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4. The poisson process in quantum stochastic calculus

Author

Pathmanathan, S.

Subjects
512, Functional analysis, Probability theory and stochastic processes
Abstract

Given a compensated Poisson process $(X_t)_{t \geq 0}$ based on $(\Omega, \mathcal{F}, \mathbb{P})$, the Wiener-Poisson isomorphism $\mathcal{W} : \mathfrak{F}_+(L^2 (\mathbb{R}_+)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$ is constructed. We restrict the isomorphism to $\mathfrak{F}_+(L^2 [0,1])$ and prove some novel properties of the Poisson exponentials $\mathcal{E}(f) := \mathcal{W}(e(f))$. A new proof of the result $\Lambda_t + A_t + A^{\dagger}_t = \mathcal{W}^{-1}\widehat{X_t} \mathcal{W}$ is also given. The analogous results for $\mathfrak{F}_+(L^2 (\mathbb{R}_+))$ are briefly mentioned. The concept of a compensated Poisson process over $\mathbb{R}_+$ is generalised to any measure space $(M, \mathcal{M}, \mu)$ as an isometry $I : L^2(M, \mathcal{M}, \mu) \to L^2 (\Omega,\mathcal{F}, \mathbb{P})$ satisfying certain properties. For such a generalised Poisson process we recall the construction of the generalised Wiener-Poisson isomorphism, $\mathcal{W}_I : \mathfrak{F}_+(L^2(M)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$, using Charlier polynomials. Two alternative constructions of $\mathcal{W}_I$ are also provided, the first using exponential vectors and then deducing the connection with Charlier polynomials, and the second using the theory of reproducing kernel Hilbert spaces. Given any measure space $(M, \mathcal{M}, \mu)$, we construct a canonical generalised Poisson process $I : L^2 (M, \mathcal{M}, \mu) \to L^2(\Delta, \mathcal{B}, \mathbb{P})$, where $\Delta$ is the maximal ideal space, with $\mathcal{B}$ the completion of its Borel $\sigma$-field with respect to $\mathbb{P}$, of a $C^*$-algebra $\mathcal{A} \subseteq \mathfrak{B}(\mathfrak{F}_+(L^2(M)))$. The Gelfand transform $\mathcal{A} \to \mathfrak{B}(L^2(\Delta))$ is unitarily implemented by the Wiener-Poisson isomorphism $\mathcal{W}_I: \mathfrak{F}_+(L^2(M)) \to L^2(\Delta)$. This construction only uses operator algebra theory and makes no a priori use of Poisson measures. A new Fock space proof of the quantum Ito formula for $(\Lambda_t + A_t + A^{\dagger}_t)_{0 \leq t \leq 1}$ is given. If $(F_{\ \! \! t})_{0 \leq t \leq 1}$ is a real, bounded, predictable process with respect to a compensated Poisson process $(X_t)_{0 \leq t \leq 1}$, we show that if $M_t = \int_0^t F_s dX_s$, then on $\mathsf{E}_{\mathrm{lb}} := \mathrm{linsp} \{ e(f) : f \in L^2_{\mathrm{lb}}[0,1] \}$, $\mathcal{W}^{-1} \widehat{M_t} \mathcal{W} = \int_0^t \mathcal{W}^{-1} \widehat{F_s} \mathcal{W} (d\Lambda_s + dA_s + dA^{\dagger}_s),$ and that $(\mathcal{W}^{-1} \widehat{M_t} \mathcal{W})_{0 \leq t \leq 1}$ is an essentially self-adjoint quantum semimartingale. We prove, using the classical Ito formula, that if $(J_t)_{0 \leq t \leq 1}$ is a regular self-adjoint quantum semimartingale, then $(\mathcal{W} \widehat{M_t} \mathcal{W}^{-1} + J_t)_{0 \leq t \leq 1}$ is an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula, and hence the quantum Ito formula. The equivalent result for the sum of a Brownian and Poisson martingale, provided that the sum is essentially self-adjoint with core $\mathsf{E}_{\mathrm{lb}}$, is also proved.

Published
2002

10. Aluminium in Brain Tissue in Non-neurodegenerative/Non-neurodevelopmental Disease: A Comparison with Multiple Sclerosis

Author

Linhart, C., Davidson, D., Pathmanathan, S., Kamaladas, T., and Exley, C.

