Mathematics and Statistics - Publications

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    On an error term related to the Jordan totient function J < inf > k < /inf > (n)
    ( 1990-01-01) Adhikari, Sukumar Das ; Sankaranarayanan, A.
    We investigate the error terms Ek(x)= ∑ n≤xJk(n)- xk+1 (k+1)ζ(k+1) for k≥2, where Jk(n) = nkΠp|n(1 - 1 pk) for k ≥ 1. For k ≥ 2, we prove ∑ n≤xEk(n)∼ xk+1 2(k+1)ζ(k+1). Also, lim inf n→∞ Ek(x) xk≤ D ζ(k+1), where D = .7159 when k = 2, .6063 when k ≥ 3. On the other hand, even though lim inf n→∞ Ek(x) xk≤- 1 2ζ(k+1), Ek(n) > 0 for integers n sufficiently large. © 1990.
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    On the frequency of Titchmarsh's phenomenon for ζ(s)-VIII
    ( 1992-04-01) Balasubramanian, R. ; Ramachandra, K. ; Sankaranarayanan, A.
    For suitable functions H = H(T) the maximum of |(ζ(σ + it)) z | taken over T≤t≤T + H is studied. For fixed σ(1/2≤σ≤l) and fixed complex constants z "expected lower bounds" for the maximum are established. © 1992 Indian Academy of Sciences.
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    On some theorems of littlewood and selberg, I
    ( 1993-01-01) Ramachandra, K. ; Sankaranarayanan, A.
    Assuming the Riemann hypothesis, we prove [formula] and [formula] with economical constants D1 = 0.46657029869824… and D2 = 3.51588780218300… . © 1993 Academic Press Inc.
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    Mean-value theorem of the riemann zeta-function over short intervals
    ( 1993-01-01) Sankaranarayanan, A. ; Srinivas, K.
    Let s = σ + it. Then, on the assumption of Riemann Hypothesis, we prove the Mean-Value Theorem for the square of the Riemann zeta-function over shorter intervals for 1/2 + A1/log log T ≤ σ ≤ 1 - δ. Here A1 is a large positive constant, δ is a small positive constant, and T ≤ t ≤ T + H where H depends on T satisfying H ≤ T. © 1993 Academic Press Inc.
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    On a method of Balasubramanian and Ramachandra (on the abelian group problem)
    ( 1997-12-01) Sankaranarayanan, A. ; Srinivas, K.
    Let an denote the number of non-isomorphic abelian groups of order n. We consider A(x) = Σn≤x an = Σj=110 Cjx1/j + E(x) where E(x) is the error term. We study E(x) through the general method of Balasubramanian and Ramachandra.
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    On Liapunov-Type Inequality for Third-Order Differential Equations
    ( 1999-05-15) Parhi, N. ; Panigrahi, S.
    In this paper, a Liapunov-type inequality has been derived for a class of third-order differential equations of the form,y‴+pty=0,wherepis a real-valued continuous function on [0,∞). The nature of the distance between consecutive two zeros or three zeros has been studied with the help of the inequality. © 1999 Academic Press.
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    Goldbach problem in polynomial values
    ( 1999-06-01) Sankaranarayanan, A.
    An even integer n ≥ 6 is called a Goldbach number if it is the sum of two odd primes. The Goldbach conjecture says that every even number n ≤ 6 is a Goldbach number. In this paper, we study the mean-square upper bound of the error term related to the Goldbach problem, in polynomial values over short intervals, uniformly with respect to the height of a polynomial of fixed degree. © 1999 Springer.
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    Liapunov-type inequality for delay-differential equations of third order
    ( 2002-09-12) Parhi, N. ; Panigrahi, S.
    A Liapunov-type inequality for a class of third order delay-differential equations is derived.
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    Disfocality and Liapunov-type inequalities for third-order equations
    ( 2003-02-01) Parhi, N. ; Panigrahi, S.
    The concept of disfocality is introduced for third-order differential equations y ‴ + p(t)y = 0. This helps to improve the Liapunov inequality when y(t) is a solution of (*) with y(a) = 0 = y′(a), y(b) = 0 = y′(b), and y(t) ≠ 0, t ε (a, b). If y(t) is a solution of (*) with y(t 1) = 0 = y (t 2) = y(t 3) = y(t 4) (t 1 < t 2 < t 3 < t 4) and y(t) ≠ 0 for t ε ∪ 3i=1(t i,t i+1), then the lower bound for (t 4-t 1) is obtained. A new criteria is obtained for disconjugacy of (*) in [a, b]. © 2003 Elsevier Science Ltd. All rights reserved.
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    Bayesian integrated functional analysis of microarray data
    ( 2004-11-22) Bhattacharjee, Madhuchhanda ; Pritchard, Colin C. ; Nelson, Peter S. ; Arjas, Elja
    Motivation: The statistical analysis of microarray data usually proceeds in a sequential manner, with the output of the previous step always serving as the input of the next one. However, the methods currently used in such analyses do not properly account for the fact that the intermediate results may not always be correct, then leading to cumulating error in the inferences drawn based on such steps. Results: Here we show that, by an application of hierarchical Bayesian methodology, this sequential procedure can be replaced by a single joint analysis, while systematically accounting for the uncertainties in this process. Moreover, we can also integrate relevant functional information available from databases into such an analysis, thereby increasing the reliability of the biological conclusions that are drawn. We illustrate these points by analysing real data and by showing that the genes can be divided into categories of interest, with the defining characteristic depending on the biological question that is considered. We contend that the proposed method has advantages at two levels. First, there are gains in the statistical and biological results from the analysis of this particular dataset. Second, it opens up new possibilities in analysing microarray data in general. © Oxford University Press 2004; all rights reserved.
