Gopala Krishna Srinivasan - Academia.edu (original) (raw)

Papers by Gopala Krishna Srinivasan

arXiv: History and Overview, 2019

In this expository note we provide a proof of Artin's theorem which states that the commutato... more In this expository note we provide a proof of Artin's theorem which states that the commutator subgroup of a free group on two generators is not finitely generated. The proof employs the infinite grid as in two other proofs in the literature mentioned in the note but takes a somewhat different approach which seems to be of didactic value.

arXiv: General Mathematics, 2018

In this note we provide a direct approach to the most basic operator in this theory namely the ex... more In this note we provide a direct approach to the most basic operator in this theory namely the exterior derivative. The crucial ingredient is a calculus lemma based on determinants. We maintain the view that in a first course at least this direct approach is preferable to the more abstract one based on characterization of the exterior derivative in terms of its properties.

Wolfram Demonstrations Project, 2007

A simple proof of Ramanujan's formula for the Fourier transform of |Γ(a + it)| 2 is given where a... more A simple proof of Ramanujan's formula for the Fourier transform of |Γ(a + it)| 2 is given where a is fixed and has positive real part and t is real. The result is extended to other values of a by solving an inhomogeneous ODE and we use it to calculate the jump across the imaginary axis.

The American Mathematical Monthly, 2007

1The integral on the right converges for complex values of x provided Re x > −1. Euler&amp... more 1The integral on the right converges for complex values of x provided Re x > −1. Euler's notation for it was [ p q ] , where x + 1 was taken to be the rational number p/q [17, vol. 1, p. 221]. ... April 2007] THE GAMMA FUNCTION: AN ECLECTIC TOUR ... This function analytically ...

Springer Proceedings in Mathematics & Statistics, 2021

In this paper we provide a unified approach to a family of integrals of Mellin-Barnes type using ... more In this paper we provide a unified approach to a family of integrals of Mellin-Barnes type using distribution theory and Fourier transforms. Interesting features arise in many of the cases which call for the application of pull-backs of distributions via smooth submersive maps defined by Hörmander. We derive by this method the integrals of Hecke and Sonine relating to various types of Bessel functions which have found applications in analytic and algebraic number theory.

Resonance, 2021

In this article, we look at some of the less explored aspects of the gamma function. We provide a... more In this article, we look at some of the less explored aspects of the gamma function. We provide a new proof of Euler’s reflection formula and discuss its significance in the theory of special functions. We also discuss the solution of Landau to a problem posed by Legendre, concerning the determination of values of the gamma function using functional identities. In 1848, Oscar Schlomilch gave an interesting additive analogue of the duplication formula. We prove a generalized version of this formula using the theory of hypergeometric functions.

Studies in Applied Mathematics, 2003

The system of evolutionary equations describing the asymptotic behavior of nonlinear waves propag... more The system of evolutionary equations describing the asymptotic behavior of nonlinear waves propagating in materials exhibiting mixed nonlinearity is derived with the resonant wave interactions inherent in the system. Our analysis differs from the results of Hunter et al., in that we have employed a different scaling, keeping in view the delayed effects of nonlinearity in certain thermodynamic systems exhibiting mixed nonlinearity. The result is to modify the transport equations obtained by Hunter et al. by the addition of certain cubic nonlinear terms. Through the method of averaging, the secular terms are eliminated. However, the averaging process is carried out in two steps; first, along manifolds of codimension two giving an advection equation, the solution of which is then averaged in a direction transverse to the above-mentioned manifold.

Studies in Applied Mathematics, 2003

In [1], the authors have demonstrated the effect on the Rankine-Hugoniot conditions for a system ... more In [1], the authors have demonstrated the effect on the Rankine-Hugoniot conditions for a system of conservation laws driven by a singular forcing function and have applied their results to a problem in water waves. We analyze here a similar problem in several space dimensions, in which the singularity in the forcing term involves a simple layer potential supported along the singularity locus. A classical theorem in electrostatics appears as a special case.

arXiv: Classical Analysis and ODEs, 2020

In this note, we look at some of the less explored aspects of the gamma function. We provide a ne... more In this note, we look at some of the less explored aspects of the gamma function. We provide a new proof of Euler's reflection formula and discuss its significance in the theory of special functions. We also discuss a result of Landau concerning the determination of values of the gamma function using functional identities. We show that his result is sharp and extend it to complex arguments. In 1848, Oskar Schlomilch gave an interesting additive analogue of the duplication formula. We prove a generalized version of this formula using the theory of hypergeometric functions.