Abstract

Human exposure to aluminium is a burgeoning issue. The brain is a sink for systemically available aluminium and a putative target of neurotoxicity. An increasing number of studies continue to confirm the presence of aluminium in human brain tissue though primarily in relation to donors who have died of a neurodegenerative or neurodevelopmental disorder. Herein, we have measured aluminium in brain tissue in donors who died of a specific disease or condition though without showing any neurodegeneration. The donors were diagnosed as not suffering from multiple sclerosis. Herein, these novel data are compared with recent data on aluminium in brain tissue in multiple sclerosis. Brain tissues from all four lobes were obtained from the Multiple Sclerosis Society Tissue Bank. Tissues were digested using microwave-assisted acid digestion and their aluminium content was measured by transversely heated graphite furnace atomic absorption spectrometry. Both are established methods in our laboratory. Detailed statistical analyses were used to compare new data with recent data for multiple sclerosis. Aluminium was found in brain tissue in each donor with a high proportion of measurements (189/291) being below 1.00 μg/g dry weight. The data for all cases (median and IQR) were 0.74 (0.48–1.28), 1.23 (0.62–1.63), 0.84 (0.45–1.14) and 1.01 (0.62–1.65) μg/g dry weight for occipital, parietal, temporal and frontal lobes, respectively. There was a statistically significant positive correlation between aluminium content of brain tissue and the age of donor. Comparison of data for this non-multiple sclerosis group with brain aluminium data for donors dying with a diagnosis of multiple sclerosis showed that the latter had a statistically significant higher content of brain aluminium. The data reinforce a previous conclusion that the aluminium content of brain tissue in multiple sclerosis is elevated and support the suggestion that human exposure to aluminium may have a role to play in the aetiology of multiple sclerosis.

Published
2024
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13. Feminism - my own version.

Author

PATHMANATHAN, S.

Subjects
FEMINISM - my own version (Poem), PATHMANATHAN, S.
Published
2024

18. Graves orbitopathy - an update

Author

Wijenayake, U. N., primary, Dissanayake, P., additional, Dematapitiya, C., additional, Pathmanathan, S., additional, Sumatillake, M., additional, and Bulugahapitiya, U., additional

Published
2021
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24. Thyroid Disorders

Author

Somasundaram, N P, primary, Sumanatilleke, M, additional, Katulanda, P, additional, Garusinghe, C, additional, Abhayarathna, S, additional, Pathmanathan, S, additional, Gunathilake, S S C, additional, Subhasinghe, C J, additional, Wijesinghe, A M, additional, Arambewela, M, additional, Dilrukshi, M D S A, additional, Vithiya, R U, additional, Ranathunga, I, additional, Wijewickrama, P S A, additional, Dissanayake, H A, additional, De Silva, A M N L, additional, Yogendranathan, N, additional, Jayasuriya,, C A, additional, Wijetunga, W M U A, additional, and Gamage, K K K, additional

Published
2020
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30. MURDER

Author

Nuhman, M. A. and Pathmanathan, S.

