@mastersthesis { Gutman19880621, author = "Gutman A.E.", school = "Dept. of Math. Anal., Faculty of Mech. and Math., Novosib. State Univ., Novosibirsk", howpublished = "Manuscript", title = "Multiplicative representation of disjointness preserving operators", year = "1988", pages = "94", doi = "10.13140/RG.2.2.25057.61280", language = "russian", annote = "A diploma thesis in mathematics. Defended at the Department of Mathematical Analysis, Novosibirsk State University. Thesis adviser: Kusraev Anatoly Georgievich (D.Sc., Leading scientific officer at the Institute of Mathematics SB AS USSR)." } @inproceedings { Gutman19890906, author = "Gutman A.E.", title = "On disjointness preserving operators in Banach--Kantorovich spaces", booktitle = "XIV School on the theory of operators in function spaces (Novgorod, September 6--14, 1989): Proceedings", address = "Novgorod", year = "1989", volume = "1", pages = "75", language = "russian", annote = "For a disjointness preserving linear operator acting from an arbitrary Banach--Kantorovich space in a universally complete Banach--Kantorovich space the equivalence is established of its $r$-$o$-continuity, $r$-continuity, majorizability, and multiplicativity on principal ideals." } @article { Gutman19900522, author = "Gutman A.E.", title = "An example of a sequentially $o$-continuous but not dominated disjointness preserving operator", journal = "Optimization", year = "1990", volume = "47(64)", pages = "116--121", language = "russian", annote = "An example is given of a linear operator acting from a Banach space into a Kantorovich space, which is sequentially $r$-continuous but not dominated." } @inproceedings { Gutman19900905, author = "Gutman A.E.", title = "On disjointness preserving operators in spaces of continuous functions", booktitle = "XV All-USSR School on the theory of operators in function spaces (Ulyanovsk, September 5--12, 1990): Proceedings", address = "Ulyanovsk", year = "1990", volume = "1", pages = "76", language = "russian", annote = "For a disjointness preserving linear operator acting from a lattice-normed space of continuous vector-valued functions into a space of weakly continuous vector-valued functions the equivalence is established of its $r$-$o$-continuity, $r$-continuity, majorizability, and multiplicativity on principal ideals." } @inproceedings { Gutman19901210, author = "Gutman A.E.", title = "Measurable Banach bundles and weight operators", booktitle = "The Fifth School of Siberian and Far-Eastern young mathematicians (Novosibirsk, December 10--16, 1990): Proceedings", address = "Novosibirsk", year = "1990", pages = "30--32", language = "russian", annote = "The notion of measurable Banach bundle with lifting is introduced and studied. Representation is constructed of Banach--Kantorovich spaces and the corresponding homomorphisms by means of measurable sections of such bundles." } @article { Gutman19890323, author = "Gutman A.E.", title = "On the representation of lattice-normed spaces", journal = "Sib. Matem. Zh.", year = "1991", volume = "32", number = "2", pages = "41--54", language = "russian", annote = "The concept is studied of complete (ample) Banach bundle and a representation is constructed of an arbitrary lattice-normed space as a space of maximal (extended) sections of such bundles. Criteria are also obtained for an operator in section spaces to admit a multiplicative representation that is a generalization of the composition of a change of variable and multiplication by a scalar-valued function (the so-called weighted shift)." } @article { Gutman19890324, author = "Gutman A.E.", title = "On the realization of lattice-normed spaces", journal = "Sib. Math. J.", year = "1991", volume = "32", number = "2", pages = "210--221", doi = "10.1007/BF00972767", annote = "The concept is studied of complete (ample) Banach bundle and a representation is constructed of an arbitrary lattice-normed space as a space of maximal (extended) sections of such bundles. Criteria are also obtained for an operator in section spaces to admit a multiplicative representation that is a generalization of the composition of a change of variable and multiplication by a scalar-valued function (the so-called weighted shift)." } @phdthesis { Gutman19910516, author = "Gutman A.E.", school = "Sobolev Inst. Math., Novosibirsk", type = "Dissertation abstract; Ph.D. in mathematics: 01.01.01", howpublished = "Manuscript", title = "Representation of lattice-normed spaces and its applications", year = "1991", pages = "14", doi = "10.13140/RG.2.2.29671.34725", language = "russian", annote = "Abstract of a Dissertation for a degree of Ph.D. in mathematics, speciality 01.01.01: mathematical analysis. The dissertation is prepared at the Department of Analysis and Geometry of the Institute of Mathematics SB AS USSR. Scientific adviser: D.Sc. A.G.Kusraev. Official opponents: D.Sc., professor A.A.Tolstonogov, Ph.D., associate professor I.A.Shvedov. Lead organization: Leningrad State University. The presentation happened on June 19, 1991 within a session of Speciality Council K 002.23.02 at the Institute of Mathematics SB AS USSR, address: Universitetskij ave., 4, Novosibirsk, 630090. The abstract distributed on May 16, 1991. Academic secretary of the council: Ph.D. V.V.Ivanov." } @phdthesis { Gutman19910619, author = "Gutman A.E.", school = "Sobolev Inst. Math., Novosibirsk", type = "Dissertation; Ph.D. in mathematics: 01.01.01", howpublished = "Manuscript", title = "Representation of lattice-normed spaces and its applications", year = "1991", pages = "110", doi = "10.13140/RG.2.2.24218.75205", language = "russian", annote = "Dissertation for a degree of Ph.D. in mathematics, speciality 01.01.01: mathematical analysis. The dissertation is prepared at the Department of Analysis and Geometry of the Institute of Mathematics SB AS USSR. Scientific adviser: D.Sc. A.G.Kusraev. Official opponents: D.Sc., professor A.A.Tolstonogov, Ph.D., associate professor I.A.Shvedov. Lead organization: Leningrad State University. The presentation happened on June 19, 1991 within a session of Speciality Council K 002.23.02 at the Institute of Mathematics SB AS USSR, address: Universitetskij ave., 4, Novosibirsk, 630090. Academic secretary of the council: Ph.D. V.V.Ivanov." } @inproceedings { Gutman19910913, author = "Gutman A.E.", title = "Lifting in the space of measurable sections", booktitle = "XV All-USSR School on the theory of operators in function spaces (Nizhny Novgorod, September 13--20, 1991): Proceedings", address = "Nizhny Novgorod", year = "1991", pages = "63", language = "russian", annote = "It is shown that the space of classes of essentially bounded measurable functions acting into a Banach space $X$ admits a lifting if and only if the space $X$ is finitely-dimensional or the domain of definition is atomic. The minimal extension of a constant measurable Banach bundle is described which ensures existence of a lifting." } @article { Gutman19930401, author = "Gutman A.E.", title = "Banach bundles in the theory of lattice-normed spaces. I. Continuous Banach bundles", journal = "Siberian Adv. Math.", year = "1993", volume = "3", number = "3", pages = "1--55", annote = "Continuous Banach bundles over extremally disconnected compacta are considered. The notion of complete (ample) Banach bundle is introduced and discussed. The question is studied on representing lattice-normed spaces as spaces of continuous sections of Banach bundles.", keywords = "extremally disconnected compactum, vector lattice, lattice-normed space, continuous Banach bundle, continuous vector-functions" } @article { Gutman19930402, author = "Gutman A.E.", title = "Banach bundles in the theory of lattice-normed spaces. II. Measurable Banach bundles", journal = "Siberian Adv. Math.", year = "1993", volume = "3", number = "4", pages = "8--40", annote = "The notions of measurable Banach bundle and lifting in a quotient space of measurable sections are introduced and discussed. The question is studied of representing lattice-normed spaces as those of measurable sections of Banach bundles.", keywords = "measure space, extremally disconnected compactum, lifting, vector lattice, lattice-normed space, continuous and measurable Banach bundle" } @article { Gutman19930403, author = "Gutman A.E.", title = "Banach bundles in the theory of lattice-normed spaces. III. Approximating sets and bounded operators", journal = "Siberian Adv. Math.", year = "1994", volume = "4", number = "2", pages = "54--75", annote = "Two questions in the general theory of lattice-normed spaces (LNSs) are considered. First, the situation is studied when a subset of an LNS is order dense in the entire LNS; the notion of order approximation is introduced and described from various points of view. Second, the situation is studied when a linear operator from one LNS to another is order bounded; several different types of boundedness are introduced and studied in detail.", keywords = "vector lattice, lattice-normed space, order approximating set, order bounded linear operator" } @article { Gutman19941102, author = "Gutman A.E.", title = "Locally one-dimensional K-spaces and $\sigma$-distributive Boolean algebras", journal = "Siberian Adv. Math.", year = "1995", volume = "5", number = "1", pages = "42--48", annote = "It is known that all band preserving operators acting in a universally complete K-space are regular if and only if the K-space is locally one-dimensional. In addition, a K-space with base $B$ is locally one-dimensional if and only if $R^{\land}=R$ in $V^{(B)}$. It seems to have been unknown so far whether there exist nondiscrete locally one-dimensional K-spaces. In the present note we give a positive answer to the question. As an auxiliary result, we establish that a K-space is locally one-dimensional if and only if its base is $\sigma$-distributive.", keywords = "locally one-dimensional K-space, discrete K-space, $\sigma$-distributive Boolean algebra, $\sigma$-inductive Boolean algebra, atomic Boolean algebra, regular operator, real numbers in a Boolean-valued universe" } @article { Gutman19941204, author = "Gutman A.E.", title = "Banach bundles in the theory of lattice-normed spaces", booktitle = "Linear operators coordinated with order", address = "Novosibirsk", publisher = "IM SB RAS", journal = "Proc. Inst. Math. SB RAS", year = "1995", volume = "29", pages = "63--211", language = "russian", annote = "The material is divided into six chapters, each of which consists of several sections. Chapter 1 contains definitions and preliminary information on the basic objects in use. The majority of sections of Chapter 2 can also be regarded preliminary, they include general information on continuous Banach bundles. Chapter 3 is central both in position and content: it concentrates the material related to ample Banach bundles and representation of lattice-normed spaces as spaces of sections. In Chapter 4, the theory of measurable Banach bundles is developed, which is made by transferring Daniell's scheme to the case of sections. In the same chapter, the notion of lifting in the space of measurable sections is introduced and studied, and the results are presented of applying the theory of ample Banach bundles to the study of measurable bundles. Chapter 5 contains applications of the results of the preceding chapters to various spaces of continuous and measurable vector-valued functions. Finally, Chapter 6 is devoted to the study of disjointness preserving operators and constructing analytic representations of such operators." } @phdthesis { Gutman19950821, author = "Gutman A.E.", school = "Sobolev Inst. Math., Novosibirsk", type = "Dissertation abstract; D.Sc. in mathematics: 01.01.01", howpublished = "Manuscript", title = "Banach bundles in the theory of lattice-normed spaces", year = "1995", pages = "16", doi = "10.13140/RG.2.2.30929.63842", language = "russian", annote = "Abstract of a Dissertation for a degree of D.Sc in mathematics, speciality 01.01.01: mathematical analysis. The dissertation is prepared at the Sobolev Institute of Mathematics SB RAS. Official opponents: D.Sc., professor A.V.Bukhvalov, D.Sc., professor S.K.Vodopyanov, D.Sc., professor S.P.Gulko. Lead organization: Nizhny Novgorod State University. The presentation happened on September 28, 1995 within a session of Dissertation Council D 002.23.02 for degrees of D.Sc in mathematics at the Sobolev Institute of Mathematics SB RAS, address: Universitetskij ave., 4, Novosibirsk, 630090. The abstract distributed on August 21, 1995. Academic secretary of the Dissertation Council: D.Sc. V.A.Sharafutdinov." } @phdthesis { Gutman19950928, author = "Gutman A.E.", school = "Sobolev Inst. Math., Novosibirsk", type = "Dissertation; D.Sc. in mathematics: 01.01.01", howpublished = "Manuscript", title = "Banach bundles in the theory of lattice-normed spaces", year = "1995", pages = "312", doi = "10.13140/RG.2.2.31670.19522", language = "russian", annote = "Dissertation for a degree of D.Sc in mathematics, speciality 01.01.01: mathematical analysis. The dissertation is prepared at the Sobolev Institute of Mathematics SB RAS. Official opponents: D.Sc., professor A.V.Bukhvalov, D.Sc., professor S.K.Vodopyanov, D.Sc., professor S.P.Gulko. Lead organization: Nizhny Novgorod State University. The presentation happened on September 28, 1995 within a session of Dissertation Council D 002.23.02 for degrees of D.Sc in mathematics at the Sobolev Institute of Mathematics SB RAS, address: Universitetskij ave., 4, Novosibirsk, 630090. Academic secretary of the Dissertation Council: D.Sc. V.A.Sharafutdinov." } @article { Gutman19941101, author = "Gutman A.E.", title = "Banach bundles in the theory of lattice-normed spaces. IV. Disjointness preserving operators", journal = "Siberian Adv. Math.", year = "1996", volume = "6", number = "2", pages = "35--102", annote = "In the present article, we study disjointness preserving operators that act in K-spaces and lattice-normed spaces. In particular, we find their analytic representations and decompositions into simpler components. We study orthomorphisms, shift operators, weighted shift operators, and arbitrary disjointness preserving operators.", keywords = "vector lattice, lattice-normed space, continuous Banach bundle, continuous section, orthomorphism, shift operator, weighted shift operator, disjointness preserving operator, analytic representation" } @book { Gutman19960101, author = "Bukhvalov A.V. and Gutman A.E. and Korotkov V.B. and Kusraev A.G. and Kutateladze S.S. and Makarov B.M.", title = "Vector lattices and integral operators", address = "Dordrecht", publisher = "Kluwer", year = "1996", pages = "ix+462", isbn = "978-94-010-6571-9", doi = "10.1007/978-94-009-0195-7", annote = "This volume is devoted to modern accomplishments in the field of vector lattices and integral operators which were achieved in Russia during the last two decades. Nonstandard methods are eleborated for the analysis of vector lattices and positive operators. Much attention is paid to studying stability under multiplication by an arbitrary bounded operator for various classes of operators which are defined in terms of order. Also, several approaches are treated to the solution of the J. von Neumann problem on the conditions for integrality of a linear operator, and full information is given on the solution of some problems posed by P.Halmos and V.Sunder. This book is intended for mathematicians, students and postgraduates interested in functional analysis, operator theory, geometry of Banach spaces and vector lattices." } @inbook { Gutman19960102, author = "Gutman