Journal of Physics A: …, 1995

We construct generalized Painlev e expansions with logarithmic terms for a general class of (`non... more We construct generalized Painlev e expansions with logarithmic terms for a general class of (`nonintegrable') scalar equations, and describe their structure in detail. These expansions were introduced without logarithms by W eiss-Tabor-Carnevale (WTC). The construction of the formal solutions is shown to involve semi-invariants of binary forms, and tools from invariant theory are applied to the determination of the type of logarithmic terms that are required for the most general singular series. The structure of the series depends strongly on whether 1 is or is not a resonance. The convergence of these series is obtained as a consequence of the general results of Littman and the rst author. The results are illustrated on a family of fth-order models for water-waves, and other examples. We also give necessary and sucient conditions for 1 to be a resonance. 1. Introduction.

Missouri Journal of Mathematical Sciences

In this note we provide a geometrical interpretation for the basic assumptions made in the method... more In this note we provide a geometrical interpretation for the basic assumptions made in the method of variation of parameters applied to second order ordinary differential equations. We also discuss a physical motivation drawn from celestial mechanics.

International Journal of Non-Linear Mechanics

Zeitschrift f�r angewandte Mathematik und Physik, 2004

We examine the uniformly imploding pellets or shells, which are driven to implode by the action o... more We examine the uniformly imploding pellets or shells, which are driven to implode by the action of compressional forces. Besides providing the precise time of implosion we discuss continuation of the solution beyond the time of implosion using WTC expansions. Our quantitative analysis confirms the observations of D. L. Book and I. B. Bernstein (1980).

Studies in Applied Mathematics, 2004

... Srinivasan, GK and Sharma, VD (2004), Energy Dissipated Across Shocks in Weak Solutions of Co... more ... Srinivasan, GK and Sharma, VD (2004), Energy Dissipated Across Shocks in Weak Solutions of Conservation Laws. Studies in Applied Mathematics, 112: 281–291. doi: 10.1111/j.0022-2526.2004.01501.x. Author Information. Indian Institute of Technology, Bombay, India. ...

Journal of Mathematical Analysis and Applications, 2003

We obtain the transport equations governing small amplitude high frequency disturbances, that inc... more We obtain the transport equations governing small amplitude high frequency disturbances, that include both quadratic and cubic nonlinearities inherent in hyperbolic systems of conservation laws. The coefficients of the nonlinear terms in the transport equation are obtained in terms of the Glimm interaction coefficients. For symmetric and isotropic systems the mean curvature of the wave front, which appears as the coefficient of the linear term in the transport equation, is shown to be related to the derivative of the ray tube area along the bicharacteristics; the amplitude of the disturbance is shown to become unbounded in the neighborhood of the point where the ray tube collapses. We also obtain a formula, akin to the one obtained by R. Rosales (1991), for the energy dissipated across shocks.

Expositiones Mathematicae, 2013

Abstract—We provide here a complete integrability of the system of six coupled ODEs arising as re... more Abstract—We provide here a complete integrability of the system of six coupled ODEs arising as reductions of the stratified Boussinesq equations by using WTC analysis. In this paper we have shown that among the several possible cases of dominant balance the ...

Canadian Mathematical Bulletin, 2011

Writing s = σ + it for a complex variable, it is proved that the modulus of the gamma function, |... more Writing s = σ + it for a complex variable, it is proved that the modulus of the gamma function, |Γ(s)|, is strictly monotone increasing with respect to σ whenever |t| > 5/4. It is also shown that this result is false for |t| ≤ 1.

Archives of Oral Biology, 1996

Substance P (SP)-, neurokinin A (NKA)-, calcitonin gene-related peptide (CGRP)-and neuropeptide Y... more Substance P (SP)-, neurokinin A (NKA)-, calcitonin gene-related peptide (CGRP)-and neuropeptide Y (NPY)-like immunoreactivities (-LI) were examined in cerebrospinal fluid (CSF), plasma and temporomandibular joint (TMJ) perfusates in rats 1 and 12 h after inoculation at the base of the tail (0.05 ml) or injection into the right TMJ (0.01 ml) of heat-killed Mycobacterium butyricum in paraffin oil. In the rats inoculated at the base of the tail (polyarthritic rats), there was a significant increase of CGRP-LI and NKA-LI. The changes in neuropeptide-LI were not as marked in the CSF of rats injected with adjuvant in one TMJ (monoarthritic rats) as in the polyarthritic group. Instead, the most significant changes in the monoarthritic rats were seen in the perfusates of both TMJs. The increases in SIP-, NKA-, CGRP-and NPY-LI were significant for both TMJs and more pronounced than in the polyarthritic rats. The results show that inoculation of adjuvant at the base of the tail induces significant changes of neuropeptide-LI predominantly in CSF, whilst an intra-articular injection induces bilateral changes in neuropeptide-LI in joint perfusate. Therefore, two different neural mechanisms may be involved early in adjuvant-induced poly-and monoarthritis.

