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*Indian* *Statistical* *Institute*,

, 907

"... Abstract 1 Here we generalize the concept of spatial tensor product, introduced by Skeide, of two product systems via a pair of normalized units. This new notion is called amalgamated tensor product of product systems, and now the amalgamation can be done using a contractive morphism. Index of amalg ..."

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Abstract 1 Here we generalize the concept of spatial tensor product, introduced by Skeide, of two product systems via a pair of normalized units. This new notion is called amalgamated tensor product of product systems, and now the amalgamation can be done using a contractive morphism. Index of amalgamation product (when done through units) adds up for normalized units but for non-normalized units, the index is one more than the sum. We define inclusion systems and use it as a tool for index computations. It is expected that this notion will have other uses. 1.

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*Indian* *Statistical* *Institute*,

, 2006

"... If A is a real symmetric matrix and P is an orthogonal projection onto a hyperplane, then we derive a formula for the Moore-Penrose inverse of P AP. As an application, we obtain a formula for the Moore-Penrose inverse of a Euclidean distance matrix (EDM) which generalizes formulae for the inverse of ..."

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If A is a real symmetric matrix and P is an orthogonal projection onto a hyperplane, then we derive a formula for the Moore-Penrose inverse of P AP. As an application, we obtain a formula for the Moore-Penrose inverse of a Euclidean distance matrix (EDM) which generalizes formulae for the inverse of a EDM in the literature. To an invertible spherical EDM, we associate a Laplacian matrix (which we define as a positive semidefinite n × n matrix of rank n − 1 and with zero row sums) and prove some properties. Known results for distance matrices of trees are derived as special cases. In particular, we obtain a formula due to Graham and Lovász for the inverse of the distance marix of a tree. It is shown that if D is a nonsingular EDM and L is the associated Laplacian, then D−1 − L is nonsingular and has a nonnegative inverse. Finally, infinitely divisible matrices are constructed using EDMs. 1

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*Indian* *Statistical* *Institute*,

, 1999

"... A regular operator T on a Hilbert C ∗-module is defined just like a closed operator on a Hilbert space, with the extra condition that the range of (I + T ∗ T) is dense. Semiregular operators are a slightly larger class of operators that may not have this property. It is shown that, like in the case ..."

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A regular operator T on a Hilbert C ∗-module is defined just like a closed operator on a Hilbert space, with the extra condition that the range of (I + T ∗ T) is dense. Semiregular operators are a slightly larger class of operators that may not have this property. It is shown that, like in the case of regular operators, one can, without any loss in generality, restrict oneself to semiregular operators on C ∗-algebras. We then prove that for abelian C ∗-algebras as well as for subalgebras of the algebra of compact operators, any closed semiregular operator is automatically regular. We also determine how a regular operator and its extensions (and restrictions) are related. Finally, using these results, we give a criterion for a semiregular operator on a liminal C ∗-algebra to have a regular extension. AMS Subject Classification No.: 46H25, 47C15

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*Indian* *Statistical* *Institute*,

, 705

"... We observe that there is no clash between the works [1] and [2]. In the Comment [1] the author has shown that one can construct a Lagrangian model of a point particle with a Magueijo-Smolin (MS) form of dispersion relation in a canonical phase space provided one modifies the Lorentz generator to J µ ..."

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We observe that there is no clash between the works [1] and [2]. In the Comment [1] the author has shown that one can construct a Lagrangian model of a point particle with a Magueijo-Smolin (MS) form of dispersion relation in a canonical phase space provided one modifies the Lorentz generator to J µν DSR = (x µ − (xp) l ηµ)p ν − (x ν − (xp) l ην)p µ. (1) On the other hand in [2] I have shown that one can keep the Lorentz generator J µν = x µ p ν − x ν p µ unchanged provided a non-canonical symplectic structure is used. In my opinion the above two formalisms are complimentary and there is no reason to treat the former [1] as an improvement, but for a bias of the author of [1] against the introduction of a non-canonical phase space. Furthermore, it is crucial to keep in mind that from the point of view of DSR, the Lagrangian in (4) [1] with the chosen form of J µν DSR is fundamental and the coordinate x µ, (with its noncanonical behavior under Lorentz transformation), is the physical coordinate and p µ is the physical momentum. According to DSR, results obtained in x µ, p µ variables (and not in X µ, P µ) should be compared with experiments. Thus (4) in [1] should be considered as the starting point and (1) in [1] is obtained in a particular parameterization. This does not mean that the DSR model is trivially related to normal particle model. This is because in order to get the correct behavior of a DSR particle one has to convert the normal particle equations (in X µ, P µ) to equations involving physical DSR coordinates x µ, p µ using (2) X µ = (F −1) µ ν xν 1 as given in [1]. This will lead to new κ-DSR physics since coordinates and momenta get mixed up under Lorentz transformation. In this way one can exploit the canonical framework to generate DSR behavior. This sort of approach is discussed extensivly in [3] in a related context where it is also shown that dynamical inputs are required in order to extrapolate kinematical equations in canonical framework to equations in DSR framework.