A.E.", chapter = "5", title = "Disjointness preserving operators", booktitle = "Vector lattices and integral operators", address = "Dordrecht", publisher = "Kluwer", year = "1996", pages = "360--454", doi = "10.1007/978-94-009-0195-7_5", annote = "In this chapter, we study disjointness preserving operators in K-spaces and lattice-normed spaces. In particular, we find their analytic representations and decompositions into simpler components. We begin with studying general properties of disjointness preserving operators; then we consider orthomorphisms, shift operators, weighted shift operators, and, finally, return to arbitrary operators and apply the accumulated experience." } @inproceedings { Gutman19940725, author = "Gutman A.E.", title = "Locally one-dimensional K-spaces and $\sigma$-distributive Boolean algebras", booktitle = "Siberian Conference on Applied and Industrial Mathematics (Novosibirsk, July 25--29, 1994): Proceedings", address = "Novosibirsk", publisher = "IM SB RAS", year = "1997", volume = "1", pages = "103--108", language = "russian", annote = "It is known that all band preserving operators acting in a universally complete K-space are regular if and only if the K-space is locally one-dimensional. In addition, a K-space with base $B$ is locally one-dimensional if and only if $R^{\land}=R$ in $V^{(B)}$. It seems to have been unknown so far whether there exist nondiscrete locally one-dimensional K-spaces. In the present note we give a positive answer to the question. As an auxiliary result, we establish that a K-space is locally one-dimensional if and only if its base is $\sigma$-distributive." } @article { Gutman19950130, author = "Gutman A.E.", title = "Locally one-dimensional K-spaces", journal = "Proc. Acad. Sci.", year = "1997", volume = "353", number = "5", pages = "590--591", language = "russian", annote = "It is known that all band preserving operators acting in a universally complete K-space are regular if and only if the K-space is locally one-dimensional. In addition, a K-space with base $B$ is locally one-dimensional if and only if $R^{\land}=R$ in $V^{(B)}$. It seems to have been unknown so far whether there exist nondiscrete locally one-dimensional K-spaces. In the present note we give a positive answer to the question. As an auxiliary result, we establish that a K-space is locally one-dimensional if and only if its base is $\sigma$-distributive." } @article { Gutman19950131, author = "Gutman A.E.", title = "Locally one-dimensional complete vector lattices", journal = "Doklady Math.", year = "1997", volume = "55", number = "2", pages = "240--241", annote = "It is known that all band preserving operators acting in a universally complete K-space are regular if and only if the K-space is locally one-dimensional. In addition, a K-space with base $B$ is locally one-dimensional if and only if $R^{\land}=R$ in $V^{(B)}$. It seems to have been unknown so far whether there exist nondiscrete locally one-dimensional K-spaces. In the present note we give a positive answer to the question. As an auxiliary result, we establish that a K-space is locally one-dimensional if and only if its base is $\sigma$-distributive." } @article { Gutman19971101, author = "Gutman A.E. and Losenkov G.A.", title = "Function representation of the Boolean-valued universe", journal = "Matem. tr.", year = "1998", volume = "1", number = "1", pages = "54--77", language = "russian", annote = "For an abstract Boolean-valued system, a function analog is proposed that is a model whose elements are functions and the basic logical operations are calculated ``pointwise.'' The new notion of continuous polyverse is introduced and studied which is a continuous bundle of models of set theory. It is shown that the class of continuous sections of a continuous polyverse is a Boolean-valued system satisfying all basic principles of Boolean-valued analysis and, conversely, every Boolean-valued algebraic system can be represented as the class of sections of a suitable continuous polyverse.", keywords = "Boolean-valued analysis, function representation, Stone space, continuous bundle, continuous section" } @article { Gutman19971102, author = "Gutman A.E. and Losenkov G.A.", title = "Function representation of the Boolean-valued universe", journal = "Siberian Adv. Math.", year = "1998", volume = "8", number = "1", pages = "99--120", annote = "For an abstract Boolean-valued system, a function analog is proposed that is a model whose elements are functions and the basic logical operations are calculated ``pointwise.'' The new notion of continuous polyverse is introduced and studied which is a continuous bundle of models of set theory. It is shown that the class of continuous sections of a continuous polyverse is a Boolean-valued system satisfying all basic principles of Boolean-valued analysis and, conversely, every Boolean-valued algebraic system can be represented as the class of sections of a suitable continuous polyverse.", keywords = "Boolean-valued analysis, function representation, Stone space, continuous bundle, continuous section" } @article { Gutman19981117, author = "Gutman A.E. and Koptev A.V.", title = "On the notion of the dual of a Banach bundle", journal = "Matem. tr.", year = "1999", volume = "2", number = "1", pages = "8--71", language = "russian", annote = "For an arbitrary continuous Banach bundle over an arbitrary topological space, the notion of a dual bundle is introduced and studied. Various necessary and sufficient conditions are presented for existence of a dual bundle, duality relations are examined between a bundle and its dual, an embedding of a bundle into its second dual is constructed, the notion of a weakly continuous section is introduced and studied. The questions under consideration are treated in the general case as well as for concrete classes of bundles and topological spaces. The results obtained are supplied with examples that justify accuracy of the results.", keywords = "continuous Banach bundle, dual Banach bundle, duality, homomorphism of Banach bundles, continuous section, weakly continuous section" } @article { Gutman19990210, author = "Gutman A.E. and Koptev A.V.", title = "On the notion of the dual of a Banach bundle", journal = "Siberian Adv. Math.", year = "1999", volume = "9", number = "1", pages = "46--98", annote = "For an arbitrary continuous Banach bundle over an arbitrary topological space, the notion of a dual bundle is introduced and studied. Various necessary and sufficient conditions are presented for existence of a dual bundle, duality relations are examined between a bundle and its dual, an embedding of a bundle into its second dual is constructed, the notion of a weakly continuous section is introduced and studied. The questions under consideration are treated in the general case as well as for concrete classes of bundles and topological spaces. The results obtained are supplied with examples that justify accuracy of the results.", keywords = "continuous Banach bundle, dual Banach bundle, duality, homomorphism of Banach bundles, continuous section, weakly continuous section" } @book { Gutman19991214, author = "Gutman A.E. and Emelyanov E.Yu. and Kusraev A.G. and Kutateladze S.S.", title = "Nonstandard analysis and vector lattices", address = "Novosibirsk", publisher = "Institute of Mathematics", year = "1999", pages = "x+380", isbn = "5-86134-068-4", language = "russian", annote = "The book is devoted to applications of nonstandard methods of analysis to the theory of vector lattices. Much attention is paid to the problem of combining infinitesimal and Boolean-valued concepts to studying classical problems of the theory of vector lattices related to constructing concrete representations of abstract functional-analytic objects: Banach--Kantorovich spaces, dominated