arXiv: History and Overview, 2019

In this expository note we provide a proof of Artin's theorem which states that the commutato... more In this expository note we provide a proof of Artin's theorem which states that the commutator subgroup of a free group on two generators is not finitely generated. The proof employs the infinite grid as in two other proofs in the literature mentioned in the note but takes a somewhat different approach which seems to be of didactic value.

arXiv: General Mathematics, 2018

In this note we provide a direct approach to the most basic operator in this theory namely the ex... more In this note we provide a direct approach to the most basic operator in this theory namely the exterior derivative. The crucial ingredient is a calculus lemma based on determinants. We maintain the view that in a first course at least this direct approach is preferable to the more abstract one based on characterization of the exterior derivative in terms of its properties.

Wolfram Demonstrations Project, 2007

A simple proof of Ramanujan's formula for the Fourier transform of |Γ(a + it)| 2 is given where a... more A simple proof of Ramanujan's formula for the Fourier transform of |Γ(a + it)| 2 is given where a is fixed and has positive real part and t is real. The result is extended to other values of a by solving an inhomogeneous ODE and we use it to calculate the jump across the imaginary axis.

The American Mathematical Monthly, 2007

1The integral on the right converges for complex values of x provided Re x > −1. Euler&amp... more 1The integral on the right converges for complex values of x provided Re x > −1. Euler's notation for it was [ p q ] , where x + 1 was taken to be the rational number p/q [17, vol. 1, p. 221]. ... April 2007] THE GAMMA FUNCTION: AN ECLECTIC TOUR ... This function analytically ...

Springer Proceedings in Mathematics & Statistics, 2021

In this paper we provide a unified approach to a family of integrals of Mellin-Barnes type using ... more In this paper we provide a unified approach to a family of integrals of Mellin-Barnes type using distribution theory and Fourier transforms. Interesting features arise in many of the cases which call for the application of pull-backs of distributions via smooth submersive maps defined by Hörmander. We derive by this method the integrals of Hecke and Sonine relating to various types of Bessel functions which have found applications in analytic and algebraic number theory.

Resonance, 2021

In this article, we look at some of the less explored aspects of the gamma function. We provide a... more In this article, we look at some of the less explored aspects of the gamma function. We provide a new proof of Euler’s reflection formula and discuss its significance in the theory of special functions. We also discuss the solution of Landau to a problem posed by Legendre, concerning the determination of values of the gamma function using functional identities. In 1848, Oscar Schlomilch gave an interesting additive analogue of the duplication formula. We prove a generalized version of this formula using the theory of hypergeometric functions.

Studies in Applied Mathematics, 2003

The system of evolutionary equations describing the asymptotic behavior of nonlinear waves propag... more The system of evolutionary equations describing the asymptotic behavior of nonlinear waves propagating in materials exhibiting mixed nonlinearity is derived with the resonant wave interactions inherent in the system. Our analysis differs from the results of Hunter et al., in that we have employed a different scaling, keeping in view the delayed effects of nonlinearity in certain thermodynamic systems exhibiting mixed nonlinearity. The result is to modify the transport equations obtained by Hunter et al. by the addition of certain cubic nonlinear terms. Through the method of averaging, the secular terms are eliminated. However, the averaging process is carried out in two steps; first, along manifolds of codimension two giving an advection equation, the solution of which is then averaged in a direction transverse to the above-mentioned manifold.

Studies in Applied Mathematics, 2003

In [1], the authors have demonstrated the effect on the Rankine-Hugoniot conditions for a system ... more In [1], the authors have demonstrated the effect on the Rankine-Hugoniot conditions for a system of conservation laws driven by a singular forcing function and have applied their results to a problem in water waves. We analyze here a similar problem in several space dimensions, in which the singularity in the forcing term involves a simple layer potential supported along the singularity locus. A classical theorem in electrostatics appears as a special case.

arXiv: Classical Analysis and ODEs, 2020

In this note, we look at some of the less explored aspects of the gamma function. We provide a ne... more In this note, we look at some of the less explored aspects of the gamma function. We provide a new proof of Euler's reflection formula and discuss its significance in the theory of special functions. We also discuss a result of Landau concerning the determination of values of the gamma function using functional identities. We show that his result is sharp and extend it to complex arguments. In 1848, Oskar Schlomilch gave an interesting additive analogue of the duplication formula. We prove a generalized version of this formula using the theory of hypergeometric functions.