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*Indian* *Statistical* *Institute*,

, 802

"... In the present paper we have directly computed the Berry curvature terms relevant for Graphene in the presence of an inhomogeneous lattice distortion. We have employed the generalized Foldy Wouthuysen framework, developed by some of us [4, 5, 6]. We show that a non-constant lattice distortion leads ..."

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In the present paper we have directly computed the Berry curvature terms relevant for Graphene in the presence of an inhomogeneous lattice distortion. We have employed the generalized Foldy Wouthuysen framework, developed by some of us [4, 5, 6]. We show that a non-constant lattice distortion leads to a valley-orbit coupling which is responsible to a valley-Hall effect. This is similar to the valley-Hall effect induced by an electric field proposed in [14] and is the analogue of the spin-Hall effect in semiconductors [16, 17]. Our general expressions for Berry curvature, for the special case of homogeneous distortion, reduce to the previously obtained results [14]. We also discuss the Berry phase in the quantization of cyclotron motion.

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*Indian* *Statistical* *Institute*,

"... The study presents the use of e-Resources by the faculty members and research scholars various engineering colleges of Visvesvaraya Technological University (VTU) Belgaum, Karnataka. The main aim of this study is to know the usage of e-Resources by faculty members and research scholars who have regi ..."

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The study presents the use of e-Resources by the faculty members and research scholars various engineering colleges of Visvesvaraya Technological University (VTU) Belgaum, Karnataka. The main aim of this study is to know the usage of e-Resources by faculty members and research scholars who have registered through the recognized Research Centers of Visvesvaraya Technological University (VTU), Belgaum, Karnataka. As a tool the survey method of questionnaire was distributed among the respondents of various departments. Out of 1000 questionnaires, 866 questionnaires were received from the respondents and 86.6 % of respondents have replied to the quires. It is observed that majority of senior level teachers and research scholars access the e-Resources for the research work rather than teaching. The study revealed that, 94 % of the users are depending on e-Resources which are more relevant for their study rather than print resources. The trend predicts that e-Resources has over taken the print resources and predicts that the print resources will be phased out in near future.

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*Indian* *Statistical* *Institute*,

"... This paper attempts to develop a model of endogenous growth with special focus on the role of health capital, public infrastructure and environmental pollution. It is an extension of the model of Agenor (2008) who does not consider environmental pollution. We analyse properties of optimal fiscal pol ..."

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This paper attempts to develop a model of endogenous growth with special focus on the role of health capital, public infrastructure and environmental pollution. It is an extension of the model of Agenor (2008) who does not consider environmental pollution. We analyse properties of optimal fiscal policy in the steady-state growth equilibrium when the level of production of the final good is the source of pollution. Tax revenue of the government is channelized into three expenditure heads-health expenditure, pollution abatement expenditure and public infrastructure expenditure. It is found that the optimum ratio of public infrastructural expenditure to national income in the steady-state equilibrium is less than the competitive output share of the public input; and it varies inversely with the magnitude of the pollution-output coefficient. There is no conflict between the social welfare maximizing solution and the growth rate maximizing solution in the steady-state equilibrium. There may exist indeterminacy in the transitional growth path converging to the unique steady-state equilibrium point that never satisfies saddle-point stability. The market economy growth rate is not necessarily less than the socially efficient growth rate in the steady-state equilibrium.

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*Indian* *Statistical* *Institute*

, 2002

"... On testing dependence between time to failure and cause of failure via conditional probabilities ..."

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On testing dependence between time to failure and cause of failure via conditional probabilities