# Professor of Mathematics

## M.Sc. University of Regina, Regina (Canada)

RESEARCH INTERESTS:

• Hind Rattan (translated as Jewel of India) Award, NRIWS of India, India, 2015.
• Marek Kuczma Best Paper Award in Functional Equations, Silesian University, Poland, 2003.
• College of Arts and Sciences Outstanding Service Award, A&S, University of Louisville, 1999-2000.
• Outstanding Scholarship, Research and Creative Activity Award, A&S, University of Louisville, 1997-1998.
• Best Paper Award in Economics, Kentucky Economics Association, 1997
• Young Investigator Award, University of Louisville, 1990
• Dean's Professional Activities Award, 1988-89, 1989-90, and 1990-91
• Fejer Award in Mathematics, University of Waterloo, Canada, 1981-83

RESEARCH WORK CITED IN RECENT BOOKS:
• H. Stetkaer, Functional Equations on Groups. World Scientific, 2013.
• L. Szekelyhidi, Functional Equations on Hypergroups. World Scientific, 2013.
• Y. J. Cho, Th. M. Rassias and R. Saadati, Stability of Functional Equations in Random Normed Spaces. Springer, 2013.
• S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, 2011.
• Pl. Kannappan, Functional Equations and Inequalities with Applications. Springer, 2009.
• M. S. Nixon and A. S. Aguado, Feature Extraction and Image Processing. Academic Press, 2008.
• W. Wu, F. Merchent and K. R. Castleman, Microscope Image Processing. Academic Press, 2008.
• L. Pardo, Statistical Inference Based on Divergence Measures. Chapman & Hall/CRC, 2006.
• T. Acharya and A. K. Roy, Image Processing: Principles and Applications. John Wiley and Sons, 2005.
• R. L. Alllen and D. W. Mills, Signal Analysis Time, Scale and Structures. John Wiley and Sons, 2004.
• S. Czerwik, Functional Equations and Inequalities in several variables. World Scientific, 2002.
• S. M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, 2001.
• E. R. Davies, Image Processing for the Food Industry. World Scientific Press, 2000.
• S. Mori, H. Nishida and H. Yamada, Optical Character Recognition. John Wiley and Sons, 1999.
• S. Ablameyko, An Introduction to Interpretation of Graphic Images. SPIE Press Press, 1997.
• J. C. Ross, Image Processing Handbook, CRC Press, 1995.
• R. Gonzales and R.E. Woods, Digital Image Processing, Academic Press, 1992.
• R. M. Haralick and L. Sapiro, Computer and Robert Vision I, Addison Wesley, 1992.
• G. X. Ritter and J.N. Wilson, Handbook of Computer Vision Algorithms in Image Algebra, CRC Press, 1996.
• H. Niemann, Pattern Analysis and Understanding, Springer-Verlag, 1989.
• J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, 1989.
• E. Castilo and M. R. Ruiz-Cobo, Functional Equations and Modelling in Science and Engineering, MDI, 1992.
• J. R. Parker, Algorithms and Image Processing and Computer vision. John Wiley and Sons Inc. 1997.
• E. R. Davies, Machine Vision: Theory, Algorithms, Practicalities. Academic Press, San Diego, 1997.
• M. R. Baye, Managerial Economics, McGraw Hill, New York 2000.
• R. B. Wilson, Nonlinear Pricing, Oxford University Press, Oxford, 1992.

SELECTED RECENT PUBLICATIONS:
BOOKS AND MONOGRAPHS:

JOURNAL ARTICLES

• Z. Wang and P. K. Sahoo, Stability of an ACQ-functional equation in various matrix normed spaces, J. Nonlinear Sci. Appl. 8 (2015), 64-85.
• H. B. Hunt, T. Riedel and P.K. Sahoo On a functional equation on groups with an involution related to quadratic polynomials in two variables, Aequationes Math. (2015).