operators, vector measures, integral operators, etc. This book is intended for those interested in modern applications of nonstandard analysis to problems of functional analysis." } @inbook { Gutman19991215, author = "Gutman A.E. and Losenkov G.A.", chapter = "2", title = "Function representation of a Boolean valued universe", booktitle = "Nonstandard analysis and vector lattices", address = "Novosibirsk", publisher = "Institute of Mathematics", year = "1999", pages = "97--125", language = "russian", annote = "Contemporary methods of Boolean-valued analysis, due to their nature, involve rather bulky logical technique. We can say that, from a pragmatic viewpoint, this technique might distract the user-analyst from a concrete aim: to apply the results of Boolean-valued analysis for solving analytical problems. Various function spaces are common in functional analysis, and so the intention is natural of replacing an abstract Boolean-valued system by some function analog, a model whose elements are functions and in which the basic logical operations are calculated ``pointwise.'' In the present chapter, a solution is proposed to the above problem. To this end, we introduce and study the new notion of continuous polyverse, the latter being a continuous bundle of models of set theory. It is shown that the class of continuous sections of a continuous polyverse is a Boolean-valued system satisfying all basic principles of Boolean-valued analysis and, conversely, every Boolean-valued algebraic system can be represented as the class of sections of a suitable continuous polyverse." } @inbook { Gutman19991216, author = "Gutman A.E. and Koptev A.V.", chapter = "3", title = "Dual Banach bundles", booktitle = "Nonstandard analysis and vector lattices", address = "Novosibirsk", publisher = "Institute of Mathematics", year = "1999", pages = "127--202", language = "russian", annote = "The solution of the problem of defining and existence of a dual continuous Banach bundle (CBB) for the case of ample bundles over extremally disconnected compact spaces is essentially based on specific properties of ample bundles and extremally disconnected compact spaces and, thus, cannot be extended to a wider class of bundles. The natural intention to extend the domain of applications for the duality theory leads to the problem of constructing a dual CBB for an arbitrary Banach bundle over an arbitrary topological space. The study of this problem is the main subject of the present chapter, where, in particular, a definition of a dual bundle is presented, with the above-formulated requirements fulfilled, and a number of necessary and sufficient conditions is suggested for existence of a dual bundle." } @book { Gutman20000410, author = "Gutman A.E. and Emelyanov E.Yu. and Kusraev A.G. and Kutateladze S.S.", title = "Nonstandard analysis and vector lattices", address = "Dordrecht", publisher = "Kluwer Academic Publishers", year = "2000", pages = "xii+307", isbn = "978-94-010-5863-6", doi = "10.1007/978-94-011-4305-9", annote = "The book is devoted to applications of nonstandard methods of analysis to the theory of vector lattices. Much attention is paid to the problem of combining infinitesimal and Boolean-valued concepts to studying classical problems of the theory of vector lattices related to constructing concrete representations of abstract functional-analytic objects: Banach--Kantorovich spaces, dominated operators, vector measures, integral operators, etc. This book is intended for those interested in modern applications of nonstandard analysis to problems of functional analysis." } @inbook { Gutman20000411, author = "Gutman A.E. and Losenkov G.A.", chapter = "2", title = "Function representation of a Boolean valued universe", booktitle = "Nonstandard analysis and vector lattices", address = "Dordrecht", publisher = "Kluwer Academic Publishers", year = "2000", pages = "81--104", doi = "10.1007/978-94-011-4305-9_2", annote = "Contemporary methods of Boolean-valued analysis, due to their nature, involve rather bulky logical technique. We can say that, from a pragmatic viewpoint, this technique might distract the user-analyst from a concrete aim: to apply the results of Boolean-valued analysis for solving analytical problems. Various function spaces are common in functional analysis, and so the intention is natural of replacing an abstract Boolean-valued system by some function analog, a model whose elements are functions and in which the basic logical operations are calculated ``pointwise.'' In the present chapter, a solution is proposed to the above problem. To this end, we introduce and study the new notion of continuous polyverse, the latter being a continuous bundle of models of set theory. It is shown that the class of continuous sections of a continuous polyverse is a Boolean-valued system satisfying all basic principles of Boolean-valued analysis and, conversely, every Boolean-valued algebraic system can be represented as the class of sections of a suitable continuous polyverse." } @inbook { Gutman20000412, author = "Gutman A.E. and Koptev A.V.", chapter = "3", title = "Dual Banach bundles", booktitle = "Nonstandard analysis and vector lattices", address = "Dordrecht", publisher = "Kluwer Academic Publishers", year = "2000", pages = "105--159", doi = "10.1007/978-94-011-4305-9_3", annote = "The solution of the problem of defining and existence of a dual continuous Banach bundle (CBB) for the case of ample bundles over extremally disconnected compact spaces is essentially based on specific properties of ample bundles and extremally disconnected compact spaces and, thus, cannot be extended to a wider class of bundles. The natural intention to extend the domain of applications for the duality theory leads to the problem of constructing a dual CBB for an arbitrary Banach bundle over an arbitrary topological space. The study of this problem is the main subject of the present chapter, where, in particular, a definition of a dual bundle is presented, with the above-formulated requirements fulfilled, and a number of necessary and sufficient conditions is suggested for existence of a dual bundle." } @book { Gutman19990919, author = "Gutman A.E. and Kolesnikov A.S.", note = "Textbook", title = "Banach--Kantorovich lattices", address = "Novosibirsk", publisher = "Novosib. State Univ.", year = "2000", pages = "80", language = "russian", annote = "The paper continues the study related to lattice-normed spaces (LNSs) and their representations as spaces of sections. First of all, we speak of the notion of lattice-ordered LNS, i.e., an LNS endowed with an order which makes the LNS into a vector lattice and its norm, into a monotone function (with respect to the absolute value). Examples of such spaces include spaces of continuous, weakly continuous, measurable, weakly measurable, and Bochner-summable functions with values in Banach lattices, spaces of order-bounded lattice-valued functions, spaces of vector-valued measures of bounded variation with values in a Banach lattice, spaces of lattice-valued measures, as well as spaces of continuous and measurable sections of bundles of Banach lattices which are studied in the paper. As an application, the problem is considered of analytic representation of the conditional expectation operator." } @book { Gutman19990920, author = "Gutman A.E. and Feofanov D.S.", note = "Textbook", title = "Analytic description of principal operator bands", address = "Novosibirsk", publisher = "Novosib. State Univ.", year = "2000", pages = "31", language = "russian", annote = "In the paper, principal bands are studied in the spaces of operators acting in vector lattices and lattice-normed spaces. The key attention is paid to the bands generated by disjointness preserving operators. The main results are criteria