无穷分析的珍贵文献（542篇）

无穷分析的珍贵文献（542篇）

  上世纪下半叶，无穷小分析（非标准分析）得以快速发展。

  在这一期间，在国际一流学术期刊上，相关研究论文出现“井喷”。

  实际情况是，有数百家大学及其学者参与其中。请见本文附件。

  反观我们国内，相关研究几乎完全是空白，… …其余的话就不用多说了。

袁萌 陈启清  1月18日

附件：

References

1. Akilov G. P. and Kutateladze S. S., Ordered Vector Spaces [in Russian], Nauka, Novosibirsk (1978).

2. AksoyA.G.andKhamsi M.A., NonstandardMethodsinFixedPointTheory, Springer-Verlag, Berlin etc. (1990). 3. Albeverio S., Fenstad J. E., et al., Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, Orlando etc. (1990).

4. Albeverio S., Luxemburg W. A. J., and Wolff M. P. H. (eds.), Advances in Analysis, Probability and Mathematical Physics: Contributions of Nonstandard Analysis, Kluwer Academic Publishers, Dordrecht (1995).

5. Albeverio S., Gordon E. I., and Khrennikov A. Yu., “Finite dimensional approximations of operators in the Hilbert spaces of functions on locally compact abelian groups,” Acta Appl. Math., 64, No. 1, 33–73 (2000).

6. Alekseev M. A., Glebski˘ı L. Yu., and Gordon E. I., On Approximations of Groups, Group Actions, and Hopf Algebras [Preprint, No. 491], Inst. Applied Fiz. Ros. Akad. Nauk, Nizhni˘ı Novgorod (1999).

7. Alexandrov A. D., Problems of Science and the Researcher’s Standpoint [in Russian], Nauka (Leningrad Branch), Leningrad (1988).

8. Alexandrov A. D., “A general view of mathematics,” in: Mathematics: Its Content, Methods and Meaning. Ed. by A. D. Alexandrov, A. N. Kolmogorov, and M. A. Lavrentev. Dover Publications, Mineola and New York (1999). (Reprint of the 2nd 1969 ed.)

9. Aliprantis C. D. and Burkinshaw O., Locally Solid Riesz Spaces, Academic Press, New York etc. (1978).

10. Aliprantis C. D. and Burkinshaw O., Positive Operators, Academic Press, New York (1985). 11. Anderson R. M., “A nonstandard representation of Brownian motion and Itˆ o integration,” Israel J. Math. Soc., 25, 15–46 (1976).

12. AndersonR.M., “Star-finiterepresentationsofmeasurespaces,” Trans.Amer. Math. Soc., 271, 667–687 (1982).

386 References

13. Andreev P. V., “On the standardization principle in the theory of bounded sets,” Vestnik Moskov. Univ. Ser. I Mat. Mekh., No. 1, 68–70 (1997).

14. Andreev P. V. and Gordon E. I., “The nonstandard theory of classes,” Vladikavkaz. Mat. Zh., 1, No. 4, 2–16 (1999). (http://alanianet.ru/omj/journal.htm) 15. Anselone P. M., Collectively Compact Operator Approximation. Theory and Applications to Integral Equations, Englewood Cliffs, Prentice Hall (1971).

16. Archimedes, Opera Omnia, Teubner, Stuttgart (1972).

17. Arens R. F. and Kaplansky I., “Topological representation of algebras,” Trans. Amer. Math. Soc., 63, No. 3, 457–481 (1948).

18. Arkeryd L. and Bergh J., “Some properties of Loeb–Sobolev spaces,” J. London Math. Soc., 34, No. 2, 317–334 (1986). 19. Arkeryd L. O., Cutland N. J., and Henson W. (eds.), Nonstandard Analysis. Theory and Applications (Proc. NATO Advanced Study Institute on Nonstandard Anal. and Its Appl., International Centre for Mathematical Studies, Edinburgh, Scotland, 30 June–13 July), Kluwer Academic Publishers, Dordrecht etc. (1997). 20. Arveson W., “Operator algebras and invariant subspaces,” Ann. of Math., 100, No. 2, 433–532 (1974). 21. Arveson W., An Invitation to C∗-Algebras, Springer-Verlag, Berlin etc. (1976). 22. Attouch H., Variational Convergence for Functions and Operators, Pitman, Boston etc. (1984). 23. Aubin J.-P. and Ekeland I., Applied Nonlinear Analysis, Wiley-Interscience, New York (1979). 24. AubinJ.-P.andFrankowskaH., Set-ValuedAnalysis, Birkh¨auser-Verlag, Boston (1990). 25. Auslander L. and Tolimieri R., “Is computing with finite Fourier transform pure or applied mathematics?” Bull. Amer. Math. Soc., 1, No. 6, 847–897 (1979). 26. Bagarello F., “Nonstandard variational calculus with applications to classical mechanics. I: An existence criterion,” Internat. J. Theoret. Phys., 38, No. 5, 1569–1592 (1999).

27. Bagarello F., “Nonstandard variational calculus with applications to classical mechanics. II: The inverse problem and more,” Internat. J. Theoret. Phys., 38, No. 5, 1593–1615 (1999). 28. Ballard D., Foundational Aspects of “Non”-Standard Mathematics, Amer. Math. Soc., Providence, RI (1994). 29. Bell J. L., Boolean-Valued Models and Independence Proofs in Set Theory, Clarendon Press, New York etc. (1985).

References 387

30. Bell J. L., A Primer of Infinitesimal Analysis, Cambridge University Press, Cambridge (1998). 31. Bell J. L. and Slomson A. B., Models and Ultraproducts: an Introduction, North-Holland, Amsterdam etc. (1969). 32. Berberian S. K., Baer ∗-Rings, Springer-Verlag, Berlin (1972). 33. BerezinF.A.andShubin M.A., TheSchr¨odingerEquation,KluwerAcademic Publishers, Dordrecht (1991). 34. Berkeley G., The Works. Vol. 1–4, Thoemmes Press, Bristol (1994). 35. Bigard A., Keimel K., and Wolfenstein S., Groupes et Anneaux R´eticul´es, Springer-Verlag, Berlin etc. (1977). (Lecture Notes in Math., 608.) 36. Birkhoff G., Lattice Theory, Amer. Math. Soc., Providence (1967). 37. Bishop E. and Bridges D., Constructive Analysis, Springer-Verlag, Berlin etc. (1985). 38. Blekhman I. I., Myshkis A. D., and Panovko A. G., Mechanics and Applied Mathematics. Logics and Peculiarities of Mathematical Applications [in Russian], Nauka, Moscow (1983). 39. Blumenthal L.M., Theory andApplicationsofDistanceGeometry, Clarendon Press, Oxford (1953). 40. Bogolyubov A. N., “Read on Euler: he is a teacher for all of us,” Nauka v SSSR, No. 6, 98–104 (1984). 41. Boole G., An Investigation of the Laws of Thought on Which Are Founded the Mathematical Theories of Logic and Probabilities, Dover Publications, New York (1957). 42. Boole G., Selected Manuscripts on Logic and Its Philosophy, Birkh¨auserVerlag, Basel (1997). (Science Networks. Historical Studies, 20.) 43. Borel E., Probability et Certitude, Presses Univ. de France, Paris (1956). 44. Bourbaki N., Set Theory [in French], Hermann, Paris (1958). 45. Boyer C. B., A History of Mathematics, John Wiley & Sons Inc., New York etc. (1968). 46. Bratteli O. and Robinson D., Operator Algebras and Quantum Statistical Mechanics, Springer-Verlag, New York etc. (1982). 47. Bukhvalov A. V., “Order bounded operators in vector lattices and in spaces of measurable functions,” J. Soviet Math., 54, No. 5, 1131–1176 (1991). 48. Bukhvalov A. V., Veksler A. I., and Ge˘ıler V. A., “Normed lattices,” J. Soviet Math., 18, 516–551 (1982). 49. Bukhvalov A. V., Veksler A. I., and Lozanovski˘ı G. Ya., “Banach lattices: some Banach aspects of their theory,” Russian Math. Surveys, 34, No.2, 159–212 (1979). 50. Burden C. W. and Mulvey C. J., “Banach spaces in categories of sheaves,” in: Applications of Sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to

388 References

Logic, Algebra and Anal., Durham, 1977), Springer-Verlag, Berlin etc., 1979, pp. 169–196. 51. Canjar R. M., “Complete Boolean ultraproducts,” J. Symbolic Logic, 52, No. 2, 530–542 (1987). 52. Cantor G., Works on Set Theory [in Russian], Nauka, Moscow (1985). 53. Capinski M. and Cutland N. J., Nonstandard Methods for Stochastic Fluid Mechanics, World Scientific Publishers, Singapore etc. (1995). (Series on Advances in Mathematics for Applied Sciences, Vol. 27.) 54. Carnot L., Reflections on Metaphysics of Infinitesimal Calculus [Russian translation], ONTI, Moscow (1933). 55. Castaing C. and Valadier M., Convex Analysis and Measurable Multifunctions, Springer-Verlag, Berlin etc. (1977). (Lecture Notes in Math., 580.) 56. Chang C. C. and Keisler H. J., Model Theory, North-Holland, Amsterdam (1990). 57. Chilin V. I., “Partially ordered involutive Baer algebras,” in: Contemporary Problems of Mathematics. Newest Advances [in Russian], VINITI, Moscow, 27, 1985, pp. 99–128. 58. Church A., Introduction to Mathematical Logic, Princeton University Press, Princeton (1956). 59. Ciesielski K., Set Theory for the Working Mathematician, Cambridge University Press, Cambridge (1997). 60. Clarke F. H., “Generalized gradients and applications,” Trans. Amer. Math. Soc., 205, No. 2, 247–262 (1975). 61. ClarkeF.H.,OptimizationandNonsmoothAnalysis,NewYork,Wiley(1983). 62. Cohen P.J., SetTheory andthe Continuum Hypothesis, Benjamin, NewYork etc. (1966). 63. Cohen P. J., “On foundations of set theory,” Uspekhi Mat. Nauk, 29, No. 5, 169–176 (1974). 64. Courant R., A Course of Differential and Integral Calculus. Vol. 1 [Russian translation], Nauka, Moscow (1967). 65. CourantR.andRobbinsH.,WhatIsMathematics? AnElementaryApproach to Ideas and Methods. Oxford University Press, Oxford etc. (1978). 66. Cozart D. and Moore L. C. Jr., “The nonstandard hull of a normed Riesz space,” Duke Math. J., 41, 263–275 (1974). 67. Cristiant C., “Der Beitrag G¨odels f¨ur die Rechfertigung der Leibnizschen Idee von der Infinitesimalen,” ¨Osterreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, Bd 192, No. 1–3, 25–43 (1983). 68. Curtis T., Nonstandard Methods in the Calculus of Variations, Pitman, London (1993). (Pitman Research Notes in Mathematics, 297.) 69. Cutland N. J., “Nonstandard measure theory and its applications,” Bull. London Math. Soc., 15, No. 6, 530–589 (1983).