• A. M. Perkins and P. K. Sahoo, A functional equation with involution related to cosine function, Aequationes Math. (2015), doi: 10.1007/s00010-015-0363-1
• E. A. Chavez and P.K. Sahoo, On functional equations of multiplicative types, J. of Math. Anal. and Appl. (2015), doi: 10.1016/j.jmaa.2015.05.060
• J. Chung and P. K. Sahoo, Heat kernel method for Levi-Civita functional equation in distributions and hyperfunctions, Bull. Aus. Math. Soc. (2015), doi: 10.1017/s0004972715000349
• B. R. Ebanks, T. Riedel and P.K. Sahoo, On the order of a derivation, Aequationes Math. (2015), doi: 10.1007/s00010-014-0328-9
• Z. Wang and P.K. Sahoo, Nonlinear fuzzy stability of a functional equation related to the characterization of inner product spaces via fixed point technique, Filomat 29:5 (2015), 1067-1080, doi 10.2298/FIL1505067W
• J. Chung and P. K. Sahoo, Stability of Wilson's functional equations with involutions, Aequationes Math. 89 (2015), 749-763.
• H. B. Hunt and P. K. Sahoo, On some functional equations on groups associated with stochastic distance measures, Aequationes Math., Aequationes Math. 89 (2015), 511-528. doi 10.1007/s00010-013-0231-9
• A. M. Perkins and P. K. Sahoo, On two functional equations with involution on groups related to sine and cosine functions, Aequationes Math. (2014), doi: 10.1007/s00010-014-0309-z
• J. Chung and P. K. Sahoo, Wilson's functional equations with involution on semigroups, Abstract and Applied Analyisis, (2014), Article ID 463918, 9 pages, doi: 10.1155/2014/463918
• J. Chung, H. B. Hunt, A. M. Perkins and P. K. Sahoo, Stability of a simple Levi-Civita functional equation on non-unital commutative semigroups, Proc. Indian Acad. Sci. (Math. Sci.) 124 (2014), 365-381.
• J. Chung and P. K. Sahoo, Stability of a logarithmic functional equation in distributions on a restricted domain. Abstract and Applied Analyisis, Volume 2013, ID 751680, 9 pages.
• V. Faiziev, R. C. Powers and P.K. Sahoo, On (psi, gamma)-stability of Jensen equation on step-three nilpotent groups. Georgian Math. Jour., 20 (2013), 657-670.
• V. Faiziev and P.K. Sahoo, Solution of Whitehead equation on groups. Mathematica Bohemica, 138 (2013), 171-180.
• P.K. Sahoo, Solved and Unsolved Problems, Problems 109, 113 and 114, Newsletter of the European Mathematical Society, 87 (2013), 56.
• V.A. Faiziev, R.C. Powers and P.K. Sahoo, Almost all triangles with distinguished acute angle satisfy a stronger triangle inequality, College Math. Jour. 44 (2013), 24-31.
• V.A. Faiziev and P.K. Sahoo, On a problem of Z. Moszner. Jour. Informatics Math. Sci. 4 (2012), 175-177.
• V. Faiziev, R. C. Powers and P.K. Sahoo, An alternative Cauchy functional equation on a semigroup. Aequationes Mathematicae, 85 (2013), 131-163.
• E.A. Chavez and P.K. Sahoo, Jensen and quadratic functional equations on semigroups, Nonlinear Analysis: Stability, Approximation, and Inequalities. (2012), 1-19.
• P.K. Sahoo, Solved and Unsolved Problems, European Mathematical Society Newsletter, 83 (2012), 59-61.
• P.K. Sahoo, On a Sincov type functional equation. Functional Equations in Mathematical Analysis (Th.M. Rassias, Ed.), Springer, 2012, pp 697-708.
• N. El-Zehiry, P.K. Sahoo and A. Elmagraby, A novel graph cut based active contour method for image segmentation. Image and Vision Computing, 29 (2011), 365-381.
• P. K. Sahoo, Quasi-mean value theorems for symmetrically differentiable functions. Tamsui Oxford Journal of Mathematical Sciences, 27 (2011), 279-301.
• E. A. Chavez and P.K. Sahoo, On a functional equation arising from number theory. Applied Mathematics Letters, 24 (2011), 344-347.
• G. H. Kim and P.K. Sahoo, Stability of a functional equation related to distance measures - I. Applied Mathematics Letters, 24 (2011), 843-849.
• G. H. Kim and P. K. Sahoo, Stability of a functional equation related to distance measures - II. Annals of Functional Analysis, 1 (2010), 26--35.