Journal of Physics A: …, 1995

We construct generalized Painlev e expansions with logarithmic terms for a general class of (`non... more We construct generalized Painlev e expansions with logarithmic terms for a general class of (`nonintegrable') scalar equations, and describe their structure in detail. These expansions were introduced without logarithms by W eiss-Tabor-Carnevale (WTC). The construction of the formal solutions is shown to involve semi-invariants of binary forms, and tools from invariant theory are applied to the determination of the type of logarithmic terms that are required for the most general singular series. The structure of the series depends strongly on whether 1 is or is not a resonance. The convergence of these series is obtained as a consequence of the general results of Littman and the rst author. The results are illustrated on a family of fth-order models for water-waves, and other examples. We also give necessary and sucient conditions for 1 to be a resonance. 1. Introduction.

Missouri Journal of Mathematical Sciences

In this note we provide a geometrical interpretation for the basic assumptions made in the method... more In this note we provide a geometrical interpretation for the basic assumptions made in the method of variation of parameters applied to second order ordinary differential equations. We also discuss a physical motivation drawn from celestial mechanics.

International Journal of Non-Linear Mechanics

Zeitschrift f�r angewandte Mathematik und Physik, 2004

We examine the uniformly imploding pellets or shells, which are driven to implode by the action o... more We examine the uniformly imploding pellets or shells, which are driven to implode by the action of compressional forces. Besides providing the precise time of implosion we discuss continuation of the solution beyond the time of implosion using WTC expansions. Our quantitative analysis confirms the observations of D. L. Book and I. B. Bernstein (1980).

Studies in Applied Mathematics, 2004

... Srinivasan, GK and Sharma, VD (2004), Energy Dissipated Across Shocks in Weak Solutions of Co... more ... Srinivasan, GK and Sharma, VD (2004), Energy Dissipated Across Shocks in Weak Solutions of Conservation Laws. Studies in Applied Mathematics, 112: 281–291. doi: 10.1111/j.0022-2526.2004.01501.x. Author Information. Indian Institute of Technology, Bombay, India. ...

Journal of Mathematical Analysis and Applications, 2003

We obtain the transport equations governing small amplitude high frequency disturbances, that inc... more We obtain the transport equations governing small amplitude high frequency disturbances, that include both quadratic and cubic nonlinearities inherent in hyperbolic systems of conservation laws. The coefficients of the nonlinear terms in the transport equation are obtained in terms of the Glimm interaction coefficients. For symmetric and isotropic systems the mean curvature of the wave front, which appears as the coefficient of the linear term in the transport equation, is shown to be related to the derivative of the ray tube area along the bicharacteristics; the amplitude of the disturbance is shown to become unbounded in the neighborhood of the point where the ray tube collapses. We also obtain a formula, akin to the one obtained by R. Rosales (1991), for the energy dissipated across shocks.

Expositiones Mathematicae, 2013

Abstract—We provide here a complete integrability of the system of six coupled ODEs arising as re... more Abstract—We provide here a complete integrability of the system of six coupled ODEs arising as reductions of the stratified Boussinesq equations by using WTC analysis. In this paper we have shown that among the several possible cases of dominant balance the ...

Canadian Mathematical Bulletin, 2011

Writing s = σ + it for a complex variable, it is proved that the modulus of the gamma function, |... more Writing s = σ + it for a complex variable, it is proved that the modulus of the gamma function, |Γ(s)|, is strictly monotone increasing with respect to σ whenever |t| > 5/4. It is also shown that this result is false for |t| ≤ 1.

Archives of Oral Biology, 1996

Substance P (SP)-, neurokinin A (NKA)-, calcitonin gene-related peptide (CGRP)-and neuropeptide Y... more Substance P (SP)-, neurokinin A (NKA)-, calcitonin gene-related peptide (CGRP)-and neuropeptide Y (NPY)-like immunoreactivities (-LI) were examined in cerebrospinal fluid (CSF), plasma and temporomandibular joint (TMJ) perfusates in rats 1 and 12 h after inoculation at the base of the tail (0.05 ml) or injection into the right TMJ (0.01 ml) of heat-killed Mycobacterium butyricum in paraffin oil. In the rats inoculated at the base of the tail (polyarthritic rats), there was a significant increase of CGRP-LI and NKA-LI. The changes in neuropeptide-LI were not as marked in the CSF of rats injected with adjuvant in one TMJ (monoarthritic rats) as in the polyarthritic group. Instead, the most significant changes in the monoarthritic rats were seen in the perfusates of both TMJs. The increases in SIP-, NKA-, CGRP-and NPY-LI were significant for both TMJs and more pronounced than in the polyarthritic rats. The results show that inoculation of adjuvant at the base of the tail induces significant changes of neuropeptide-LI predominantly in CSF, whilst an intra-articular injection induces bilateral changes in neuropeptide-LI in joint perfusate. Therefore, two different neural mechanisms may be involved early in adjuvant-induced poly-and monoarthritis.