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Mechanical decision for a class of integral inequalities

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Abstract

A class of integral inequalities is transformed into homogeneous symmetric polynomial inequalities beyond Tarski model, where the number of elements of the polynomial, say n, is also a variable and the coefficients are functions of n. This is closely associated with some open problems formulated recently by Yang et al. Using Timofte’s dimension-decreasing method for symmetric polynomial inequalities, combined with the inequality-proving package BOTTEMA and a program of implementing the method known as successive difference substitution, we provide a procedure for deciding the nonnegativity of the corresponding polynomial inequality such that the original integral inequality is mechanically decidable; otherwise, a counterexample will be given. The effectiveness of the algorithm is illustrated by some more examples.

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References

  1. Yang L, Xia B C. Automated Proving and Discovering for Inequalities (Series in Mathematical Mechanization)(in Chinese). Beijing: Science Press, 2008

    Google Scholar 

  2. Chow Y S, Teicher H. Probability Theory. New York: Springer, 1978

    MATH  Google Scholar 

  3. Corduneanu C. Integral Equations and Stability of Feedback Systems. London and New York: Springer-Verlag, 1973

    MATH  Google Scholar 

  4. Courant R, Hilbert D. Methods of Mathematical Physics. New York: Interscience Publishing Company, 1953

    Google Scholar 

  5. Liptser R S, Shiryayev A N. Stastistics of Random Processes. New York: Springer, 1977

    Google Scholar 

  6. Wang L, Yu W S, Zhang L. On the number of positive solutions to a class of integral equations. Control Cybern, 2003, 32: 383–395

    MATH  Google Scholar 

  7. Timofte V. On the positivity of symmetric polynomial functions. Part I: General results. J Math Anal Appl, 2003, 284: 174–190

    Article  MATH  MathSciNet  Google Scholar 

  8. Timofte V. On the positivity of symmetric polynomial functions. Part II: Lattice general results and positivity criteria for degrees 4 and 5. J Math Anal Appl, 2005, 304: 652–667

    Article  MATH  MathSciNet  Google Scholar 

  9. Timofte V. On the positivity of symmetric polynomial functions. Part III: Extremal polynomials of degrees 4. J Math Anal Appl, 2005, 307: 565–578

    Article  MATH  MathSciNet  Google Scholar 

  10. Yang L. A dimension-decreasing algorithm with generic program for automated inequality proving (in Chinese). Chin High Tech Lett, 1998, 8: 20–25

    Google Scholar 

  11. Yang L. Recent advances in automated theorem proving on inequalities. J Comput Sci Tech, 1999, 14: 434–446

    Article  MATH  MathSciNet  Google Scholar 

  12. Yang L, Xia S H. Automated proving for a class of constructive geometric inequalities (in Chinese). Chin J Comput, 2003, 26: 769–778

    MathSciNet  Google Scholar 

  13. Yang L. Solving harder problems with lesser mathematics. In: Proceedings of the 10th Asian Technology Conference in Mathematics. ATCM Inc, Blacksburg, 2005. 37–46

    Google Scholar 

  14. Yang L. Difference substitution and automated inequality proving (in Chinese). J Guangzhou Univ (Nat Sci Ed), 2006, 5: 1–7

    Google Scholar 

  15. Yang L. Deciding the nonnegativity of multivariate polynomials without cell-decomposition. In: Preproceedings of the 7th International workshop on Automated Deduction in Geometry. East China Normal University, Shanghai, 2008. 143–153

    Google Scholar 

  16. Yang L, Yao Y. Difference substitution matrices and decision on nonnegativity of polynomials (in Chinese). J Syst Sci Math Sci, 2009, 29: 1169–1177

    MATH  MathSciNet  Google Scholar 

  17. Yao Y. Infinite product convergence of column stochastic mean matrix and machine decision for positive semi-definite forms (in Chinese). Sci Sin Math, 2010, 53: 251–264 (also see http://arxiv.org/abs/0904.4030 for English version)

    Google Scholar 

  18. Rudin W. Principles of Mathematical Analysis. 3rd ed. New York: McGraw-Hill, 1976

    MATH  Google Scholar 

  19. Tarski A. A Decision Method for Elementary Algebra and Geometry. Berkeley: The University of California Press, 1951

    MATH  Google Scholar 

  20. Chou S C. Mechanical Geometry Theorem Proving. Dordrecht: Reidel, 1988

    MATH  Google Scholar 

  21. Wu W T. On the decision problem and the mechanization of theorem-proving in elementary geometry. Sci China, 1978, 21: 150–172