• S. M. Jung, M. S. Moslehian and P. K. Sahoo, Stability of generalized Jensen equation on restricted domains. Journal of Mathematical Inequalities, 4 (2010), 191-206.
• P.K. Sahoo, On the Ulam-Hyers stability of a Sincov type functional equation. Journal of Informatics and Mathematical Sciences, 1 (2) (2009), 81-90.
• P. K. Sahoo, On a functional equation associated with the trapezoidal rule - II. Bulletin of the Institute of Mathematics Academia Sinica (New Series), 4 (1) (2009), 9-23.
• V. Faiziev and P. K. Sahoo, On (psi, gamma )-stability of a Cauchy equation on some noncommutative groups. Publicationes Mathematicae Debreceen, 75 (2009), 67-83.
• P. K. Sahoo, On a functional equation associated with the trapezoidal rule - II. Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 4 (2009), 9-23.
• V. A. Faiziev and P. K. Sahoo, On (psi, gamma)-stability of quadratic equation on groups. Tamsui Oxford Journal of Mathematical Sciences, 24 (2008), 311-337.
• K. B. Houston and P.K. Sahoo, Two functional equations and their solutions. Applied Mathematics Letters, 21 (2008), 974-977.
• V. A. Faiziev and P. K. Sahoo, On the stability of the quadratic equation on groups. Bull. Belgian Math. Soc. Simon Stevin 15 (2008), 135-151.
• M. S. Moslehian, F. Rahbarnia and P. K. Sahoo, Approximate double centralizers are exact double centralizers. Bol. Soc. Mat. Mexicana (3) 13 (2007), 111-122.
• V. A. Faiziev and P. K. Sahoo, On the stability of Drygas functional equation on groups. Banach J. Mathematical Analysis, 1 (2007), 43-55.
• P. K. Sahoo, Some results related to the integral mean value theorem. International Journal of Mathematical Education in Science and Technology, 38 (2007), 818 - 822.
• K. Nikodem, T. Riedel and P. K. Sahoo, The stability problem of the Hermite-Hadamard inequality. Mathematical Inequalities and Applications, 10 (2007), 359-363.
• P. K. Sahoo, On a functional equation associated with the trapezoidal rule. Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 2 (2007), 67-82.
• K. B. Houston and P.K. Sahoo, On a functional equation related to determinant of symmetric two-by-two matrices. Sarajevo J. Mathematics, 3 (2007), 61-71.
• V. Faiziev and P.K. Sahoo, On Drygas functional equation on groups. Internat. J. Applied Mathematics & Statistics, 7 (2007), 59-69.
• V. Faiziev and P.K. Sahoo, Stability of Drygas functional equation on T(3, R). Internat. J. Applied Mathematics & Statistics, 7 (2007), 59-69.
• V. Faiziev and P.K. Sahoo, On the stability of Jensen's functional equation on groups. Proceedings of Indian Math Soc., 117 (2007), 31-48.
• V. Faiziev and P.K. Sahoo, On the space of (psi,gamma)--pseudojensen mappings on groups. Journal of Nonlinear Functional Analysis and Applications, 11 (2006), 759-791.
• S. M. Jung and P.K. Sahoo, Hyers-Ulam-Rassias stability of a functional equation of Davison in rings. Nonlinear Functional Analysis and Applications, 11 (2006), 891-896.
• B. Nahata, K. Ostaszewski and P.K. Sahoo, Rising health care expenditures: A demand side analysis. The ICFAI Journal of Risk & Insurance, 3 (2006), 53-64.
• P.K. Sahoo and G. Arora, Image thresholding using two-dimensional Tsallis-Havrda-Charvat entropy. Pattern Recognition Letters, 27 (2006), 520-528.
• P.K. Sahoo, A generalized cubic functional equation. Acta Mathematica Sinica, English Series, 21 (2005), 1159-1166.
• V. Faiziev and P.K. Sahoo, On stability of Jensen type functional equation on groups. Bulletin of Korean Mathematical Society, 42 (2005), 757-776.
• T. Riedel and P.K. Sahoo, Characterizations of Stolarsky mean. Aequationes Mathematicae, 70 (2005), 51-62.
• V. Faiziev and P.K. Sahoo, Remark on the second bounded cohomology of amalgamated product of groups. Sarajevo J. Math., 1 (2005), 27-48.