Published
1987

32. The poisson process in quantum stochastic calculus

Author

Pathmanathan, S

Abstract

Given a compensated Poisson process $(X_t)_{t \geq 0}$ based on $(\Omega, \mathcal{F}, \mathbb{P})$, the Wiener-Poisson isomorphism $\mathcal{W} : \mathfrak{F}_+(L^2 (\mathbb{R}_+)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$ is constructed. We restrict the isomorphism to $\mathfrak{F}_+(L^2 [0,1])$ and prove some novel properties of the Poisson exponentials $\mathcal{E}(f) := \mathcal{W}(e(f))$. A new proof of the result $\Lambda_t + A_t + A^{\dagger}_t = \mathcal{W}^{-1}\widehat{X_t} \mathcal{W}$ is also given. The analogous results for $\mathfrak{F}_+(L^2 (\mathbb{R}_+))$ are briefly mentioned. The concept of a compensated Poisson process over $\mathbb{R}_+$ is generalised to any measure space $(M, \mathcal{M}, \mu)$ as an isometry $I : L^2(M, \mathcal{M}, \mu) \to L^2 (\Omega,\mathcal{F}, \mathbb{P})$ satisfying certain properties. For such a generalised Poisson process we recall the construction of the generalised Wiener-Poisson isomorphism, $\mathcal{W}_I : \mathfrak{F}_+(L^2(M)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$, using Charlier polynomials. Two alternative constructions of $\mathcal{W}_I$ are also provided, the first using exponential vectors and then deducing the connection with Charlier polynomials, and the second using the theory of reproducing kernel Hilbert spaces. Given any measure space $(M, \mathcal{M}, \mu)$, we construct a canonical generalised Poisson process $I : L^2 (M, \mathcal{M}, \mu) \to L^2(\Delta, \mathcal{B}, \mathbb{P})$, where $\Delta$ is the maximal ideal space, with $\mathcal{B}$ the completion of its Borel $\sigma$-field with respect to $\mathbb{P}$, of a $C^*$-algebra $\mathcal{A} \subseteq \mathfrak{B}(\mathfrak{F}_+(L^2(M)))$. The Gelfand transform $\mathcal{A} \to \mathfrak{B}(L^2(\Delta))$ is unitarily implemented by the Wiener-Poisson isomorphism $\mathcal{W}_I: \mathfrak{F}_+(L^2(M)) \to L^2(\Delta)$. This construction only uses operator algebra theory and makes no a priori use of Poisson measures. A new Fock space proof of the quantum Ito formula for $(\Lambda_t + A_t + A^{\dagger}_t)_{0 \leq t \leq 1}$ is given. If $(F_{\ \! \! t})_{0 \leq t \leq 1}$ is a real, bounded, predictable process with respect to a compensated Poisson process $(X_t)_{0 \leq t \leq 1}$, we show that if $M_t = \int_0^t F_s dX_s$, then on $\mathsf{E}_{\mathrm{lb}} := \mathrm{linsp} \{ e(f) : f \in L^2_{\mathrm{lb}}[0,1] \}$, $\mathcal{W}^{-1} \widehat{M_t} \mathcal{W} = \int_0^t \mathcal{W}^{-1} \widehat{F_s} \mathcal{W} (d\Lambda_s + dA_s + dA^{\dagger}_s),$ and that $(\mathcal{W}^{-1} \widehat{M_t} \mathcal{W})_{0 \leq t \leq 1}$ is an essentially self-adjoint quantum semimartingale. We prove, using the classical Ito formula, that if $(J_t)_{0 \leq t \leq 1}$ is a regular self-adjoint quantum semimartingale, then $(\mathcal{W} \widehat{M_t} \mathcal{W}^{-1} + J_t)_{0 \leq t \leq 1}$ is an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula, and hence the quantum Ito formula. The equivalent result for the sum of a Brownian and Poisson martingale, provided that the sum is essentially self-adjoint with core $\mathsf{E}_{\mathrm{lb}}$, is also proved.

Published
2016

35. Beyond Borders.

Author

PATHMANATHAN, S.

Subjects
BEYOND Borders (Poem), PATHMANATHAN, S.
Published
2024

36. ANTIBACTERIAL EFFECTS VARIOUS TYPES OF HONEY AND CITRUS JUICE ON STREPTOCOCCUS PYOGENES: A SYSTEMATIC REVIEW.

Author

Mohamad, N. S., Pathmanathan, S. G., Ismail, Z., Sani, A., Aripin, K. N. B. N., Rashid, Z. Z., and Mohamed, N. A.

Subjects
*CINAHL database, *CITRUS, *FRUIT juices, *HONEY, *MEDLINE, *STREPTOCOCCUS, *SYSTEMATIC reviews, *PLANT extracts
Abstract

The home remedy of taking honey along with a citrus juice of lemon, lime or calamansi to soothe sore throat has long been practiced in many cultures across the world, since ancient times. This paper aims to systematically review the antibacterial effect of honey and citrus juice on Streptococcus pyogenes by means of a systematic search in EBSCOhost, Medline, Scopus and ISI Web of Science databases for reports of studies investigating the antibacterial effects of honey and citrus fruit juice on S. pyogenes. A total of 415 publications were initially identified, out of which, 20 were finally chosen and reviewed by looking at the tittles, abstracts and full paper using pre-determined inclusion and exclusion criteria in relation to honey (n=16) and citrus (n=4). The majority of the studies showed that both honey and citrus have significant antimicrobial effect on S. pyogenes. There are still not many available data though on the combined effect of honey and citrus on the bacterium. This knowledge gap offers an opportunity to investigate those effects with the purpose of supporting traditional practice with scientific evidences. [ABSTRACT FROM AUTHOR]

Published
2018

49. Camurati-Engelmann disease

Author

Pathmanathan, S, primary, Somasundaram, Noel P, additional, De Silva, Anil, additional, Jeyakumar, J, additional, Perera, MV, additional, and Samarasekara, Anandi, additional

Published
2012
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