References 389

70. Cutland N. J., “Infinitesimal methods in control theory, deterministic and stochastic,” Acta Appl. Math., 5, No. 2, 105–137 (1986). 71. Cutland N. J. (ed.), Nonstandard Analysis and Its Applications, Cambridge University Press, Cambridge (1988). 72. Dacunha-Castelle D. and Krivine J.-L., “Applications des ultraproduits a l’etude des espaces et des algebres de Banach,” Studia Math., 41, 315–334 (1972). 73. Dales H. and Woodin W., An Introduction to Independence for Analysts, Cambridge University Press, Cambridge (1987). 74. Dauben J. W., Abraham Robinson. The Creation of Nonstandard Analysis, a Personal and Mathematical Odyssey, Princeton University Press, Princeton (1995). 75. Davis M., Applied Nonstandard Analysis, John Wiley & Sons, New York etc. (1977). 76. Day M., Normed Linear Spaces, Springer-Verlag, New York and Heidelberg (1973). 77. de Jonge E. and van Rooij A. C. M., Introduction to Riesz Spaces, Mathematisch Centrum, Amsterdam (1977). 78. Dellacherie C., Capacities and Stochastic Processes [in French], SpringerVerlag, Berlin etc. (1972). 79. Demyanov V. F. and Vasilev L. V., Nondifferentiable Optimization [in Russian], Nauka, Moscow (1981). 80. Demyanov V. F. and Rubinov A. M., Constructive Nonsmooth Analysis, Verlag Peter Lang, Frankfurt/Main (1995). 81. Diener F. and Diener M., “Les applications de l’analyse non standard,” Recherche, 20, No. 1, 68–83 (1989). 82. Diener F. and Diener M. (eds.), Nonstandard Analysis in Practice, SpringerVerlag, Berlin etc. (1995). 83. Diestel J., Geometry of Banach Spaces: Selected Topics, Springer-Verlag, Berlin etc. (1975). 84. Diestel J. and Uhl J. J., Vector Measures, Amer. Math. Soc., Providence, RI (1977). (Math. Surveys; 15.) 85. Digernes T., Husstad E., andVaradarajanV., “FiniteapproximationsofWeyl systems,” Math. Scand., 84, 261–283 (1999). 86. Digernes T., Varadarajan V., and Varadhan S., “Finite approximations to quantum systems,” Rev. Math. Phys., 6, No. 4, 621–648 (1994). 87. Dinculeanu N., Vector Measures, VEB Deutscher Verlag der Wissenschaften, Berlin (1966). 88. Dixmier J., C∗-Algebras and Their Representations [in French], GauthierVillars, Paris (1964).

390 References

89. Dixmier J., C∗-Algebras, North-Holland, Amsterdam, New York, and Oxford (1977). 90. Dixmier J., Les Algebres d’Operateurs dans l’Espace Hilbertien (Algebres de von Neumann), Gauthier-Villars, Paris (1996). 91. Dolecki S., “A general theory of necessary optimality conditions,” J. Math. Anal. Appl., 78, No. 12, 267–308 (1980). 92. DoleckiS., “Tangencyanddifferentiation: marginalfunctions,” Adv. inAppl. Math., 11, 388–411 (1990). 93. Dragalin A. G., “An explicit Boolean-valued model for nonstandard arithmetic,” Publ. Math. Debrecen, 42, No. 3–4, 369–389 (1993). 94. Dunford N. and Schwartz J. T., Linear Operators. Vol. 1: General Theory, John Wiley & Sons Inc., New York (1988). 95. Dunford N. and Schwartz J. T., Linear Operators. Vol. 2: Spectral Theory. Selfadjoint Operators in Hilbert Space, John Wiley & Sons Inc., New York (1988). 96. Dunford N. andSchwartz J. T., LinearOperators. Vol.3: Spectral Operators, John Wiley & Sons Inc., New York (1988). 97. Dunham W., Euler: The Master of Us All, The Mathematical Association of America, Washington (1999). (The Dolciani Mathematical Expositions, 22.) 98. Eda K., “A Boolean power and a direct product of abelian groups,” Tsukuba J. Math., 6, No. 2, 187–194 (1982). 99. Eda K., “On a Boolean power of a torsion free abelian group,” J. Algebra, 82, No. 1, 84–93 (1983). 100. Ekeland I. and TemamR., Convex Analysisand VariationalProblems, NorthHolland, Amsterdam (1976). 101. Eklof P., “Theory of ultraproducts for algebraists,” in: Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977. 102. Ellis D., “Geometry in abstract distance spaces,” Publ. Math. Debrecen, 2, 1–25 (1951). 103. Emelyanov `E. Yu., “Invariant homomorphisms of nonstandard enlargements of Boolean algebras and vector lattices,” Siberian Math. J., 38, No. 2, 244– 252 (1997). 104. EngelerE.,MetamathematicsofElementaryMathematics[inGerman], Springer-Verlag, Berlin etc. (1983). 105. Ershov Yu. L. and Palyutin E. A., Mathematical Logic [in Russian], Nauka, Moscow (1987). 106. Esenin-Volpin A. S., “Analysis of potential implementability,” in: Logic Studies [in Russian], Izdat. Akad. Nauk SSSR, Moscow, 1959, pp. 218–262. 107. Espanol L., “Dimension of Boolean valued lattices and rings,” J. Pure Appl. Algebra, No. 42, 223–236 (1986).

References 391

108. Euclid’s “Elements.” Books 7–10 [Russian translation], Gostekhizdat, Moscow and Leningrad (1949). 109. Euler L., Introduction to Analysis of the Infinite. Book I [Russian translation], ONTI, Moscow (1936); [English translation], Springer-Verlag, New York etc. (1988). 110. Euler L., Differential Calculus [Russian translation], Gostekhizdat, Leningrad (1949). 111. Euler L., Integral Calculus. Vol. 1 [Russian translation], Gostekhizdat, Moscow (1950). 112. Euler L., Opera Omnia. Series Prima: Opera Mathematica. Vol. 1–29, Birkh¨auser-Verlag, Basel etc. 113. Euler L., Foundations of Differential Calculus, Springer-Verlag, New York (2000). 114. Faith C., Algebra: Rings, Modules, and Categories. Vol. 1, Springer-Verlag, Berlin etc. (1981). 115. Fakhoury H., “Repr´esentations d’op´erateurs `a valeurs dans L1(X,Σ,μ),” Math. Ann., 240, No. 3, 203–212 (1979). 116. Farkas E. and Szabo M., “On the plausibility of nonstandard proofs in analysis,” Dialectica, 38, No. 4, 297–310 (1974). 117. Fattorini H. O., The Cauchy Problem, Addison-Wesley (1983). 118. Foster A. L., “Generalized ‘Boolean’ theory of universal algebras. I. Subdirect sums and normal representation theorems,” Math. Z., 58, No. 3, 306–336 (1953). 119. FosterA.L., “Generalized‘Boolean’theory ofuniversal algebras.II.Identities and subdirect sums of functionally complete algebras,” Math. Z., 59, No. 2, 191–199 (1953). 120. Fourman M. P., “The logic of toposes,” in: Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977. 121. Fourman M. P. and Scott D. S., “Sheaves and logic,” in: Applications of Sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Durham, 1977), Springer-Verlag, Berlin etc., 1979, pp. 302–401. 122. Fourman M. P., Mulvey C. J., and Scott D. S. (eds.), Applications of Sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Durham, 1977), Springer-Verlag, Berlin etc., 1979. 123. Fraenkel A. and Bar-Hillel I., Foundations of Set Theory, North-Holland, Amsterdam (1958). 124. Fuchs L., Partially Ordered Algebraic Systems, Pergamon Press, Oxford (1963). 125. Fuller R. V., “Relations among continuous and various noncontinuous functions,” Pacific J. Math., 25, No. 3, 495–509 (1968).

392 References

126. Gandy R. O., “Limitations to mathematical knowledge,” in: Logic Colloquium-80, North-Holland, New York and London, 1982, pp. 129–146. 127. Georgescu G. and Voiculescu I., “Eastern model theory for Boolean-valued theories,” Z. Math. Logik Grundlag. Math., No. 31, 79–88 (1985). 128. GlazmanI.M.andLyubich Yu.I., Finite-DimensionalLinearAnalysis, M.I.T. Press, Cambridge (1974). 129. G¨odel K., “What is Cantor’s continuum problem?” Amer. Math. Monthly, 54, No. 9, 515–525 (1947). 130. G¨odel K., “Compatibility of the axiom of choice and the generalized continuum hypothesis with the axioms of set theory,” Uspekhi Mat. Nauk, 8, No. 1, 96–149 (1948). 131. Goldshte˘ın V. M., Kuzminov V. I., and Shvedov I. A., “The K¨unneth formula for Lp-cohomologies of warped products,” Sibirsk. Mat. Zh., 32, No. 5, 29–42 (1991). 132. Goldshte˘ın V. M., Kuzminov V. I., and Shvedov I. A., “On approximation of exact and closed differential forms by compactly-supported forms,” Sibirsk. Mat. Zh., 33, No. 2, 49–65 (1992). 133. Goldblatt R., Topoi: Categorical Analysis of Logic, North-Holland, Amsterdam etc. (1979). 134. Goldblatt R., Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, Springer-Verlag, New York etc. (1998). 135. Goodearl K. R., Von Neumann Regular Rings, Pitman, London (1979). 136. Gordon E. I., “Real numbers in Boolean-valued models of set theory and K-spaces,” Dokl. Akad. Nauk SSSR, 237, No. 4, 773–775 (1977). 137. GordonE.I., “K-spaces inBoolean-valuedmodelsofsettheory,”Dokl. Akad. Nauk SSSR, 258, No. 4, 777–780 (1981). 138. Gordon E. I., “To the theorems of identity preservation in K-spaces,” Sibirsk. Mat. Zh., 23, No. 3, 55–65 (1982). 139. Gordon E. I., RationallyCompleteSemiprime CommutativeRingsin Boolean Valued Models of Set Theory, Gorki˘ı, VINITI, No. 3286-83 Dep (1983). 140. Gordon E. I., “Nonstandard finite-dimensional analogs of operators in L2(Rn),” Sibirsk. Mat. Zh., 29, No. 2, 45–59 (1988). 141. Gordon E. I., “Relatively standard elements in E. Nelson’s internal set theory,” Sibirsk. Mat. Zh., 30, No. 1, 89–95 (1989). 142. Gordon E. I., “Hyperfinite approximations of locally compact Abelian groups,” Dokl. Akad. Nauk SSSR, 314, No. 5, 1044–1047 (1990). 143. Gordon E. I., Elements of Boolean Valued Analysis [in Russian], Gorki˘ı State University Press, Gorki˘ı (1991). 144. Gordon E. I., “Nonstandard analysis and compact Abelian groups,” Sibirsk. Mat. Zh., 32, No. 2, 26–40 (1991).