• B. Nahata, K. Ostaszewski and P.K. Sahoo, Rising health care expenditures: A demand side analysis. The Journal of Insurance Issues, 28 (2005), 88-102.
• S.M. Jung and P.K. Sahoo, Superstability of the generalized orthogonal equation on restricted domains. Proc. Indian Acad. Sci. (Math. Sci.) 114 (2004), 1-15.
• J.K. Chung, P.K. Sahoo and L. Szekelyhidi, On quadratic differences that depend on the product of arguments--Revisited. Aequationes Mathematicae, 67 (2004), 214-224.
• P.K. Sahoo and G. Arora, A thresholding method based on two-dimensional Renyi's entropy, Pattern Reccogntion, 37 (2004), 1149-1161.
• P.K. Sahoo, On a functional equation characterizing polynomials of degree three. Bull. Inst. Math. Academia Sinica. 32 (2004), 35-44.
• Soon-Mo Jung and P.K. Sahoo, Stability of a functional equation that arises in the theory of conditionally specified distributions. Kyungpook Math. J. 44 (2004), 111-116.
• V. Faiziev and P.K. Sahoo, On the space of pseudojensen functions on groups. St. Peterburg Journal of Mathematics, 14 (2003), 1043-1065.
• P.K. Sahoo and L. Szekelyhidi, On the general solution of a functional equation on Z+Z. Archives of Mathematics, 81 (2003), 233-239.
• M. Das, T. Riedel and P.K. Sahoo, Hyers-Ulam stability of Flett's points. Applied Mathematics Letters, 16 (2003), 269-271.
• PL. Kannappan and P.K. Sahoo, On three functional equations related to the Bose-Einstein entropy. Entropy Measures, Maximum Entropy Principle and Emerging Applications, 119 (2003), 267-285.
• P.K. Sahoo, Hyers-Ulam stability of an Abel functional equation. Stability of Functional Equations of Ulam-Hyers-Rassias Type, (2003), 143-149.
• P.K. Sahoo, On a functional equation connected with a property of complex numbers. Journal of Interdisciplinary Mathematics, 6 (2003), 231-238.
• S.M. Jung and P.K. Sahoo, Stability of a functional equation of Drygas. Aequationes Mathematicae, 64 (2002), 263-273.
• J.K. Chung and P.K. Sahoo, General solution of some functional equations related to the determinant of symmetric matrices. Demonstratio Mathematica, 35 (2002), 539-544.
• S. M. Jung and P. K. Sahoo, On the stability of the Pompeiu functional equation. Mathematical Inequalities and Applications, 5 (2002), 257-261.
• V. Faiziev, Th. M. Rassias and P.K. Sahoo, The space of ($\psi, \gamma$)-additive mappings on semigroups. Transactions of the AMS, 354 (2002), 4455-4472.
• V. Faiziev and P. K. Sahoo, Pseudocharacters and the problem of expressibility for some groups. Journal of Algebra, 250 (2002), 603-637.
• S. M. Jung and P. K. Sahoo, Stability of a functional equation for square root spirals. Applied Mathematics Letters, 15 (2002), 435-438.
• P. K. Sahoo and L. Szekelyhidi, A functional equation on $\mathbb{Z}_n oplus \mathbb{Z}_n$. Acta Mathematica Hungarica, 94 (2002), 93-98.
• R. C. Powers and P. K. Sahoo, Triangle congruence and Moulton plane. Elemente der Mathematik, 56 (2001), 95-101.
• S. M. Jung and P. K. Sahoo, A functional equation characterizing cubic polynomials and its stability, International Journal of Mathematics and Mathematical Sciences, 27 (2001), 301-308.
• P. K. Sahoo and L. Szekelyhidi, On a functional equation related to digital filtering. Aequationes Mathematicae, 62 (2001), 280-285.
• M. Das, T. Riedel and P. K. Sahoo, Flett's mean value theorem for approximately differentiable functions. Radovi Matematicki, 10 (2001), 157-164.
• R. C. Powers, T. Riedel and P. K. Sahoo, Flett's meam value theorem on topological vector spaces, International Journal of Mathematics and Mathematical Sciences, 27 (2001), 689-694.