    Google Scholar 

  22. Wu W T. Mechanical Theorem Proving in Geometries: Basic Principle. (Jin X, Wang D, trans.) New York: Springer, 1984

    Google Scholar 

  23. Wu W T. Mathematical Mechanization (Series in Mathematical Mechanization)(in Chinese). Beijing: Science Press, 2003

    Google Scholar 

  24. Buchberger B. Grobner bases: an algorithmic method in polynomial ideal theory. In: Multidimensional Systems Theory. Dordrecht: Reidel, 1985. 184–232

    Google Scholar 

  25. Kapur D. Geometry theorem proving using Hilbert’s Nullstellensata. In: Proc SYMSAC’86. New York: ACM Press, 1986. 202–208

    Google Scholar 

  26. Kutzler B, Stifter S. Automated geometry theorem proving using Buchberger’s algorithm. In: Proc SYMSAC’86. New York: ACM Press, 1986. 209–214

    Google Scholar 

  27. Yang L, Zhang J Z, Hou X R. Nonlinear Algebraic Equation Systems and Automated Theorem Proving (Series in Nonlinear Sciences)(in Chinese). Shanghai: Shanghai Scientific and Technological Education Publishing House, 1996

    Google Scholar 

  28. Zhang J Z, Yang L, Deng M K. The parallel numerical method of mechanical theorem proving. Theo Comput Sci, 1990, 74: 253–271

    Article  MATH  MathSciNet  Google Scholar 

  29. Chou S C, Zhang J Z, Gao X S. Machine Proofs in Geometry: Automated Production of Readable Proofs for Geometry Theorems. Singapore: World Scientific, 1994

    MATH  Google Scholar 

  30. Arnon D S, Collins G E, McCallum S. Cylindrical algebraic decomposition I: the basic algorithm. SIAM J Comput, 1984, 13: 865–877

    Article  MathSciNet  Google Scholar 

  31. Arnon D S, Collins G E, McCallum S. Cylindrical algebraic decomposition II: an adjacency algorithm for the plane. SIAM J Comput, 1984, 13: 878–889

    Article  MathSciNet  Google Scholar 

  32. Brown C W. Simple CAD construction and its applications. J Symb Comput, 2001, 31: 521–547

    Article  MATH  Google Scholar 

  33. Collins G E. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Brakhage H, ed. Automata Theory and Formal Languages, LNCS Vol. 33. Berlin Heidelberg: Springer, 1975. 134–165

    Google Scholar 

  34. Collins G E, Hong H. Partial cylindrical algebraic decomposition for quantifier elimination. J Symb Comput, 1991, 12: 299–328

    Article  MATH  MathSciNet  Google Scholar 

  35. Gerhold S, Kauers M. A procedure for proving special function inequalities involving a discrete parameter. In: Proc ISSAC’05. New York: ACM Press, 2005. 156–162

    Google Scholar 

  36. Kauers M. Computer proofs for polynomial identities in arbitrary many variables. In: Proc ISSAC’04. New York: ACM Press, 2004. 199–204

    Google Scholar 

  37. Yang L, Feng Y, Yao Y. A class of mechanically decidable problems beyond Tarski’s model. Sci China Ser A-Math, 2007, 50: 1611–1620

    Article  MATH  MathSciNet  Google Scholar 

  38. Yang L, Hou X R, Xia B C. A complete algorithm for automated discovering of a class of inequality-type theorems. Sci China Ser F-Inf Sci, 2001, 44: 33–49

    Article  MATH  MathSciNet  Google Scholar 

  39. Yang L, Hou X R, Zeng Z B. A complete discrimination system for polynomials. Sci China Ser E-Tech Sci, 1996, 39: 628–646

    MATH  MathSciNet  Google Scholar 

  40. Wen J J, Cheng S S, Gao C B. Optimal sublinear inequalities involving geometric and power means. Math Bohem, 2009, 134: 133–149

    MathSciNet  Google Scholar 

  41. Yao Y, Xu J. Descartes’ law of signs for generalized polynomials and its application to dimension-decreasing method (in Chinese). Acta Math Sin, 2009, 52: 625–630

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Shanghai Key Laboratory of Trustworthy Computing, Software Engineering Institute, East China Normal University, Shanghai, 200062, China

    Lu Yang & WenSheng Yu

  2. The Key Laboratory of Complex Systems and Intelligence Science, Institute of Automation, Chinese Academy of Sciences, Beijing, 100190, China

    WenSheng Yu & RuYi Yuan

  3. Laboratory for Automated Reasoning and Programming, Chengdu Institute of Computer Applications, Chinese Academy of Sciences, Chengdu, 610041, China

    Lu Yang

Authors
  1. Lu Yang
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  2. WenSheng Yu
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  3. RuYi Yuan
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Correspondence to WenSheng Yu.

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Yang, L., Yu, W. & Yuan, R. Mechanical decision for a class of integral inequalities. Sci. China Inf. Sci. 53, 1800–1815 (2010). https://doi.org/10.1007/s11432-010-4037-2

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