References 393

145. Gordon E. I., “On Loeb measures,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 2, 25–33 (1991). 146. Gordon E. I., Nonstandard Methods in Commutative Harmonic Analysis, Amer. Math. Soc., Providence, RI (1997). 147. Gordon E. I. and Lyubetski˘ı V. A., “Some applications of nonstandard analysis in the theory of Boolean valued measures,” Dokl. Akad. Nauk SSSR, 256, No. 5, 1037–1041 (1981). 148. Gordon E. I. and Morozov S. F., Boolean Valued Models of Set Theory [in Russian], Gorki˘ı State University, Gorki˘ı (1982). 149. Gra¨tzer G., General Lattice Theory, Birkh¨auser-Verlag, Basel (1978). 150. Grayson R. J., “Heyting-valued models for intuitionistic set theory,” Applications of Sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Durham, 1977), Springer-Verlag, Berlin etc., 1979, pp. 402–414. 151. Gurari˘ı V. P., Group Methods in Commutative Harmonic Analysis [in Russian], VINITI, Moscow (1988). 152. Gutman A. E., “Banach fibering in lattice normed space theory,” in: Linear Operators Compatible with Order [in Russian], Sobolev Institute Press, Novosibirsk, 1995, pp. 63–211. 153. GutmanA.E., “Locallyone-dimensional K-spacesand σ-distributiveBoolean algebras,” Siberian Adv. Math., 5, No. 2, 99–121 (1995). 154. Gutman A. E., Emelyanov `E. Yu., Kusraev A. G., and Kutateladze S. S., Nonstandard Analysis and Vector Lattices, Kluwer Academic Publishers, Dordrecht (2000). 155. Hallet M., Cantorian Set Theory and Limitation of Size, Clarendon Press, Oxford (1984). 156. Halmos P. R., Lectures on Boolean Algebras, Van Nostrand, Toronto, New York, and London (1963). 157. Hanshe-Olsen H. and St¨ormer E., Jordan Operator Algebras, Pitman Publishers Inc., Boston etc. (1984). 158. Harnik V., “Infinitesimals from Leibniz to Robinson time—to bring them back to school,” Math. Intelligencer, 8, No. 2, 41–47 (1986). 159. HatcherW., “Calculusisalgebra,” Amer. Math. Monthly, 89, No. 6, 362–370 (1989). 160. Heinrich S., “Ultraproducts of L1-predual spaces,” Fund. Math., 113, No. 3, 221–234 (1981). 161. Helgason S., The Radon Transform, Birkh¨auser-Verlag, Boston (1999). 162. Henle J. M. and Kleinberg E. M., Infinitesimal Calculus, Alpine Press, Cambridge and London (1979). 163. Henson C. W., “On the nonstandard representation of measures,” Trans. Amer. Math. Soc., 172, No. 2, 437–446 (1972).

394 References

164. Henson C. W., “When do two Banach spaces have isometrically isomorphic nonstandard hulls?” Israel J. Math., 22, 57–67 (1975). 165. Henson C. W., “Nonstandard hulls of Banach spaces,” Israel J. Math., 25, 108–114 (1976). 166. Henson C. W., “Unbounded Loeb measures,” Proc. Amer. Math. Soc., 74, No. 1, 143–150 (1979). 167. Henson C. W., “Infinitesimals in functional analysis,” in: Nonstandard Analysis and Its Applications, Cambridge University Press, Cambridge etc., 1988, pp. 140–181. 168. Henson C. W. and Keisler H. J., “On the strength of nonstandard analysis,” J. Symbolic Logic, 51, No. 2, 377–386 (1986). 169. Henson C. W. andMoore L. C.Jr. “Nonstandard hulls oftheclassical Banach spaces,” Duke Math. J., 41, No. 2, 277–284 (1974). 170. Henson C. W. and Moore L. C. Jr., “Nonstandard analysis and the theory of Banach spaces,” in: Nonstandard Analysis. Recent Developments, SpringerVerlag, Berlin etc., 1983, pp. 27–112. (Lecture Notes in Math., 983.) 171. HensonC.W., KaufmannM.,andKeislerH.J.,“Thestrengthofnonstandard methods in arithmetic,” J. Symbolic Logic, 49, No. 34, 1039–1057 (1984). 172. Hermann R., “Supernear functions,” Math. Japon., 31, No. 2, 320 (1986). 173. HernandezE.G.,“Boolean-valuedmodelsofsettheorywithautomorphisms,” Z. Math. Logik Grundlag. Math., 32, No. 2, 117–130 (1986). 174. Hewitt E. and Ross K., Abstract Harmonic Analysis. Vol. 1 and 2, Springer -Verlag, Berlin etc. (1994). 175. Hilbert D. and Bernays P., Foundations of Mathematics. Logical Calculi and the Formalization of Arithmetic [in Russian], Nauka, Moscow (1979). 176. Hilbert’s Problems [in Russian], Nauka, Moscow (1969). 177. Hiriart-UrrutyJ.-B., “Tangentcones, generalizedgradientsandmathematical programming in Banach spaces,” Math. Oper. Res., 4, No. 1, 79–97 (1979). 178. Hobbes T., Selected Works. Vol. 1 [Russian translation], Mysl, Moscow (1965). 179. Hoehle U., “Almost everywhere convergence and Boolean-valued topologies,” in: Topology, Proc. 5th Int. Meet., Lecce/Italy 1990, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 29, 1992, pp. 215–227. 180. Hofmann K. H. and Keimel K., “Sheaf theoretical concepts in analysis: Bundles and sheaves of Banach spaces, Banach C(X)-modules,” in: Applications of Sheaves, Proc. Res. Sympos., Durham, 1977, Springer-Verlag, Berlin etc., 1979, pp. 415–441. 181. Hofstedter D. R., G¨odel, Escher, Bach: an Eternal Golden Braid, Vintage Books, New York (1980). 182. Hogbe-Nlend H., Theorie des Bornologie et Applications, Springer-Verlag, Berlin etc. (1971).

References 395

183. Horiguchi H., “A definition of the category of Boolean-valued models,” Comment. Math. Univ. St. Paul., 30, No. 2, 135–147 (1981). 184. Horiguchi H., “The category of Boolean-valued models and its applications,” Comment. Math. Univ. St. Paul., 34, No. 1, 71–89 (1985). 185. Hrb´ aˇcek K., “Axiomaticfoundations fornonstandardanalysis,” Fund. Math., 98, No. 1, 1–24 (1978). 186. Hrb´ aˇcek K., “Nonstandard set theory,” Amer. Math. Monthly, 86, No. 8, 659–677 (1979). 187. Hurd A. E. (ed.), Nonstandard Analysis. Recent Developments, SpringerVerlag, Berlin (1983). 188. HurdA.E.andLoebH.,AnIntroductiontoNonstandardAnalysis,Academic Press, Orlando etc. (1985). 189. Ilin V. A., Sadovnichi˘ı V. A., and Sendov Bl. Kh., Mathematical Analysis [in Russian], Nauka, Moscow (1979). 190. Ionescu Tulcea A. and Ionescu Tulcea C., Topics in the Theory of Lifting, Springer-Verlag, Berlin etc. (1969). 191. Ivanov V. V., “Geometric properties of monotone functions and probabilities of random fluctuations,” Siberian Math. J., 37, No. 1, 102–129 (1996). 192. Ivanov V. V., “Oscillations of means in the ergodic theorem,” Dokl. Akad. Nauk, 347, No. 6, 736–738 (1996). 193. Jaffe A., “Ordering the universe: the role of mathematics,” SIAM Rev., 26, No. 4, 473–500 (1984). 194. Jarnik V., Bolzano and the Foundations of Mathematical Analysis, Society of Czechosl. Math. Phys., Prague (1981). 195. Jech T. J., Lectures in Set Theory with Particular Emphasis on the Method of Forcing, Springer-Verlag, Berlin (1971). 196. Jech T. J., The Axiom of Choice, North-Holland, Amsterdam etc. (1973). 197. Jech T. J., “Abstract theory of abelian operator algebras: an application of forcing,” Trans. Amer. Math. Soc., 289, No. 1, 133–162 (1985). 198. Jech T. J., “First order theory of complete Stonean algebras (Boolean-valued real and complex numbers),” Canad. Math. Bull., 30, No. 4, 385–392 (1987). 199. Jech T. J., “Boolean-linear spaces,” Adv. in Math., 81, No. 2, 117–197 (1990). 200. Jech T. J., Set Theory, Springer-Verlag, Berlin (1997). 201. Johnstone P. T., Topos Theory, Academic Press, London etc. (1977). 202. Johnstone P. T., Stone Spaces, Cambridge University Press, Cambridge and New York (1982). 203. Jordan P., von Neumann J., and Wigner E., “On an algebraic generalization of the quantum mechanic formalism,” Ann. Math., 35, 29–64 (1944). 204. Kachurovski˘ı A. G., “Boundedness of mean fluctuations in the statistic ergodic theorem,” Optimization, No. 48(65), 71–77 (1990).