• S. M. Jung and P. K. Sahoo, Hyers-Ulam stability of the quadratic equation of Pexider type. Journal of Korean Mathematical Society, 38 (2001), 645--656.
• J.K. Chung, S. M. Jung and P. K. Sahoo, On general solution and stability of a functional equation on group. Journal of the Korean Mathematics Society, 38 (2001), 37-47.
• T. Riedel, M. Sablik and P. K. Sahoo, On a functional equation in Actuarial Mathematics. Journal of Mathematical Analysis and Applications, 253 (2001), 16-34.
• S. M. Jung and P. K. Sahoo, On stability of a mean value type functional equation. Demonstratio Mathematica, 33 (2000), 793-796.
• P. K. Sahoo, On generalized Shannon functional inequality and its applications. Survey on Classical Inequalities, 517 (2000), 203-224.
• S. M. Jung and P. K. Sahoo, On the Hyers-Ulam stability of a functional equation of Davison, Kyungpook Mathematical Journal, 40 (2000), 87-92.
• P.K. Sahoo and T. Santiwipanont, Some informational properties of the Dirichlet distribution. Journal of Combinatorics, Information \& Systems Sciences, 24 (1999), 159-168.
• S. M. Jung and P.K. Sahoo, Hyers-Ulam-Rassias stability of an equation of Davison. Journal Mathematical Analysis and Applications, 238 (1999), 297-304.
• R. Davitt, R.C. Powers, T. Riedel and P.K. Sahoo, Flett mean value theorem for holomorphic functions. Mathematics Magazine, 72 (1999), 304-307.
• P.K. Sahoo, On a functional equation characterizing stochastic distance. The Bulletin of Korean Math. Soc., 36 (1999), 287-303.
• B. Nahata, K. Ostaszewski and P.K. Sahoo, Buffet Pricing. Journal of Business, 72 (1999), 215-228.
• R.C. Powers, T. Riedel and P.K. Sahoo, Limit properties of differential mean values. Journal of Mathematical Analysis and Applications, 227 (1998), 216-226.
• E. Deeba, P.K. Sahoo and S. Xie, On a class of functional equations in distributions. Journal of Mathematical Analysis and Applications, 223 (1998), 334-346.
• PL. Kannappan. T. Riedel and P.K. Sahoo, On a generalization of a functional equation associated with Simpson's rule. Prace Matematyczne, 196 (1998), 85-101.
• J.K. Chung, PL. Kannappan and P.K. Sahoo, On second difference in product form. Aequationes Mathematicae, 55 (1998), 1-17.
• E. Deeba, E.L. Koh, P.K. Sahoo and S. Xie, On a distributional analog of a sum form functional equation. Acta Mathematica Hungarica, 78 (1998), 321-332.
• PL. Kannappan and P.K. Sahoo, On generalizations of the Pompeiu functional equation. International Journal of Mathematics and Mathematical Sciences, 21 (1998), 117-124.
• P.K. Sahoo, D.W. Slaaf and T.A. Albert, Threshold selection using a minimal histogram entropy difference. Optical Engineering, 36 (1997), 1976-1981.
• P.K. Sahoo, C. Wilkins and J. Yeager, Threshold selection using Renyi's entropy. Pattern Recognition, 30 (1997), 71-84.
• T. Riedel and P.K. Sahoo, On two functional equations connected with the characterizations of the distance measures. Aequationes Mathematicae, 54 (1997), 242-263.
• T. Riedel and P.K. Sahoo, A functional equation characterizing low degree polynomials in two variables. Demonstratio Mathematica, 30 (1997), 85-94.
• B.R. Ebanks, PL. Kannappan, P.K. Sahoo and W. Sander, Characterization of sum form information measures on open domains. Aequationes Mathematicae, 54 (1997), 1-30.
• J.K. Chung, PL. Kannappan and P.K. Sahoo, On trigonometric functional equations of rectangular type. Aequationes Mathematicae, 53 (1997), 36-53.
• J.K. Chung, B.R. Ebanks, C.T. Ng, P.K. Sahoo and W.B. Zeng, Quadratic differences that depend on the product of arguments. Results in Mathematics, 31 (1997), 53-74.
• PL. Kannappan, T. Riedel and P.K. Sahoo, On a functional equation associated with Simpson's rule. Results in Mathematics, 31 (1997), 115-126.