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    Bayesian association-based fine mapping in small chromosomal segments
    ( 2005-01-01) Sillanpää, Mikko J. ; Bhattacharjee, Madhuchhanda
    A Bayesian method for fine mapping is presented, which deals with multiallelic markers (with two or more alleles), unknown phase, missing data, multiple causal variants, and both continuous and binary phenotypes. We consider small chromosomal segments spanned by a dense set of closely linked markers and putative genes only at marker points. In the phenotypic model, locus-specific indicator variables are used to control inclusion in or exclusion from marker contributions. To account for covariance between consecutive loci and to control fluctuations in association signals along a candidate region we introduce a joint prior for the indicators that depends on genetic or physical map distances. The potential of the method, including posterior estimation of trait-associated loci, their effects, linkage disequilibrium pattern due to close linkage of loci, and the age of a causal variant (time to most recent common ancestor), is illustrated with the well-known cystic fibrosis and Friedreich ataxia data sets by assuming that haplotypes were not available. In addition, simulation analysis with large genetic distances is shown. Estimation of model parameters is based on Markov chain Monte Carlo (MCMC) sampling and is implemented using WinBUGS. The model specification code is freely available for research purposes from http://www.rni.helsinki.fi/~mjs/.
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    Homogeneity of subjective cellular automata
    ( 2005-06-01) Subrahmonian Moothathu, T. K.
    We bring out some similarities among one-dimensional surjective cellular automata. Four main results are the following: (i) all periodic points of a cellular automata are shift-periodic if and only if the set of periodic points of any fixed period is finite, (ii) forward recurrent points as well as backward recurrent points are residual for every onto cellular automata, (iii) every onto cellular automata is semi-open, and (iv) all transitive cellular automata are weak mixing and hence maximally sensitive (which improves an existing result).
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    String dust cosmological model in higher-dimensional space-time
    ( 2005-09-01) Khadekar, G. S. ; Patki, Vrishali ; Radha, R.
    We have investigated the bulk viscous fluid string dust cosmological model in the higher dimensional space-time. To obtain a determinate solution, it is assumed that the coefficient of bulk viscosity is a power function of the energy density τ = τcρm(t) and the scalar of expansion is proportional to shear scalar, which leads to a relation between metric potentials A = KRn where A and R are functions of time. It is also observed that models appear to be singular at x = 1/b log (1 - b/a) and x = - 1/c in the presence and absence of bulk viscosity and for n = 1, the model represent an isotropic universe. The physical and geometrical aspects of the model are also discussed. © World Scientific Publishing Company.
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    Set of periods of additive cellular automata
    ( 2006-03-07) Moothathu, T. K.Subrahmonian
    It is shown that the set of periods of any additive cellular automata F, where the addition is done modulo a prime p, can be determined using some simple conditions on the coefficients in the linear expression of F. In particular, we establish that the set of periods has only four possibilities: {1,m} for some m where 1≤m < p, N\{pm:m∈N}, N\{2pm:m∈N∪{0}} or the whole set N={1,2,3,...}. Using our results, the set of periods of any additive cellular automata, where the addition is done modulo a square-free positive integer, is easily obtained. © 2005 Elsevier B.V. All right reserved.
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    Association mapping of complex trait loci with context-dependent effects and unknown context variable
    ( 2006-11-29) Sillanpää, Mikko J. ; Bhattacharjee, Madhuchhanda
    A novel method for Bayesian analysis of genetic heterogeneity and multilocus association in random population samples is presented. The method is valid for quantitative and binary traits as well as for multiallelic markers. In the method, individuals are stochastically assigned into two etiological groups that can have both their own, and possibly different, subsets of trait-associated (disease-predisposing) loci or alleles. The method is favorable especially in situations when etiological models are stratified by the factors that are unknown or went unmeasured, that is, if genetic heterogeneity is due to, for example, unknown genes X environment or genes X gene interactions. Additionally, a heterogeneity structure for the phenotype does not need to follow the structure of the general population; it can have a distinct selection history. The performance of the method is illustrated with simulated example of genes X environment interaction (quantitative trait with loosely linked markers) and compared to the results of single-group analysis in the presence of missing data. Additionally, example analyses with previously analyzed cystic fibrosis and type 2 diabetes data sets (binary traits with closely linked markers) are presented. The implementation (written in WinBUGS) is freely available for research purposes from http://www.rni.helsinki.fi/~mjs/. Copyright © 2006 by the Genetics Society of America.
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    Higher moments of certain L-functions on the critical line
    ( 2007-07-01) Sankaranarayanan, A.
    We study the sixth-power moments of certain L-functions belonging to a sub-class of the Selberg's class on the critical line and, using this, we conclude an upper bound for the fourth-power moments of certain L-functions related to GL 3 on the critical line. This is an analogue of the upper bound for the twelfth-power moment of the Riemann zeta-function on the critical line. © 2007 Springer Science+Business Media, Inc.
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    Nonlinear renewal equations
    ( 2008-01-01) Perthame, Benoît ; Tumuluri, Suman Kumar
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    Orbits of Darboux-like real functions
    ( 2008-01-01) Subrahmonian Moothathu, T. K.
    We show that, with respect to the dynamics of iteration, Darbouxlike functions from ℝ to ℝ can exhibit some strange properties which are impossible for continuous functions. To be precise, we show that (i) there is an extendable function from ℝ to ℝ which is 'universal for orbits' in the sense that it possesses every orbit of every function from ℝ to ℝ up to an arbitrary small translation, and which has orbits asymptotic to any real sequence, (ii) there is a function f: ℝ → ℝ such that for every n ∈ ℕ, fn is almost continuous and the graph of fn is dense in ℝ2, in spite of the fact that all f-orbits are finite. To prove (i) we assume the Continuum Hypothesis.