396 References

205. Kachurovski˘ı A. G., “The rate of convergence in ergodic theorems,” Uspekhi Mat. Nauk, 51, No. 4, 73–124 (1996). 206. Kadison R. V. and Ringrose J. R., Fundamentals of the Theory of Operator Algebras, Vol. 1 and 2, Amer. Math. Soc., Providence, RI (1997). Vol. 3 and 4, Birkh¨auser-Verlag, Boston (1991–1992). 207. Kalina M., “On the ergodic theorem within nonstandard models,” Tatra Mt. Math. Publ., 10, 87–93 (1997). 208. Kalton N. J., “The endomorphisms of Lp (0 ≤ p ≤ 1),” Indiana Univ. Math. J., 27, No. 3, 353–381 (1978). 209. Kalton N. J., “Linear operators on Lp for 0 <p<1,” Trans. Amer. Math. Soc., 259, No. 2, 319–355 (1980). 210. Kalton N. J., “Representation of operators between function spaces,” Indiana Univ. Math. J., 33, No. 5, 639–665 (1984). 211. Kalton N. J., “Endomorphisms of symmetric function spaces,” Indiana Univ. Math. J., 34, No. 2, 225–247 (1985). 212. Kamo S., “Nonstandard natural number systems and nonstandard models,” J. Symbolic Logic, 46, No. 2, 365–376 (1987). 213. Kanovei V. and Reeken M., “Internal approach to external sets and universes,” Studia Logica, part 1: 55, 227–235 (1995); part II: 55, 347–376 (1995); part III: 56, 293–322 (1996). 214. Kanovei V. and Reeken M., “Mathematics in a nonstandard world. I,” Math. Japon., 45, No. 2, 369–408 (1997). 215. Kanovei V. and Reeken M., “A nonstandard proof of the Jordan curve theorem,” Real Anal. Exchange, 24, No. 1, 161–169 (1998). 216. Kanovei V. and Reeken M., “Extending standard models of ZFC to models of nonstandard set theories,” Studia Logica, 64, No. 1, 37–59 (2000). 217. Kanove˘ı V. G., “On well-posedness of the Euler method of decomposition of sinus to infinite product,” Uspekhi Mat. Nauk, 43, No. 4, 57–81 (1988). 218. Kanove˘ı V. G., “Nonsoluble hypothesis in the theory of Nelson’s internal sets,” Uspekhi Mat. Nauk, 46, No. 6, 3–50 (1991). 219. Kanove˘ı V. G., “A nonstandard set theory in the -language,” Mat. Zametki, 70, No. 1, 46–50 (2001). 220. Kant I., Selected Works. Vol. 3 [Russian translation], Mysl, Moscow (1964). 221. Kantorovich L. V., “On semi-ordered linear spaces and their applications to the theory of linear operations,” Dokl. Akad. Nauk SSSR, 4, No. 1–2, 11–14 (1935). 222. Kantorovich L. V., “General forms of some classes of linear operations,” Dokl. Akad. Nauk SSSR, 3, No. 9, 101–106 (1936). 223. Kantorovich L. V., “On one class of functional equations,” Dokl. Akad. Nauk SSSR, 4, No. 5, 211–216 (1936).

References 397

224. Kantorovich L. V., “On some classes of linear operations,” Dokl. Akad. Nauk SSSR, 3, No. 1, 9–13 (1936). 225. Kantorovich L. V., “To the general theory of operations in semi-ordered spaces,” Dokl. Akad. Nauk SSSR, 1, No. 7, 271–274 (1936). 226. Kantorovich L. V., “On functional equations,” Transactions of Leningrad State University, 3, No. 7, 17–33 (1937). 227. Kantorovich L. V. and Akilov G. P., Functional Analysis, Pergamon Press, Oxford and New York (1982). 228. Kantorovich L. V., Vulikh B. Z., and Pinsker A. G., Functional Analysis in Semiordered Spaces [in Russian], Gostekhizdat, Moscow and Leningrad (1950). 229. Kaplansky I., “Projections in Banach algebras,” Ann. of Math. (2), 53, 235–249 (1951). 230. Kaplansky I., “Algebras of type I,” Ann. of Math. (2), 56, 460–472 (1952). 231. Kaplansky I., “Modules over operator algebras,” Amer. J. Math., 75, No. 4, 839–858 (1953). 232. Kawai T., “Axiomsystems of nonstandard set theory,” LogicSymposia, Proc. Conf. Hakone 1979/1980, Springer-Verlag, Berlin etc., 1981, pp. 57–65. 233. Kawebuta Sh., “A ‘non-standard’ approach to the field equations in the functional form,” Osaka J. Math., 23, No. 1, 55–67 (1986). 234. Keisler H. J., Elementary Calculus: An Approach Using Infinitesimals, Prindle, Weber, and Schmidt, Boston, Mass. (1976). 235. Keisler H. J., “Aninfinitesimal approach tostochasticanalysis,” Mem. Amer. Math. Soc., 48, 184 p. (1984). 236. Keldysh M. V., “On completeness of eigenfunctions of some classes of nonself-adjoint operators,” Uspekhi Mat. Nauk, 26, No. 4, 15–41 (1971). 237. Khurgin Ya. I. and Yakovlev V. P., Compactly Supported Functions in Physics and Technology [in Russian], Nauka, Moscow (1971). 238. Kleene S., Mathematical Logic, John Wiley & Sons, New York etc. (1967). 239. Kline M., Mathematical Thought From Ancient to Modern Times, Oxford University Press, Oxford (1972). 240. Kolesnikov E. V., Kusraev A. G., and Malyugin S. A., On Dominated Operators [Preprint, No. 26], Sobolev Institute of Mathematics, Novosibirsk (1988). 241. Kollatz L., Functional Analysis and Numerical Mathematics [in German], Springer-Verlag, Berlin etc. (1964). 242. Kolmogorov A. N. and Dragalin A. G., Mathematical Logic. Supplementary Chapters [in Russian], Moscow University Press, Moscow (1984). 243. Kopperman R., Model Theory and Its Applications, Allynand Bacon, Boston (1972). 244. Kopytov V. M., Lattice-Ordered Groups [in Russian], Nauka, Moscow (1984).

398 References

245. Kornfeld I. P., Sinai Ya. G., and Fomin S. V., Ergodic Theory, SpringerVerlag, Berlin (1982). 246. Korol A. I. and Chilin V. I., “Measurable operators in a Boolean valued model of set theory,” Dokl. Akad. Nauk UzSSR, No. 3, 7–9 (1989). 247. KramosilI., “ComparingalternativedefinitionsofBoolean-valuedfuzzy sets,” Kybernetika, 28, No. 6, 425–443 (1992). 248. Krasnoselski˘ı M. A., Positive Solutions of Operator Equations, P. Noordhoff Ltd., Groningen (1964). 249. Kreisel G., “Observations of popular discussions on foundations,” in: Axiomatic Set Theory. Proc. Symposia in Pure Math., Amer. Math. Soc., Providence, RI, 1971, 1, pp. 183–190. 250. Krupa A., “On various generalizations of the notion of an F-power to the case of unbounded operators,” Bull. Polish Acad. Sci. Math., 38, 159–166 (1990). 251. Kudryk T. S. and Lyantse V. `E., “Operator-valued charges on finite sets,” Mat. Stud., 7, No. 2, 145–156 (1997). 252. Kudryk T. S., Lyantse V. `E., and Chujko G. I., “Nearstandardness of finite sets,” Mat. Stud., 2, 25–34 (1993). 253. Kudryk T. S., Lyantse V. `E., and Chujko G. I., “Nearstandard operators,” Mat. Stud., 3, 29–40 (1994). 254. Kuratowski K. and Mostowski A., Set Theory, North-Holland, Amsterdam etc. (1967). 255. Kuribayashi Y., “Fourier transform using nonstandard analysis,” RIMS Kokyuroku, 975, 132–144 (1996). 256. Kusraev A. G., “Subdifferentials of nonsmooth operators and necessary conditions for an extremum,” Optimization (Novosibirsk), No.24, 75–117(1980). 257. Kusraev A. G., “On a general method of subdifferentiation,” Dokl. Akad. Nauk SSSR, 257, No. 4, 822–826 (1981). 258. Kusraev A. G., “Boolean valued analysis of duality between universally complete modules,” Dokl. Akad. Nauk SSSR, 267, No. 5, 1049–1052 (1982). 259. Kusraev A. G., “General disintegration formulas,” Dokl. Akad. Nauk SSSR, 265, No. 6, 1312–1316 (1982). 260. Kusraev A. G., “On the subdifferentials of composites of sets and functions,” Sibirsk. Mat. Zh., 23, No. 2, 116–127 (1982). 261. Kusraev A. G., “On Boolean valued convex analysis,” Mathematische Optimiering. Theorie und Anwendungen. Wartburg/Eisenach, 1983, pp. 106–109. 262. Kusraev A. G., “On the subdifferential of a sum,” Sibirsk. Mat. Zh., 25, No. 4, 107–110 (1984). 263. Kusraev A. G., Vector Duality and Its Applications [in Russian], Nauka, Novosibirsk (1985).