• R. Powers, T. Riedel and P.K. Sahoo, On a model of plane geometry. Elemente der Mathematik, 51 (1996), 171-175.
• P.K. Sahoo, Three open problems in functional equations. American Mathematical Monthly, 102 (1995), 741-742.
• P.K. Sahoo, Determination of all additive sum form information measures of k positive discrete probability distributions. Journal of Mathematical Analysis and Applications, 194 (1995), 235-249.
• J.K. Chung, B.R. Ebanks, C.T. Ng, P.K. Sahoo and W.B. Zeng, On a generalized rectangular and rhobic functional equations. Publicationes Mathematicae Debrecen, 46 (1995), 249-270.
• T. Riedel and P.K. Sahoo, On a generalization of a functional equation associated with the distance between the probability distributions. Publicationes Mathematicae Debrecen, 46 (1995), 125-135.
• PL. Kannappan, P.K. Sahoo and M.S. Jacobson, A characterization of low degree polynomials. Demonstratio Mathematica, 28 (1995), 87-96.
• PL. Kannappan and P.K. Sahoo, Characterization of polynomials and divided difference. The Proceedings of the Indian Academy of Sciences, 105 (1995), 287-290.
• J.K. Chung, PL. Kannappan and P.K. Sahoo, On trigonometric functional equations of rectangular type. Aquationes Mathematicae, 105 (1995), 287-290.
• J.K. Chung, B.R. Ebanks, C.T. Ng and P.K. Sahoo, On a quadratic-trignometric functional equation and some applications. The Trasactions of the AMS, 347 (1995), 1131-1161.
• J.K. Chung, B.R. Ebanks, C.T. Ng and P.K. Sahoo, On a functional equation connected with Rao's quadratic entropy. The Proceedings of the AMS, 120 (1994), 843-848.
• J.K. Chung, B.R. Ebanks, C.T. Ng, P.K. Sahoo and W.B. Zeng, On a functional equation of Abel. Results in Mathematics, 26 (1994), 241-252.
• R.C. Powers, T. Riedel and P.K. Sahoo, Some models of geometries and a functional equation. Colloquium Mathematicum. 44 (1993), 165-173.
• PL. Kannappan and P.K. Sahoo, Sum form equation of multiplicative type. Acta Mathematica Hungarica, 61 (1993), 205-219.
• J.K. Chung, B.R. Ebanks and P.K. Sahoo, On a functional equation of Swiatak on groups. Aequationes Mathematicae, 45 (1993), 246-266.
• PL. Kannappan and P.K. Sahoo, Rotation invariant separable functions are Gaussian. SIAM Journal of Mathematical Analysis, 23 (1992), 1341-1351.
• B. Nahata, K. Ostaszewski, and P.K. Sahoo, Direction of price changes in third-degree price discrimination. American Economics Review, 80 (1990), 1254-1258.

FOR A COMPLETE LIST OF MY PAPERS Reviewed in MathSciNet CLICK HERE .

THESES SUPERVISED :
• Glenna Grider, Image Compression using iterated functions system, MA, 1992.
• Letao Wang, D'Alembert type functional equations on groups, MA, 1993.
• Carrye Wilknis, Threshold selection with Renyi's entropy, MA, 1994.
• David Eichenberger, On a functional equation characterizing polynomials, MA, 1996.
• David Zimmermann, Some algebraic and analytic properties of quarternions, MA, 1999.
• Tippawan Santiwipanont, On beta distributions, MA, 1999.
• Joseph Twagilimana, Mean-variance model in portfolio analysis, MA, 2002.
• Joyee Dutt, Comparative study of buffet pricing and two-part tariff, MA, 2002.
• Kelly Houston, Characterizations of determinant and permanent of symmetric matrices, BSc, 2007.
• Noha El-Zehiry, Graph cut optimization for the bimodal piecewise smooth Mumford-Shah functional, MA, 2007.
• Sara Evans, Bayesian Regression Analysis, MA, 2012.
• Heather Hunt, Several functional equations defined on groups arising from stochastic distance measures, Ph.D, Summer 2014.
• Allison Perkins, Functional equations with involution related to sine and cosine functions, Ph.D, Summer 2014.