run time O^*(n^(k/2+c log k)) (where c is a constant), answering an open question of Koutis and Williams [Ioannis Koutis and Ryan Williams, 2016]. As corollaries, we show O^*(binom{n}{downarrow k/2})-time exact counting algorithms for several combinatorial problems: k-Tree, t-Dominating Set, m-Dimensional k-Matching. - For k-MMD problem, we give a randomized algorithm of run time 4.32^k * poly(n,k). Our algorithm uses only poly(n,k) space. This matches the run time of a recent algorithm [Cornelius Brand et al., 2018] for k-MMD which requires exponential (in k) space. Other results include fast deterministic algorithms for (k,n)-MLC and k-MMD problems for depth three circuits

Controlled Organocatalytic Ring-Opening Polymerization of ε-Thionocaprolactone

For the first time, the controlled ring-opening polymerization (ROP) of ε-thionocaprolactone (tnCL) is conducted. The organocatalytic ROP of tnCL occurs without carbonyl scrambling, leading to homopoly(ε-thionocaprolactone) (PtnCL). The ROP by base catalysts alone is proposed to proceed via a nucleophilic mechanism, while the addition of an H-bond donating thiourea (TU) is shown to provide excellent reaction control. The increased reaction control provided by the TU occurs in the virtual absence of binding between tnCL and TU, and a mechanistic account for this observation is discussed. The monomer ring strain is measured and found to be similar to δ-valerolactone (VL). Copolymers with VL are synthesized, and the resulting analysis of the copolymer materials properties provides the only known physical characterizations of poly(thio(no)ester-co-ester)s

SLE in Pregnancy

Systemic lupus erythematosus is specially a disease of young women in their child bearing age. It has been found that SLE disease activity can repair in pregnancy even though patient is in complete remission state before the pregnancy. Besides this, pregnancy complications are higher in SLE patients. Another important issue is the use of drugs to control SLE because some of these drugs are potentially terotogenic. Fetal outcome is also a challenging issue. Therefore, multidisciplinary approach has key role in the management of Lupus pregnancy.Key words: Systemic lupus erythematosus (SLE); Lupus pregnancy; Lupus flare; Anti-phospholipid Antibody Syndrome;Neonatal Lupus Syndrome; Congenital heart block.DOI: 10.3329/bsmmuj.v3i1.5517BSMMU J 2010; 3(1): 54-5