References 399

264. Kusraev A. G., “Linear operators in lattice-normed spaces,” in: Studies on Geometry in the Large and Mathematical Analysis. Vol. 9 [in Russian], Trudy Inst. Mat. (Novosibirsk), Novosibirsk, 1987, pp. 84–123. 265. KusraevA.G.,“BooleanvaluedanalysisandJB-algebras,”SiberianMath.J., 35, No. 1, 114–122 (1994). 266. Kusraev A. G., “Dominated operators,” in: Linear Operators Compatible with Order [in Russian], Sobolev Institute Press, Novosibirsk, 1995, pp. 212– 292. 267. Kusraev A. G., Boolean Valued Analysis of Duality Between Involutive Banach Algebras [in Russian], North Ossetian University Press, Vladikavkaz (1996). 268. Kusraev A. G., Dominated Operators, Kluwer Academic Publishers, Dordrecht (2000). 269. Kusraev A. G. and Kutateladze S. S.,“Analysis of subdifferentials by means of Boolean valued models,” Dokl. Akad. Nauk SSSR, 265, No. 5, 1061–1064 (1982). 270. Kusraev A. G. and Kutateladze S. S., Notes on Boolean Valued Analysis [in Russian], Novosibirsk University Press, Novosibirsk (1984). 271. Kusraev A. G. and Kutateladze S. S., “On combination of nonstandard methods. I,” Sibirsk. Mat. Zh., 31, No. 5, 111–119 (1990). 272. Kusraev A. G. and Kutateladze S. S., “Nonstandard methods for Kantorovich spaces,” Siberian Adv. Math., 2, No. 2, 114–152 (1992). 273. Kusraev A. G. and Kutateladze S. S., “Nonstandard methods in geometric functional analysis,” Amer. Math. Soc. Transl. Ser. 2, 151, 91–105 (1992). 274. Kusraev A. G. and Kutateladze S. S., “The Kre˘ın–Milman theorem and Kantorovich spaces,” Optimization (Novosibirsk), No. 51 (68), 5–18 (1992). 275. Kusraev A. G. and Kutateladze S. S., 55 Unsolved Problems of Nonstandard Analysis [in Russian], Novosibirsk University Press, Novosibirsk (1993). 276. Kusraev A. G. and Kutateladze S. S., Nonstandard Methods of Analysis, Kluwer Academic Publishers, Dordrecht etc. (1994). 277. Kusraev A. G. and Kutateladze S. S., “Boolean-valued introduction to the theory of vector lattices,” Amer. Math. Soc. Transl. Ser. 2, 163, 103–126 (1995). 278. Kusraev A. G. and Kutateladze S. S., “Nonstandard methods in functional analysis,” in: Interaction Between Functional Analysis, Harmonic Analysis, and Probability Theory, Marcel Dekker Inc., New York, 1995, pp. 301–306. 279. Kusraev A. G. and Kutateladze S. S., Subdifferentials: Theory and Applications, Kluwer Academic Publishers, Dordrecht etc. (1995). 280. Kusraev A. G. and Kutateladze S. S., “On combined nonstandard methods in the theory of positive operators,” Mat. Stud., 7, No. 1, 33–40 (1997).

400 References

281. Kusraev A. G. and Kutateladze S. S., Boolean Valued Analysis, Kluwer Academic Publishers, Dordrecht etc. (1999). 282. Kusraev A. G. and Kutateladze S. S., “On combined nonstandard methods in functional analysis,” Vladikavkaz. Mat. Zh., 2, No. 1 (2000). (http://alanianet.ru/omj/journal.htm) 283. Kutateladze S. S., “Infinitesimal tangentcones,” Sibirsk. Mat. Zh., 27, No.6, 67–76 (1985). 284. Kutateladze S. S., “Microlimits, microsums, and Toeplitz matrices,” Optimization (Novosibirsk), No. 35, 16–23 (1985). 285. Kutateladze S. S., “Nonstandard analysis of tangential cones,” Dokl. Akad. Nauk SSSR, 284, No. 3, 525–527 (1985). 286. Kutateladze S. S., “A variant of nonstandard convex programming,” Sibirsk. Mat. Zh., 27, No. 4, 84–92 (1986). 287. Kutateladze S. S., “Cyclic monads and their applications,” Sibirsk. Mat. Zh., 27, No. 1, 100–110 (1986). 288. Kutateladze S. S., Fundamentals of Nonstandard Mathematical Analysis. Vol. 1: Naive Foudations of Infinitesimal Methods; Vol. 2: Set-Theoretic Foundations of Nonstandard Analysis; Vol. 3: Monads in General Topology [in Russian] (1986). 289. Kutateladze S. S., “On nonstandard methods in subdifferential calculus,” in: Partial Differential Equations [in Russian], Nauka, Novosibirsk, 1986, pp. 116–120. 290. Kutateladze S. S., “Epiderivatives determined by a set of infinitesimals,” Sibirsk. Mat. Zh., 28, No. 4, 140–144 (1987). 291. Kutateladze S. S., “Infinitesimals and a calculus of tangents,” in: Studies in Geometry in the Large and Mathematical Analysis [in Russian], Nauka, Novosibirsk, 1987, pp. 123–135. 292. Kutateladze S. S., “On topological notions pertaining to continuity,” Sibirsk. Mat. Zh., 28, No. 1, 143–147 (1987). 293. Kutateladze S. S., “Monads of proultrafilters and extensional filters,” Sibirsk. Mat. Zh., 30, No. 1, 129–133 (1989). 294. Kutateladze S. S., “On fragments of positive operators,” Sibirsk. Mat. Zh., 30, No. 5, 111–119 (1989). 295. Kutateladze S. S., “The stances of nonstandard analysis,” in: Contemporary Problems of Analysis and Geometry [in Russian], Nauka, Novosibirsk, 1989, pp. 153–182. (Trudy Inst. Mat., 14.) 296. Kutateladze S. S., “Credenda of nonstandard analysis,” Siberian Adv. Math., 1, No. 1, 109–137 (1991). 297. Kutateladze S. S., “Nonstandard tools for convex analysis,” Math. Japon., 43, No. 2, 391–410 (1996).

References 401

298. Kutateladze S. S. (ed.), Vector Lattices and Integral Operators, Kluwer Academic Publishers, Dordrecht (1996). 299. Kutateladze S. S., Formalisms of Nonstandard Analysis [in Russian], Novosibirsk University Press, Novosibirsk (1999). 300. Kutateladze S. S., Fundamentals of Functional Analysis, Kluwer Academic Publishers, Dordrecht (1996). 301. Kuzmina I. S., “Lusitania and its creator,” in: Naukav SSSR, No. 1, 107–110 (1985). 302. Lacey H. E., The Isometric Theory of Classical Banach Spaces, SpringerVerlag, Berlin etc. (1974). 303. Lambek J., Lectures on Rings and Modules, Blaisdell, Waltham (1966). 304. Landers D. and Rogge L., Nonstandard Analysis [in German], SpringerVerlag, Berlin etc. (1994). 305. Lang S., Algebra, Addison-Wesley, Reading (1965). 306. Langwitz D., “Nicht-standard-mathematik, begr¨undel durch eine Verallgemeinerung der K¨orpererweiterung,” Exposit. Math., 1, 307–333 (1983). 307. Larsen R., Banach Algebras, an Introduction, Dekker, New York (1973). 308. Lavrentev M. A., “Nikola˘ı Nikolaevich Luzin,” Uspekhi Mat. Nauk, 29, No. 5, 177–182 (1979). 309. Lavrentev M. A., Science. Technical Progress. Personnel [in Russian], Nauka, Novosibirsk (1980). 310. League J.-M., “Science, technology, and the world,” Nauka i Zhizn, No. 11, 3–11 (1986). 311. Leibniz G. W., “Nova Methodus pro Maximis et Minimis, Itemque Tangentibus, quae nec Fractals nec Irrationales Quattitates, et Singulare pro Illus Calculi Genns,” Uspekhi Mat. Nauk, 3, No. 1, 166–173 (1948). 312. Leibniz G. W., Selected Works. Vol. 1 [Russian translation], Mysl, Moscow (1983). 313. Leibniz G. W., Selected Works. Vol. 2 [Russian translation], Mysl, Moscow (1984). 314. Levin B. Ya., Distribution of Roots of Entire Functions [in Russian], Gostekhizdat, Moscow (1956). 315. Levin V. L., Convex Analysis in Spaces of Measurable Functions and Its Application in Mathematics and Economics [in Russian], Nauka, Moscow (1985). 316. Levy A., Basic Set Theory, Springer-Verlag, Berlin etc. (1979). 317. Li N., “The Boolean-valued model of the axiom system of GB,” Chinese Sci. Bull., 36, No. 2, 99–102 (1991). 318. Lindenstrauss J., “Extension of compact operators,” Mem. Amer. Math. Soc., 48, 112 p. (1964).

402 References

319. Lindenstrauss J. and Tzafriri L., Classical Banach Spaces. Vol. 1: Sequence Spaces, Springer-Verlag, Berlin etc. (1977). 320. Lindenstrauss J. and Tzafriri L., Classical Banach Spaces. Vol. 2: Function Spaces, Springer-Verlag, Berlin etc. (1979). 321. Locher J. L. (ed.), The World of M. C. Escher, Abradale Press, New York (1988). 322. Loeb P. A., “Conversion from nonstandard to standard measure spaces and applications to probability theory,” Trans. Amer. Math. Soc., 211, 113–122 (1975). 323. Loeb P. A., “An introduction to nonstandard analysis and hyperfinite probability theory,” in: Probabilistic Analysis and Related Topics. Vol. 2 (Ed. A. T. Bharucha-Reid), Academic Press, New York, 1979, pp. 105–142. 324. Loeb P. and Wolff M. P. H. (eds.), Nonstandard Analysis for the Working Mathematician, Kluwer Academic Publishers, Dordrecht etc. (2000). 325. Loewen P. D., Optimal Control via Nonsmooth Analysis, Amer. Math. Soc., Providence, RI (1993). 326. Lowen R., “Mathematics and fuzziness,” in: Fuzzy Sets Theory and Applications (Louvain-la-Neuve, 1985), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., 177, Reidel, Dordrecht, and Boston, 1986, pp. 3–38. 327. Lutz R. and Gose M., Nonstandard Analysis. A Practical Guide with Applications, Springer-Verlag, Berlin etc. (1981). (Lecture Notes in Math., 881.) 328. Luxemburg W. A. J., “A general theory of monads,” in: Applications of Model Theory to Algebra, Analysis and Probability, Holt, Rinehart, and Winston, New York, 1969, pp. 18–86. 329. Luxemburg W. A. J., “A nonstandard approach to Fourier analysis,” in: Contributions to Nonstandard Analysis, North-Holland, Amsterdam, 1972, pp. 16–39. 330. LuxemburgW. A.J.(ed.), ApplicationsofModelTheorytoAlgebra, Analysis and Probability, Holt, Rinehart, and Winston, New York (1969). 331. Luxemburg W. A. J. and Robinson A. (eds.), Contribution to Non-Standard Analysis, North-Holland, Amsterdam (1972). 332. Luxemburg W. A. J. and Zaanen A. C., Riesz Spaces. Vol. 1, North-Holland, Amsterdam and London (1971). 333. Luzin N. N., “Real function theory: State of the art,” in: Proceedings of the All-Russia Congress of Mathematicians (Moscow, April 27–May 4, 1927), Glavnauka, Moscow and Leningrad, 1928, pp. 11–32. 334. Luzin N. N., The Theory of Real Functions [in Russian], Uchpedgiz, Moscow (1940). 335. Luzin N. N., Collected Works. Vol. 3 [in Russian], Izdat. Akad. Nauk SSSR, Moscow (1959).