for an operator to belong to the band generated by an operator in question. Every criterion provides a special analytic representation of the band under consideration." } @article { Gutman20010219, author = "Gutman A.E. and Ryabko D.B.", title = "The nonstandard hull of a normed space in a Boolean-valued universe", journal = "Matem. tr.", year = "2001", volume = "4", number = "2", pages = "42--52", language = "russian", annote = "In this article, we extend some results of infinitesimal analysis on normed spaces and the field of reals to the functional representation of a Boolean-valued universe. In particular, we prove equivalence of the following three conditions for an arbitrary polyverse over $Q$: a point $q\in Q$ is not $\sigma$-isolated; the stalk of the polyverse at $q$ is countably saturated; and the nonstandard hull of every normed space in the stalk of the polyverse at $q$ is complete.", keywords = "Boolean-valued analysis, infinitesimal analysis, nonstandard analysis, polyverse, nonstandard hull" } @article { Gutman20010220, author = "Gutman A.E. and Ryabko D.B.", title = "The nonstandard hull of a normed space in a Boolean-valued universe", journal = "Siberian Adv. Math.", year = "2002", volume = "12", number = "2", pages = "38--47", annote = "In this article, we extend some results of infinitesimal analysis on normed spaces and the field of reals to the functional representation of a Boolean-valued universe. In particular, we prove equivalence of the following three conditions for an arbitrary polyverse over $Q$: a point $q\in Q$ is not $\sigma$-isolated; the stalk of the polyverse at $q$ is countably saturated; and the nonstandard hull of every normed space in the stalk of the polyverse at $q$ is complete.", keywords = "Boolean-valued analysis, infinitesimal analysis, nonstandard analysis, polyverse, nonstandard hull" } @article { Gutman20011224, author = "Gutman A.E. and Ryabko D.B.", title = "Completeness criterion for the nonstandard hull of a normed space in a Boolean-valued universe", journal = "Proc. Acad. Sci.", year = "2002", volume = "384", number = "2", pages = "153--155", language = "russian", annote = "In this article, we extend some results of infinitesimal analysis on normed spaces and the field of reals to the functional representation of a Boolean-valued universe. In particular, we prove equivalence of the following three conditions for an arbitrary polyverse over $Q$: a point $q\in Q$ is not $\sigma$-isolated; the stalk of the polyverse at $q$ is countably saturated; and the nonstandard hull of every normed space in the stalk of the polyverse at $q$ is complete." } @article { Gutman20011225, author = "Gutman A.E. and Ryabko D.B.", title = "Completeness criterion for the nonstandard hull of a normed space in a Boolean-valued universe", journal = "Doklady Math.", year = "2002", volume = "65", number = "3", pages = "337--338", annote = "In this article, we extend some results of infinitesimal analysis on normed spaces and the field of reals to the functional representation of a Boolean-valued universe. In particular, we prove equivalence of the following three conditions for an arbitrary polyverse over $Q$: a point $q\in Q$ is not $\sigma$-isolated; the stalk of the polyverse at $q$ is countably saturated; and the nonstandard hull of every normed space in the stalk of the polyverse at $q$ is complete." } @article { Gutman20020204, author = "Gutman A.E. and Ryabko D.B.", title = "Function representation of Kantorovich spaces by means of a Boolean-valued models", journal = "Vladikavk. Math. J.", year = "2002", volume = "4", number = "1", pages = "34--49", language = "russian", annote = "In the paper, the notion is introduced of outer section of a polyverse (a function representation of a Boolean-valued universe) and a new function representation is obtained of K-spaces and vector lattices as outer sections. In particular, an isomorphism is constructed between an arbitrary vector lattice and an outer subset of the field of reals in the corresponding Boolean-valued universe. Within the new function representation, analogs are discovered of the main motions and facts of the theory of vector lattices. It is also clarified which of the properties of K-spaces under consideration admit ``pointwise criteria.''" } @inproceedings { Gutman20020518, author = "Gutman A.E. and Ryabko D.B.", howpublished = "Electronic", title = "Functional representation of a Dedekind-complete Reisz space in a Boolean valued universe", booktitle = "The 4th Conference on Function Spaces at SIUE (USA, May 14--19, 2002): Proceedings", address = "USA, Illinois, Edwardsville", year = "2002", pages = "16", annote = "Using a functional model of a Boolean-valued universe, the basic notions are introduced of infinitesimal analysis within a Boolean-valued universe, thus enriching the synthesis of the two main branches of nonstandard analysis, infinitesimal and Boolean-valued. The results obtained are applied to the theory of Riesz spaces." } @inproceedings { Gutman20030923, author = "Ageev A.N. and Gutman A.E.", title = "Artificial intelligence assets and business management", booktitle = "Informatics and telecommunication problems. International Scientific and Technical Conference (Novosibirsk, September 23--24, 2003): Proceedings", address = "Novosibirsk", publisher = "Siberian State University of Telecommunications and Information Sciences", year = "2003", pages = "167--170", language = "russian", annote = "Disclaimer: The text of the report is written solely by the first author. (The second author is included only because he designed and developed the MAM program mentioned in the report.)" } @article { Gutman20031114, author = "Gutman A.E. and Sotnikov A.I.", title = "On order complete sigma-algebras", journal = "Sci. Proc. TyvSU. Kyzyl: TyvSU", year = "2003", volume = "1", pages = "83--86", language = "russian", annote = "A criterion is obtained for existence of a $\sigma$-algebra $\Sigma$ of subsets of a given set $X$ such that $\Sigma$ is order complete and atomic, but is not discrete, i.e., $\Sigma$ does not coincide with the totality of the unions of various subsets of some partition of $X$." } @article { Gutman20031015, author = "Gutman A.E. and Sotnikov A.I.", title = "Order properties of the space of finitely additive transition functions", journal = "Sib. Matem. Zh.", year = "2004", volume = "45", number = "1", pages = "80--102", language = "russian", annote = "The basic order properties, as well as some metric and algebraic properties, are studied of the set of finitely additive transition functions on an arbitrary measurable space, as endowed with the structure of an ordered normed algebra, and some connections are revealed with the classical spaces of linear operators, vector measures, and measurable vector-valued functions. In particular, the question is examined of splitting the space of transition functions into the sum of the subspaces of countably additive and purely finitely additive transition functions.", keywords = "transition function, purely finitely additive measure, lifting of a measure space, vector measure, measurable vector-valued function, ordered vector space, vector lattice, Riesz space, K-space, Banach lattice, ordered Banach algebra" } @article { Gutman20031016, author = "Gutman A.E. and Sotnikov A.I.", title = "Order properties of the space of finitely additive transition functions", journal = "Sib. Math. J.", year = "2004", volume = "45", number = "1", pages = "69--85", doi = "10.1023/B:SIMJ.0000013013.03647.65", annote = "The basic order properties, as well as some metric and algebraic properties, are studied of the set of finitely additive transition functions on an arbitrary measurable space, as endowed with the structure of an ordered normed algebra, and some connections