References 403

336. Luzin N. N., Differential Calculus [in Russian], Vysshaya Shkola, Moscow (1961). 337. “Luzin N. N. is an outstanding mathematician and teacher,” Vestnik Akad. Nauk SSSR, No. 11, 95–102 (1984). 338. Lyantse V. `E., “Is it possible to ignore nonstandard analysis?” in: General Theory of Boundary Problems [in Russian], Naukova Dumka, Kiev, 1983, pp. 108–112. 339. Lyantse V. `E., “On nonstandard analysis,” in: Mathematics of Today [in Russian], Vishcha Shkola, Kiev, 1986, pp. 26–44. 340. Lyantse V. `E., “Nearstandardness on a finite set,” Dissertationes Math., 369, 63 p. (1997). 341. Lyantse V. `E. and Kudrik T. S., “On functions of discrete variable,” in: Mathematics of Today [in Russian], Vishcha Shkola, Kiev, 1987. 342. Lyantse W. and Kudryk T., Introduction to Nonstandard Analysis, VNTL Publishers, Lviv (1997). 343. Lyantse V. `E. and Yavorskyj Yu. M., “Nonstandard Sturm–Liouville difference operator,” Mat. Stud., 10, No. 1, 54–68 (1998). 344. Lyantse V. `E. and Yavorskyj Yu. M., “Nonstandard Sturm–Liouville difference operator. II,” Mat. Stud., 11, No. 1, 71–82 (1999). 345. Lyubetski˘ı V. A., “On some algebraic problems of nonstandard analysis,” Dokl. Akad. Nauk SSSR, 280, No. 1, 38–41 (1985). 346. Lyubetski˘ı V. A. and Gordon E. I., “Boolean completion of uniformities,” in: Studies on Nonclassical Logics and Formal Systems [in Russian], Nauka, Moscow, 1983, pp. 82–153. 347. Lyubetski˘ı V. A. and Gordon E. I., “Immersing bundles into a Heyting valued universe,” Dokl. Akad. Nauk SSSR, 268, No. 4, 794–798 (1983). 348. MacLane S., Categories for the Working Mathematician, Springer-Verlag, New York (1971). 349. Maltsev A. I., Algebraic Systems. Posthumous Edition, Springer-Verlag, New York and Heidelberg (1973). 350. Malykhin V. I., “New aspects of general topology pertaining to forcing,” Uspekhi Mat. Nauk, 43, No. 4, 83–94 (1988). 351. Malykhin V. I. and Ponomar¨ev V. I., “General topology (set-theoretic direction),” in: Algebra. Topology. Geometry [in Russian], VINITI, Moscow, 1975, 13, pp. 149–229. 352. Manin Yu. I., A Course in Mathematical Logic, Springer-Verlag, Berlin etc. (1977). 353. Marinakis K., “The infinitely small in space and in time,” Trans. Hellenic Inst. Appl. Sci., No. 8, 62 p. (1970). 354. Martin-L¨of P., “The definition of random sequences,” Inform. and Control., 9, 602–619 (1966).

404 References

355. MazonJ.M. and Segura de Le´onS., “Order bounded orthogonally additive operators,” Rev. Roumaine Math. Pures Appl., 35, No. 4, 329–354 (1990). 356. Mazya V. G., The Spaces of S. L. Sobolev [in Russian], Leningrad University Publ. House, Leningrad (1985). 357. Mazya V. G. and Khavin V. P., “A nonlinear potential theory,” Uspekhi Mat. Nauk, 27, No. 6, 67–138 (1972). 358. Mckee T., “Monadic characterizable in nonstandard topology,” Z. Math. Logik Grundlag. Math., 26, No. 5, 395–397 (1940). 359. Medvedev F. A., “Cantor’s set theory and theology,” in: Historio-Mathematical Studies [in Russian], Nauka, Moscow, 1985, 29, pp. 209–240. 360. Medvedev F. A., “Nonstandard analysis and the history of classical analysis,” in: Laws of the Development of Modern Mathematics [in Russian], Nauka, Moscow, 1987, pp. 75–84. 361. Melter R., “Boolean valued rings and Boolean metric spaces,” Arch. Math., No. 15, 354–363 (1964). 362. Mendelson E., Introduction to Mathematical Logic, Van Nostrand, Princeton etc. (1963). 363. Milvay C. J., “Banach sheaves,” J. Pure Appl. Algebra, 17, No. 1, 69–84 (1980). 364. MochoverM.andHirschfeldJ., LecturesonNon-StandardAnalysis, SpringerVerlag, Berlin etc. (1969). (Lecture Notes in Math., 94.) 365. Molchanov I. S., “Set-valued estimators for mean bodies related to Boolean models,” Statistics, 28, No. 1, pp. 43–56 (1996). 366. Molchanov V. A., The One-Dimensional Mathematical Analysis in Nonstandard Presentation [in Russian], K. A. Fedin Saratov Ped. Inst., Saratov (1989). 367. Molchanov V. A., “The use of double nonstandard enlargements in topology,” Sibirsk. Mat. Zh., 30, No. 3, 64–71 (1989). 368. Molchanov V.A., IntroductionintotheCalculusofInfinitesimals[inRussian], K. A. Fedin Saratov Ped. Inst., Saratov (1990). 369. Molchanov V. A., Nonstandard Convergences in the Spaces of Mappings [in Russian], K. A. Fedin Saratov Ped. Inst., Saratov (1991). 370. Monk J. D. and Bonnet R. (eds.), Handbook of Boolean Algebras. Vol. 1–3, North-Holland, Amsterdam etc. (1989). 371. Moore L. C. Jr., “Hyperfinite extensions of bounded operators on a separable Hilbert space,” Trans. Amer. Math. Soc., 218, 285–295 (1976). 372. Mostowski A., Constructible Sets with Applications, North-Holland, Amsterdam (1969). 373. Murphy G., C∗-Algebras and Operator Theory, Academic Press, Boston, (1990).

References 405

374. Namba K., “Formal systems and Boolean valued combinatorics,” in: SoutheastAsianConference onLogic(Singapore, 1981), Stud. LogicFound. Math., 111, North-Holland, Amsterdam and New York, 1983, pp. 115–132. 375. Natterer F., The Mathematics of Computerized Tomography, Teubner and John Wiley & Sons, Stuttgart and Chichester, etc. (1986). 376. Na˘ımark M. A., Normed Rings, Noordhoff, Groningen (1959). 377. Na˘ımark M. A., Representation Theory of Groups [in Russian], Nauka, Moscow (1976). 378. Nelson E., “Internal set theory. A new approach to nonstandard analysis,” Bull. Amer. Math. Soc., 83, No. 6, 1165–1198 (1977). 379. Nelson E., Radically Elementary Probability Theory, Princeton University Press, Princeton (1987). 380. Nelson E., “The syntax of nonstandard analysis,” Ann. Pure Appl. Logic, 38, No. 2, 123–134 (1988). 381. Nepe˘ıvoda N. N., Applied Logic [in Russian], Novosibirsk University Press, Novosibirsk (2000). 382. Neubrunn I., Rie´can B., and Riecanou Z., “An elementary approach to some applications of nonstandard analysis,” Rend. Circl. Math. Palermo, No. 3, 197–200 (1984). 383. Neumann J., von, Collected Works. Vol. 1 and 2, Pergamon Press, New York, Oxford, London, and Paris (1961). 384. Newton I., The Mathematical Papers of Isaac Newton [in Russian], ONTI, Moscow and Leningrad (1937). 385. Ng Siu-Ah, Hypermodels in Mathematical Finance. Modelling via Infinitesimal Analysis, World Scientific, Singapore etc. (2001). 386. Nishimura H., “Boolean valued and Stone algebra valued measure theories,” Math. Logic Quart., 40, No. 1, 69–75 (1994). 387. Novikov P. S., Constructive Mathematical Logic from the Point of View of Classical Logic [in Russian], Nauka, Moscow (1977). 388. Ozawa M., “Boolean valued analysis approach to the trace problem of AW∗algebras,” J. London Math. Soc. (2), 33, No. 2, 347–354 (1986). 389. Ozawa M., “Embeddable AW∗-algebras and regular completions,” J. London Math. Soc., 34, No. 3, 511–523 (1986). 390. Ozawa M., “Boolean-valued interpretation of Banach space theory and module structures of von Neumann algebras,” Nagoya Math. J., 117, 1–36 (1990). 391. Pedersen G. K., Analysis Now, Springer-Verlag, Berlin etc. (1995). 392. Penot J.-P., “Compact nets, filters and relations,” J. Math. Anal. Appl., 93, No. 2, 400–417 (1983). 393. P´eraire Y., “Une nouvelle th´eorie des infinitesimaux,” C. R. Acad. Sci. Paris Ser. I., 301, No. 5, 157–159 (1985).

406 References

394. P´eraire Y., “A general theory of infinitesimals,” Sibirsk. Mat. Zh., 31, No. 3, 107–124 (1990). 395. P´eraire Y., “Th´eorie relative des ensembles internes,” Osaka J. Math., 29, No. 2, 267–297 (1992). 396. P´eraire Y., “Some extensions of the principles of idealization transfer and choice in the relative internal set theory,” Arch. Math. Logic., 34, 269–277 (1995). 397. Pietsch A., Operator Ideals, VEB Deutschen Verlag der Wissenschaften, Berlin (1978). 398. Pinus A. G., Boolean Constructions in Universal Algebras, Kluwer Academic Publishers, Dordrecht etc. (1993). 399. Pirce R. S., “Modules over commutative regular rings,” Mem. Amer. Math. Soc., No. 70 (1967). 400. Pontryagin L. S., Analysis of Infinitesimals [in Russian], Nauka, Moscow (1980). 401. PontryaginL. S., Mathematical AnalysisforSchoolchildren [inRussian], Nauka, Moscow (1980). 402. Pontryagin L. S., Continuous Groups [in Russian], Nauka, Moscow (1984). 403. Postnikov A. G., Introduction to Analytic Number Theory [in Russian], Nauka, Moscow (1971). 404. Prokhorova M. F., “On relative nearstandardness in IST,” Siberian Math. J., 39, No. 3, 518–521 (1998). 405. Raebiger F. and Wolff M. P. H., “On the approximation of positive operators and the behavior of the spectra of the approximants,” in: Integral Equations Operator Theory, 1997, 28, pp. 72–86. 406. Raebiger F. and Wolff M. P. H., “Spectral and asymptotic properties of dominated operators,” J. Austral. Math. Soc. Ser. A, 63, 16–31 (1997). 407. Rasiowa E. and Sikorski R., The Mathematics of Metamathematics, PWN, Warsaw (1962). 408. Reader on the History of Mathematics [in Russian], Prosveshchenie, Moscow (1977). 409. Reed M. and Simon B., Methods of Modern Mathematical Physics, Academic Press, New York and London (1972). 410. Reinhardt H.-J., Analysis of Approximation Methods for Differential and Integral Equations, Springer-Verlag, New York, Berlin, Heidelberg, and Tokyo (1985). 411. Rema P. S., “Boolean metrization and topological spaces,” Math. Japon., 9, No. 9, 19–30 (1964). 412. Repicky M., “Cardinal characteristics of the real line and Boolean-valued models,” Comment. Math. Univ. Carolin., 33, No. 1, 184 (1992).

References 407

413. Reshetnyak Yu. G., A Course of Mathematical Analysis. Vol. 1, Part 1 [in Russian], Sobolev Institute Press, Novosibirsk (1999). 414. Revuzhenko A. F., Mechanics of Elastico-Plastic Media and Nonstandard Analysis [in Russian], Novosibirsk University Press, Novosibirsk (2000). 415. Rickart Ch., General Theory of Banach Algebras, Van Nostrand, Princeton (1960). 416. Riesz F. and Sz¨okefalvi-Nagy B., Lectures on Functional Analysis [in French], Acad´emi Kiad´o, Budapest (1972). 417. RobertA.,AnalyseNon-Standard, Queen’sUniversityPress,Kingston(1984). 418. Robert A., “One approache naive de l’analyse non-standard,” Dialectica, 38, 287–290 (1984). 419. Robinson A., Introduction to the Theory of Models and to the Metamathematics of Algebra, North-Holland, Amsterdam (1963). 420. Robinson A., “The metaphysics of the calculus,” Problems in the Philosophy of Mathematics, North-Holland, Amsterdam, 1967, 1, pp. 28–46. 421. Robinson A., Non-Standard Analysis, Princeton Univ. Press, Princeton (1996). 422. Rockafellar R. T., Convex Analysis, Princeton University Press, Princeton (1970). 423. Rockafellar R. T., The Theory of Subgradients and Its Applications to Problems of Optimization: Convex and Nonconvex Functions, Springer-Verlag, Berlin etc. (1981). 424. Rockafellar R. T. and Wets R. J.-B., Variational Analysis, Springer-Verlag, Birkh¨auser-Verlag (1998). 425. Rosser J. B., Logic for Mathematicians, McGraw-Hill voon Company Inc., New York etc. (1953). 426. Rosser J. B., Simplified Independence Proofs. Boolean Valued Models of Set Theory, Academic Press, New York and London (1969). 427. Rubio J. E., Optimization and Nonstandard Analysis, Marcel Dekker, New York and London (1994). (Pure and Applied Mathematics, 184.) 428. Rudin W., Functional Analysis, McGraw-Hill Inc., New York (1991). 429. Ruzavin G. I., Philosophical Problems of the Foundations of Mathematics [in Russian], Nauka, Moscow (1983). 430. SakaiS. C∗-Algebras and W∗-Algebras, Springer-Verlag, Berlin etc. (1971). 431. Salbany S. and Todorov T., “Nonstandard analysis in topology: nonstandard and standard compactifications,” J. Symbolic Logic, 65, No. 4, 1836–1840 (2000). 432. Saracino D. and Weispfenning V., “On algebraic curves over commutative regular rings,” in: Model Theory and Algebra (a Memorial Tribute to Abraham Robinson), Springer-Verlag, New York etc., 1969. (Lecture Notes in Math., 498.)

408 References

433. SarymsakovT.A.etal., Ordered Algebras[inRussian], Fan, Tashkent(1983). 434. Schaefer H. H., Banach Lattices and Positive Operators, Springer-Verlag, Berlin etc. (1974). 435. Schr¨oder J., “Das Iterationsverfahren bei allgemeinierem Abshtandsbegriff,” Math. Z., Bd 66, 111–116 (1956). 436. Schwartz L., Analysis. Vol. 1 [in French], Hermann, Paris (1967). 437. Schwarz H.-U., Banach Lattices and Operators, Teubner, Leipzig (1984). 438. Schwinger J., “Unitary operator bases. The special canonical group,” Proc. Natl. Acad. Sci. USA, 46, 570–579, 1401–1405 (1960). 439. Severi F., “Italian algebraic geometry, its rigor, methods, and problems,” Mathematics, 3, No. 1, 111–141 (1959). 440. Sextus Empiricus, Selected Works. Vol. 1 [Russian translation], Mysl, Moscow (1976). 441. Shamaev I. I., “On decomposition and representation of regular operators,” Sibirsk. Mat. Zh., 30, No. 2, 192–202 (1989). 442. Shamaev I. I., “An operator disjoint from lattice homomorphisms, Maharam operators, and integral operators,” Optimization (Novosibirsk), No. 46, 154– 159 (1990). 443. Shamaev I. I., Topological Methods for the Theory of Regular Operators in Kantorovich Spaces: ScD Thesis [in Russian], Yakutsk (1994). 444. Shoenfield J. R., Mathematical Logic, Addison-Wesley, Reading (1967). 445. Shoenfield J. R., “Axioms of set theory,” in: Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977. 446. Shotaev G. N., “On bilinear operators in lattice-normed spaces,” Optimization (Novosibirsk), No. 37, 38–50 (1986). 447. Sikorski R., Boolean Algebras, Springer-Verlag, Berlin etc. (1964). 448. Sikorski˘ı M. R., “Some applications of Boolean-valued models to study operators on polynormed spaces,” Sov. Math., 33, No. 2, 106–110 (1989). 449. Smith K., “Commutative regular rings and Boolean-valued fields,” J. Symbolic Logic, 49, No. 1, 281–297 (1984). 450. Smithson R., “Subcontinuity for multifunctions,” Pacific J. Math., 61, No. 4, 283–288 (1975). 451. Sobolev V. I., “On a poset valued measure of a set, measurable functions and some abstract integrals,” Dokl. Akad. Nauk SSSR, 91, No. 1, 23–26 (1953). 452. Solovay R. M., “A model of set theory in which every set of reals is Lebesgue measurable,” Ann. of Math. (2), 92, No. 2, 1–56 (1970). 453. Solovay R. and Tennenbaum S., “Iterated Cohen extensions and Souslin’s problem,” Ann. Math., 94, No. 2, 201–245 (1972). 454. Solov¨ev Yu. P. and Troitski˘ı E. V.,C∗-Algebras and Elliptic Operators in Differential Topology [in Russian], Factorial, Moscow (1996).

References 409

455. Sourour A. R., “Operators with absolutely bounded matrices,” Math. Z., 162, No. 2, 183–187 (1978). 456. Sourour A. R., “The isometries of Lp(Ω,X),” J. Funct. Anal., 30, No. 2, 276–285 (1978). 457. Sourour A. R., “A note on integral operators,” Acta Sci. Math., 41, No. 43, 375–379 (1979). 458. Sourour A. R., “Pseudo-integral operators,” Trans. Amer. Math. Soc., 253, 339–363 (1979). 459. Sourour A. R., “Characterization and order properties of pseudo-integral operators,” Pacific J. Math., 99, No. 1, 145–158 (1982). 460. Stern J., “Some applications of model theory in Banach space theory,” Ann. Math. Logic, 9, No. 1, 49–121 (1976). 461. Stern J., “The problem of envelopes for Banach spaces,” Israel J. Math., 24, No. 1, 1–15 (1976). 462. Stewart I., “Frog and Mouse revisited,” Math. Intelligencer, 8, No. 4, 78–82 (1986). 463. Stroyan K. D., “Infinitesimal analysis of curves and surfaces,” in: Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977. 464. Stroyan K. D., “Infinitesimal calculus in locally convex spaces. I,” Trans. Amer. Math. Soc., 240, 363–384 (1978); “Locally convex infinitesimal calculus. II,” J. Funct. Anal., 53, No. 1, 1–15 (1983). 465. Stroyan K. D. and Bayod J. M., Foundations of Infinitesimal Stochastic Analysis, North-Holland, Amsterdam etc. (1986). 466. Stroyan K. D. and Luxemburg W. A. J., Introduction to the Theory of Infinitesimals, Academic Press, New York etc. (1976). 467. Struik D. J., A Concise History of Mathematics, Dover Publications, New York (1987). 468. Stummel F., “Diskrete Konvergenz Linearer Operatoren. I,” Math. Ann., 190, 45–92 (1970). 469. Stummel F., “Diskrete Konvergenz Linearer Operatoren. II,” Math. Z., 120, 231–264 (1971). 470. Sunder V. S., An Invitation to Von Neumann Algebras, Springer-Verlag, New York etc. (1987). 471. Sz´ekely G. J., Paradoxes in Probability Theory and Mathematical Statistics [in Russian], Mir, Moscow (1990). 472. Tacon D. G., “Nonstandard extensions of transformations between Banach spaces,” Trans. Amer. Math. Soc., 260, No. 1, 147–158 (1980). 473. Takeuti G., Two Applications of Logic to Mathematics, Iwanami and Princeton University Press, Tokyo and Princeton (1978). 474. Takeuti G., “A transfer principle in harmonic analysis,” J. Symbolic Logic, 44, No. 3, 417–440 (1979).

410 References

475. Takeuti G., “Boolean valued analysis,” in: Applications of Sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Durham, 1977), Springer-Verlag, Berlin etc., 1979, pp. 714–731. 476. Takeuti G., “Quantum set theory,” in: Current Issues in Quantum Logic (Erice, 1979), Plenum Press, New York and London, 1981, pp. 303–322. 477. Takeuti G., “C∗-algebras and Boolean valued analysis,” Japan J. Math., 9, No. 2, 207–246 (1983). 478. Takeuti G., “Von Neumann algebras and Boolean valued analysis,” J. Math. Soc. Japan, 35, No. 1, 1–21 (1983). 479. Takeuti G. and Titani S., “Heyting-valued universes of intuitionistic set theory,” Logic Symposia, Proc. Conf. Hakone 1979/1980, Springer-Verlag, Berlin and New York, 1981, pp. 189–306. (Lecture Notes in Math., 89.) 480. Takeuti G. and Titani S., “Globalization of intuitionistic set theory,” Ann. Pure Appl. Logic, 33, No. 2, 195–211 (1987). 481. Takeuti G. and Zaring W. M., Introduction to Axiomatic Set Theory, Springer-Verlag, New York etc. (1971). 482. Takeuti G. and Zaring W. M., Axiomatic Set Theory, Springer-Verlag, New York (1973). 483. Tall D., “The calculus of Leibniz—an alternative modern approach,” Math. Intelligencer, 2, No. 1, 54–60 (1979/1980). 484. Tanaka K. and Yamazaki T., “A nonstandard construction of Haar measure and weak Koenig’s lemma,” J. Symbolic Logic, 65, No. 1, 173–186 (2000). 485. Thibault L., “Epidifferentielles de fonctions vectorielles,” C. R. Acad. Sci. Paris S´er. I Math., 290, No. 2, A87–A90 (1980). 486. ThibaultL., “Subdifferentialsofnonconvexvector-valuedmappings,” J.Math. Anal. Appl., 86, No. 2 (1982). 487. Tikhomorov V. M., “Convex analysis,” in: Analysis. II. Convex Analysis and Approximation Theory, Encycl. Math. Sci. 14, 1-92 (1990). 488. Topping D. M., “Jordan algebras of self-adjoint operators,” Mem. Amer. Math. Soc., 53, 48 p. (1965). 489. Trenogin V. A., Functional Analysis [in Russian], Nauka, Moscow (1980). 490. Troitski˘ı V. G., “Nonstandard discretization and the Loeb extension of a family of measures,” Siberian Math. J., 34, No. 3, 566–573 (1993). 491. Tsalenko M. Sh. and Shulge˘ıfer E. F., Fundamentals of Category Theory [in Russian], Nauka, Moscow (1974). 492. Uspenski˘ı V. A., “Seven considerations on the philosophical topics of mathematics,” in: Laws of the Development of Modern Mathematics [in Russian], Nauka, Moscow, 1987, pp. 106–155. 493. Uspenski˘ı V. A., What Is Nonstandard Analysis? [in Russian], Nauka, Moscow (1987).

References 411

494. Van de Berg I., Nonstandard Asymptotic Analysis, Springer-Verlag, Berlin etc. (1987). 495. van der Hoeven J., “On the computation of limsups,” J. Pure Appl. Algebra, 117–118, 381–394 (1997). 496. Varadarajan V., Quantization of Semisimple Groups and Some Applications [Preprint] (1996). 497. Veksler A. I., “On a new construction of the Dedekind completion of some lattices and l-groups with division,” Siberian Math. J., 10, No. 6, 1206–1213 (1969). 498. Veksler A. I., “Banach cyclic spaces and Banach lattices,” Dokl. Akad. Nauk SSSR, 213, No. 4, 770–773 (1973). 499. Veksler A. I. and Ge˘ıler V. A., “On the order and disjoint completeness of semiordered linear spaces,” Sibirsk. Mat. Zh., 13, No. 1, 43–51 (1972). 500. Veksler A. I. and Gordon E. I., “Nonstandard extension of not-everywheredefined positive operators,” Sibirsk. Mat. Zh., 35, No. 4, 640–647 (1994). 501. Venkataraman K., “Boolean valued almost periodic functions: existence of the mean,” J. Indian Math. Soc., 43, No. 1–4, 275–283 (1979). 502. Venkataraman K., “Boolean valued almost periodic functions on topological groups,” J. Indian Math. Soc., 48, No. 1–4, 153–164 (1984). 503. Vershik A. M. and Gordon E. I., “The groups locally embedded into the class of finite groups,” Algebra and Analysis, 9, No. 1, 72–86 (1997). 504. Vilenkin N., “The commander of ‘Lusitania’,” Znanie—Sila, No. 1, 27–29 (1984). 505. Vladimirov D. A., Boolean Algebras in Analysis, Kluwer Academic Publishers, Dordrecht (2001). 506. Vladimirov V. S., Volovich I. V., and Zelenov E. I., p-Adic Analysis and Mathematical Physics [in Russian], Nauka, Moscow (1994). 507. Voltaire, Verses and Proses [Russian translation], Moskovski˘ı Rabochi˘ı, Moscow (1997). 508. von Neumann J., Collected Works. Vol. 1: Logic, Theory of Sets and Quantum Mechanics, Pergamon Press, Oxford etc. (1961). 509. von Neumann J., Collected Works. Vol. 3: Rings of Operators, Pergamon Press, New York, Oxford, London, and Paris (1961). 510. von Neumann J., Collected Works. Vol. 4: Continuous Geometry and Other Topics, Pergamon Press, Oxford, London, New York, and Paris (1962). 511. VopˇenkaP., “General theoryof.-models,” Comment. Math. Univ. Carolin., 7, No. 1, 147–170 (1967). 512. Vopˇenka P., “The limits of sheaves over extremally disconnected compact Hausdorff spaces,” Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys., 15, No. 1, 1–4 (1967).

412 References

513. Vopˇenka P., Mathematics in the Alternative Set Theory, Teubner, Leipzig (1979). 514. Vulikh B. Z., Introduction to the Theory of Partially Ordered Spaces, Noordhoff, Groningen (1967). 515. Vygodski˘ıM. Ya., Fundamentals ofthe Calculusof Infinitesimals[inRussian], GTTI, Moscow and Leningrad (1933). 516. WangHaoand McNaughtonR., AxiomaticSystemsofSet Theory [inFrench], Gauthier-Villars, Paris; E. Nauwelaerts, Louvain (1953). 517. Ward D. E., “Convex subcones of the contingent cone in nonsmooth calculus and optimization,” Trans. Amer. Math. Soc., 302, No. 2, 661–682 (1987). 518. Ward D. E., “The quantification tangent cones,” Canad. J. Math., 40, No. 3, 666–694 (1988). 519. Ward D. E., “Corrigendum to ‘Convex subcones of the contingent cone in nonsmooth calculus and optimization,’” Trans. Amer. Math. Soc., 311, No. 1, 429–431 (1989). 520. Wattenberg F., “Nonstandard measure theory: Avoiding pathological sets,” Trans. Amer. Math. Soc., 250, 357–368 (1979). 521. Weil A., “Euler,” Amer. Math. Monthly, 91, No. 9, 537–542 (1984). 522. Weis L., “Decompositions of positive operators and some of their applications,” in: Functional Analysis: Survey and Recent Results. Vol. 3: Proc. 3rd Conf., Paderborn, 24–29 May, 1983, Amsterdam, 1984. 523. Weis L., “On the representation of order continuous operators by random measures,” Trans. Amer. Math. Soc., 285, No. 2, 535–563 (1984). 524. Weis L., “The range of an operator in C(X) and its representing stochastic kernel,” Arch. Math., 46, 171–178 (1986). 525. Weispfenning V., “Model-completeness and elimination of quantifiers for subdirect products of structures,” J. Algebra, 36, No. 2, 252–277 (1975). 526. Westfall R., Never at Rest. A Bibliography of Isaak Newton, Cambridge University Press, Cambridge (1982). 527. Wolff M. P. H., “An introduction to nonstandard functional analysis,” in: L. O. Arkeryd, N. J. Cutland, and C. Ward Henson (eds): Nonstandard Analysis, Theory and Applications, Kluwer Academic Publishers, Dordrecht, 1997, pp. 121–151. 528. Wolff M. P. H., “On the approximation of operators and the convergence of the spectra of approximants,” in: Operator Theory: Advances and Applications, 1998, 103, pp. 279–283. 529. Wright J. D. M., “Vector lattice measures on locally compact spaces,” Math. Z., 120, No. 3, 193–203 (1971). 530. YamaguchiJ.,“Boolean[0,1]-valuedcontinuousoperators,”Internat.J.Comput. Math., 68, No. 1–2, 71–79 (1998).

References 413

531. Yessenin-Volpin A. S., “The ultra intuitionistic criticism and the traditional program for foundations of mathematics,” in: Intuitionism and Proof Theory, North-Holland, Amsterdam and London, 1970, pp. 3–45. 532. Yood B., Banach Algebras. An Introduction, Carleton University Press, Ottawa (1988). 533. YoungL.,LecturesontheCalculusofVariationsandOptimalControlTheory, Chelsea Publishing Co., New York (1980). 534. Yushkevich A. P., “Leibniz and the foundation of infinitesimal calculus,” Uspekhi Mat. Nauk, 3, No. 1, 150–164 (1948). 535. Zaanen A. C., Riesz Spaces. Vol. 2, North-Holland, Amsterdam etc. (1983). 536. Zaanen A. C., Introduction to Operator Theory in Riesz Spaces, SpringerVerlag, Berlin etc. (1997). 537. Zadeh L., The Concept of a Linguistic Variable and Its Application to Approximate Reasoning, Elsevier, New York etc. (1973). 538. Zeldovich Ya. B. and Sokolov D. D., “Fractals, similarity, and intermediate asymptotics,” Uspekhi Fiz. Nauk, 146, No. 3, 493–506 (1985). 539. Zhang Jin-wen, “A unified treatment of fuzzy set theory and Boolean valued set theory fuzzy set structures and normal fuzzy set structures,” J. Math. Anal. Appl., 76, No. 1, 297–301 (1980). 540. Zivaljevic R., “Infinitesimals, microsimplexes and elementary homology theory,” Amer. Math. Monthly, 93, No. 7, 540–544 (1986).

541. Zorich V. A., Mathematical Analysis. Part 1 [in Russian], Nauka, Moscow (1981).

542. Zvonkin A. K. and Shubin M. A., “Nonstandard analysis and singular perturbations of ordinary differential equations,” Uspekhi Mat. Nauk, 39, No. 2, 77–127 (1984).