are revealed with the classical spaces of linear operators, vector measures, and measurable vector-valued functions. In particular, the question is examined of splitting the space of transition functions into the sum of the subspaces of countably additive and purely finitely additive transition functions.", keywords = "transition function, purely finitely additive measure, lifting of a measure space, vector measure, measurable vector-valued function, ordered vector space, vector lattice, Riesz space, K-space, Banach lattice, ordered Banach algebra" } @article { Gutman20040812, author = "Gutman A.E. and Feofanov D.S.", title = "Description of principal bands generated by disjointness preserving operators", journal = "Vladikavk. Math. J.", year = "2004", volume = "6", number = "3", pages = "26--35", language = "russian", annote = "In the paper, principal bands are studied in the spaces of operators acting in vector lattices and lattice-normed spaces. The key attention is paid to the bands generated by disjointness preserving operators. The main results are criteria for an operator to belong to the band generated by an operator in question. Every criterion provides a special analytic representation of the band under consideration." } @article { Gutman20050202, author = "Gutman A.E. and Koptev A.V. and Popov A.I.", title = "Finite representability in stalks of ample Banach bundles", journal = "Vladikavk. Math. J.", year = "2005", volume = "7", number = "1", pages = "39--45", language = "russian", annote = "It is shown that the stalks of ample Banach bundles inherit (and strengthen in some cases) finite representability of a normed space in ``adjacent'' stalks. Each of the facts established in the paper can be regarded as an analog of the corresponding property of ultraproducts of Banach spaces.", keywords = "local theory of Banach spaces, ultraproduct of Banach spaces, finite representability of a normed space, ample continuous Banach bundle" } @book { Gutman20050905, author = "Gutman A.E. and Emelyanov E.Yu. and Koptev A.V. and Kusraev A.G. and Kutateladze S.S. and Malyugin S.A.", title = "Nonstandard analysis and vector lattices. 2nd ed., corr. and enl.", address = "Novosibirsk", publisher = "Institute of Mathematics", year = "2005", pages = "x+400", isbn = "5-86134-127-3", language = "russian", annote = "The book is devoted to applications of nonstandard methods of analysis to the theory of vector lattices. Much attention is paid to the problem of combining infinitesimal and Boolean-valued concepts to studying classical problems of the theory of vector lattices related to constructing concrete representations of abstract functional-analytic objects: Banach--Kantorovich spaces, dominated operators, vector measures, integral operators, etc. This book is intended for those interested in modern applications of nonstandard analysis to problems of functional analysis." } @inbook { Gutman20050906, author = "Gutman A.E. and Losenkov G.A.", chapter = "2", title = "Function representation of a Boolean valued universe", booktitle = "Nonstandard analysis and vector lattices. 2nd ed., corr. and enl.", address = "Novosibirsk", publisher = "Institute of Mathematics", year = "2005", pages = "95--123", language = "russian", annote = "Contemporary methods of Boolean-valued analysis, due to their nature, involve rather bulky logical technique. We can say that, from a pragmatic viewpoint, this technique might distract the user-analyst from a concrete aim: to apply the results of Boolean-valued analysis for solving analytical problems. Various function spaces are common in functional analysis, and so the intention is natural of replacing an abstract Boolean-valued system by some function analog, a model whose elements are functions and in which the basic logical operations are calculated ``pointwise.'' In the present chapter, a solution is proposed to the above problem. To this end, we introduce and study the new notion of continuous polyverse, the latter being a continuous bundle of models of set theory. It is shown that the class of continuous sections of a continuous polyverse is a Boolean-valued system satisfying all basic principles of Boolean-valued analysis and, conversely, every Boolean-valued algebraic system can be represented as the class of sections of a suitable continuous polyverse." } @inbook { Gutman20050907, author = "Gutman A.E. and Koptev A.V.", chapter = "3", title = "Dual Banach bundles", booktitle = "Nonstandard analysis and vector lattices. 2nd ed., corr. and enl.", address = "Novosibirsk", publisher = "Institute of Mathematics", year = "2005", pages = "125--201", language = "russian", annote = "The solution of the problem of defining and existence of a dual continuous Banach bundle (CBB) for the case of ample bundles over extremally disconnected compact spaces is essentially based on specific properties of ample bundles and extremally disconnected compact spaces and, thus, cannot be extended to a wider class of bundles. The natural intention to extend the domain of applications for the duality theory leads to the problem of constructing a dual CBB for an arbitrary Banach bundle over an arbitrary topological space. The study of this problem is the main subject of the present chapter, where, in particular, a definition of a dual bundle is presented, with the above-formulated requirements fulfilled, and a number of necessary and sufficient conditions is suggested for existence of a dual bundle." } @article { Gutman20051003, author = "Gutman A.E. and Kusraev A.G. and Reshetnyak Yu.G.", howpublished = "Electronic", title = "On the $n$th birthday of Semen Samsonovich Kutateladze for $n=60$", journal = "Sib. Electron. Math. Rep.", year = "2005", volume = "2", pages = "A.12--A.33", language = "russian", annote = "An address to professor S.S.Kutateladze on the occasion of his 60th birthday, which includes a list of his publications containing 393 bibliographic records. Prof. S.S.Kutateladze is known due to his contribution to nonstandard analysis, the theory of ordered vector spaces, subdifferential calculus, and other topics of functional analysis." } @article { Gutman20070820, author = "Gutman A.E. and Koptev A.V.", title = "Spaces of $CD_0$-sections and $CD_0$-homomorphisms of Banach bundles", journal = "Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform.", year = "2007", volume = "7", number = "4", pages = "27--48", language = "russian", annote = "In the paper, the space $CD_0(Q,X)=C(Q,X)+c_0(Q,X)$ is considered whose elements are the sums of continuous and discrete sections of a Banach bundle $X$ over a compact Hausdorff space $Q$ without isolated points. As is known, this space is isometric to the space $C(Q^{\bullet},X^{\bullet})$ of continuous sections of some Banach bundle $X^{\bullet}$ over the compact space $Q^{\bullet}=Q\times\{0,1\}$ (with a special topology). We clarify the connection between $X$ and $X^{\bullet}$ on the example of subbundles as well as bundles obtained by continuous change of variables and by restriction onto a topological subspace. In addition, we introduce and study the space $CD_0[X,Y]$ of $CD_0$-homomorphisms of Banach bundles $X$ and $Y$ and show that is possesses some properties analogous to those of the space of $CD_0$-sections.", keywords = "continuous Banach bundle, section of a Banach bundle, Banach $C(Q)$-module, homomorphism of Banach bundles, homomorphism of $C(Q)$-modules" } @inproceedings { Gutman20070920, author = "Gutman A.E. and Koptev A.V.", howpublished = "Electronic", title = "The space of $CD_0$-sections of a Banach bundle", booktitle = "Mathematics in the Modern World. The Russian Conference Dedicated to the 50th Anniversary of the Sobolev Institute of Mathematics (Novosibirsk, September 17--23, 2007): Proceedings", address = "Novosibirsk", publisher = "Sobolev Institute of Mathematics SB RAS", year = "2007", pages = "61--62", language = "russian", annote = "The space $CD_0(Q)=C(Q)+c_0(Q)$ introduced as an example of a Banach lattice with some unusual order-topological properties became the subject of further intensive study. One of the results of the study was the representation of $CD_0(Q)$ as the space $C(Q^{\bullet})$ of continuous functions on the set $Q^{\bullet}=Q\times\{0,1\}$ endowed with a special compact Hausdorff topology. In this paper, we extend the result to the case of sections of Banach bundles." } @article { Gutman20070926, author = "Gutman A.E. and Koptev A.V.", title = "Spaces of $CD_0$-functions and Alexandroff duplicate", journal = "Vladikavk. Math. J.", year = "2007", volume = "9", number = "3", pages = "11--21", language = "russian", annote = "We first briefly expose some crucial phases in studying the space $CD_0(Q)=C(Q)+c_0(Q)$ whose elements are the sums of continuous and ``discrete'' functions defined on a compact Hausdorff space $Q$ without isolated points. In this part, special emphasis is on describing the compact space $Q^{\bullet}$ representing the Banach lattice $CD_0(Q)$ as $C(Q^{\bullet})$. In addition, a large fragment of the article is dedicated to the analogous frame related to the space $CD_0(Q,X)$ of ``continuous-discrete'' sections of a Banach bundle $X$ and the space of $CD_0$-homomorphisms of Banach bundles.", keywords = "Banach lattice, AM-space, Alexandroff duplicate, continuous Banach bundle, section of a Banach bundle, Banach $C(Q)$-module, homomorphism of Banach bundles, homomorphism of $C(Q)$-modules" } @booklet { Gutman20071210, author = "Gutman A.E. and Kusraev A.G. and Kutateladze S.S.", note = "Preprint / IAMI VSC RAS; N 3", title = "The Wickstead problem", address = "Vladikavkaz", year = "2007", pages = "44", doi = "10.13140/RG.2.2.22573.79849", annote = "In 1977 Anthony Wickstead raised the question of the conditions for all band preserving linear operators to be order bounded in a vector lattice. This article overviews the main ideas and results on the Wickstead problem and its variations, focusing primarily on the case of band preserving operators in a universally complete vector lattice.", keywords = "band preserving operator, universally complete vector lattice, $\sigma$-distributive Boolean algebra, local Hamel basis, transcendence basis, derivation, Boolean valued representation" } @booklet { Gutman20071214, author = "Gutman A.E. and Kusraev A.G. and Kutateladze S.S.", note = "Electronic preprint / arXiv:0712.2378 [math.FA]", howpublished = "Electronic", title = "The Wickstead problem", address = "arXiv.org", year = "2007", pages = "44", annote = "In 1977 Anthony Wickstead raised the question of the conditions for all band preserving linear operators to be order bounded in a vector lattice. This article overviews the main ideas and results on the Wickstead problem and its variations, focusing primarily on the case of band preserving operators in a universally complete vector lattice.", keywords = "band preserving operator, universally complete vector lattice, $\sigma$-distributive Boolean algebra, local Hamel basis, transcendence basis, derivation, Boolean valued representation" } @article { Gutman20080213, author = "Gutman A.E. and Kusraev A.G. and Kutateladze S.S.", howpublished = "Electronic", title = "The Wickstead problem", journal = "Sib. Electron. Math. Rep.", year = "2008", volume = "5", pages = "293--333", annote = "In 1977 Anthony Wickstead raised the question of the conditions for all band preserving linear operators to be order bounded in a vector lattice. This article overviews the main ideas and results on the Wickstead problem and its variations, focusing primarily on the case of band preserving operators in a universally complete vector lattice.", keywords = "band preserving operator, universally complete vector lattice, $\sigma$-distributive Boolean algebra, local Hamel basis, transcendence basis, derivation, Boolean valued representation" } @article { Gutman20080618, author = "Gutman A.E. and Kutateladze S.S.", title = "On the theory of grossone", journal = "Sib. Matem. Zh.", year = "2008", volume = "49", number = "5", pages = "1054--1063", language = "russian", annote = "A trivial formalization is given for the informal reasonings presented in a series of papers by Ya.D.Sergeyev on a positional numeral system with an infinitely large base, grossone; the system which is groundlessly opposed by its originator to the classical nonstandard analysis.", keywords = "nonstandard analysis, infinitesimal analysis, positional numeral system" } @article { Gutman20080619, author = "Gutman A.E. and Kutateladze S.S.", title = "On the theory of grossone", journal = "Sib. Math. J.", year = "2008", volume = "49", number = "5", pages = "835--841", doi = "10.1007/s11202-008-0082-0", annote = "A trivial formalization is given for the informal reasonings presented in a series of papers by Ya.D.Sergeyev on a positional numeral system with an infinitely large base, grossone; the system which is groundlessly opposed by its originator to the classical nonstandard analysis.", keywords = "nonstandard analysis, infinitesimal analysis, positional numeral system" } @booklet { Gutman20080808, author = "Gutman A.E. and Kutateladze S.S.", note = "Electronic preprint / arXiv:0808.1164 [math.GM]", howpublished = "Electronic", title = "A trivial formalization of the theory of grossone", address = "arXiv.org", year = "2008", pages = "6", annote = "A trivial formalization is given for the informal reasonings presented in a series of papers by Ya.D.Sergeyev on a positional numeral system with an infinitely large base, grossone; the system which is groundlessly opposed by its originator to the classical nonstandard analysis.", keywords = "nonstandard analysis, infinitesimal analysis, positional numeral system" } @article { Gutman20081118, author = "Gutman A.E. and Koptev A.V.", howpublished = "Electronic", title = "Spaces of $CD_0$-functions and $CD_0$-sections of Banach bundles", journal = "Sib. Electron. Math. Rep.", year = "2009", volume = "6", pages = "219--242", annote = "We first briefly expose some crucial phases in studying the space $CD_0(Q)=C(Q)+c_0(Q)$ whose elements are the sums of continuous and ``discrete'' functions defined on a compact Hausdorff space $Q$ without isolated points. (In this part, special emphasis is on describing the compact space $Q^{\bullet}$ representing the Banach lattice $CD_0(Q)$ as $C(Q^{\bullet})$.) The rest of the article is dedicated to the analogous frame related to the space $CD_0(Q,X)$ of ``continuous-discrete'' sections of a Banach bundle $X$ and the space of $CD_0$-homomorphisms of Banach bundles.", keywords = "Banach lattice, AM-space, Alexandroff duplicate, continuous Banach bundle, section of a Banach bundle, Banach $C(Q)$-module, homomorphism of Banach bundles, homomorphism of $C(Q)$-modules" } @article { Gutman20090602, author = "Gutman A.E. and Lisovskaya S.A.", title = "The boundedness principle for lattice-normed spaces", journal = "Sib. Matem. Zh.", year = "2009", volume = "50", number = "5", pages = "1050--1059", language = "russian", annote = "Three classical facts of the theory of normed spaces are considered: the boundedness principle, the Banach--Steinhaus theorem, and the uniform boundedness principle for a compact convex set. By means of Boolean-valued analysis, exact analogs of the theorems are proven for the case of lattice-normed spaces.", keywords = "Banach--Steinhaus theorem, Banach--Kantorovich space, cyclically compact set, Boolean-valued analysis" } @article { Gutman20090603, author = "Gutman A.E. and Lisovskaya S.A.", title = "The boundedness principle for lattice-normed spaces", journal = "Sib. Math. J.", year = "2009", volume = "50", number = "5", pages = "830--837", doi = "10.1007/s11202-009-0093-5", annote = "Three classical facts of the theory of normed spaces are considered: the boundedness principle, the Banach--Steinhaus theorem, and the uniform boundedness principle for a compact convex set. By means of Boolean valued analysis, exact analogs of the theorems are proven for the case of lattice-normed spaces.", keywords = "Banach--Steinhaus theorem, Banach--Kantorovich space, cyclically compact set, Boolean valued analysis" } @inproceedings { Gutman20090918, author = "Gutman A.E. and Lisovskaya S.A.", howpublished = "Electronic", title = "The boundedness principle for lattice-normed spaces", booktitle = "Contemporary Analysis and Geometry. International Conference (Novosibirsk, September 14--20, 2009): Proceedings", address = "Novosibirsk", publisher = "Institute of Mathematics", year = "2009", pages = "29", language = "russian", annote = "Three classical facts of the theory of normed spaces are considered: the boundedness principle, the Banach--Steinhaus theorem, and the uniform boundedness principle for a compact convex set. By means of Boolean valued analysis, exact analogs of the theorems are proven for the case of lattice-normed spaces over a uniformly complete Kantorovich space." } @article { Gutman20091214, author = "Gutman A.E. and Kutateladze S.S. and Reshetnyak Yu.G.", title = "Cofinite numbers, nonstandard analysis, and mechanics", journal = "Sib. Zh. Ind. Mat.", year = "2010", volume = "13", number = "1(41)", pages = "55--58", language = "russian", annote = "We demonstrate the mathematical insignificance of the versions of nonstandard analysis proposed in the articles by A.F.Revuzhenko.", keywords = "nonstandard analysis, Revuzhenko, pseudoscience" } @article { Gutman20091215, author = "Gutman A.E. and Kutateladze S.S. and Reshetnyak Yu.G.", title = "Cofinite numbers, nonstandard analysis, and mechanics", journal = "J. Appl. Ind. Math.", year = "2010", volume = "4", number = "2", pages = "191--193", doi = "10.1134/S1990478910020079", annote = "We demonstrate the mathematical insignificance of the versions of nonstandard analysis proposed in the articles by A.F.Revuzhenko.", keywords = "nonstandard analysis, Revuzhenko, pseudoscience" } @inproceedings { Gutman20110701, author = "Gutman A.E.", title = "Prefix rewriting systems as object-oriented data models", booktitle = "Ershov Informatics Conference 2011. Knowledge and Ontologies *ELSEWHERE* 2011. The Third Workshop (Novosibirsk, July 1, 2011): Proceedings", address = "Novosibirsk", publisher = "Price-Courier", year = "2011", pages = "5--14", annote = "A deterministic longest-prefix rewriting system is a rewriting system such that there are no rewriting rules $X\rightarrow Y$, $X\rightarrow Z$ with $Y\ne Z$, and only longest prefixes of words are subject to rewriting. Given such a system, analogs are defined and examined of some concepts related to object-oriented data systems: inheritance of classes and objects, instances of classes, class and instance attributes, conceptual dependence and consistency, conceptual scheme, types and subtypes, etc. A special attention is paid to the effective verification of various properties of the rewriting systems under consideration. In particular, the algorithms are presented for answering the following questions: Are all words finitely rewritable? Do there exist recurrent words? Is the system conceptually consistent? Given two words $X$ and $Y$, does $X$ conceptually depend on $Y$? Does the type of $X$ coincide with that of $Y$? Is the type of $X$ a subtype of that of $Y$?", keywords = "prefix rewriting, term rewriting, object-oriented data system, information system, consistency verification, ontology of a data model" } @article { Gutman20111123, author = "Gutman A.E.", title = "An example of using $\Delta_1$ terms in Boolean-valued analysis", journal = "Vladikavk. Math. J.", year = "2012", volume = "14", number = "1", pages = "47--63", doi = "10.23671/VNC.2012.14.10953", language = "russian", annote = "Syntactic tools related to $\Delta_1$ terms are demonstrated by application to Boolean valued analysis. As an example, the question is considered of what approaches to defining the field $R$ of reals and what complete Boolean algebras $B$ provide the explicit inclusion of $R^{\land}$ into $R$ inside the Boolean-valued universe $V^{(B)}$.", keywords = "set theory, conservative extension, real number, Boolean valued analysis, canonical embedding, $\sigma$-distributive Boolean algebra, $\Sigma_1$ formula." } @inproceedings { Gutman20121115, author = "Gutman A.E.", howpublished = "Electronic", title = "Representation and analysis of object-oriented data by means of rewriting systems", booktitle = "Mal'tsev Meeting. International Conference (Novosibirsk, November 12--16, 2012)", address = "Novosibirsk", year = "2012", pages = "26", language = "russian", annote = "In the framework of deterministic prefix rewriting systems, the notions are introduced and studied which are typical for the object-oriented approach to data organization: inheritance of classes and objects, instances of classes, class and instance attributes, conceptual dependence and consistency, conceptual scheme, types and subtypes, etc. A special attention is paid to the effective verification of various properties of the rewriting systems under consideration." } @article { Gutman20130201, author = "Vodopyanov S.K. and Gordon E.I. and Gutman A.E. and Koptev A.V. and Kutateladze S.S. and Malyugin S.A. and Reshetnyak Yu.G.", howpublished = "Electronic", title = "Anatoly Georgievich Kusraev is 60", journal = "Sib. Electron. Math. Rep.", year = "2013", volume = "10", pages = "A.13--A.29", language = "russian", annote = "An address to professor A.G.Kusraev on the occasion of his 60th birthday, which includes a list of his publications containing 274 bibliographic records. Prof. A.G.Kusraev is known due to his contribution to neighboring topics of functional analysis and optimization, nonstandard analysis, the theory of vector duality, subdifferential calculus, the theory of dominated operators, and other contemporary topics of order analysis." } @article { Gutman20130319, author = "Vodopyanov S.K. and Gordon E.I. and Gutman A.E. and Koptev A.V. and Kutateladze S.S. and Malyugin S.A. and Reshetnyak Yu.G.", title = "Contribution of Anatoly Kusraev to subdifferential calculus and Boolean valued analysis (on the occasion of his 60th birthday)", journal = "Vladikavk. Math. J.", year = "2013", volume = "15", number = "1", pages = "90--97", language = "russian", annote = "An address to professor A.G.Kusraev on the occasion of his 60th birthday, which includes descriptions and statements of his main results in subdifferential calculus and Boolean-valued analysis: the method of general position, formulas for subdifferentials in the most wide statement, the method of Boolean-valued representation of algebraic systems, Boolean-valued representation of lattice-normed spaces, bounded operators, and Maharam operators, injective Banach lattices, etc." } @inproceedings { Gutman20130823, author = "Gutman A.E.", title = "Positive lifting in a measurable bundle of Banach lattices", booktitle = "Differential Equations. Function Spaces. Approximation Theory. International Conference dedicated to the 105th anniversary of the birthday of S.L.Sobolev (Novosibirsk, August 18--24, 2013): Proceeding", address = "Novosibirsk", publisher = "Institute of Mathematics SB RAS", year = "2013", pages = "377", language = "russian", annote = "It is shown that in the definition of lifting for classes of measurable sections of a measurable bundle of Banach lattices, the requirement of preservation of the lattice operations is superfluous and can be replaced with positivity." } @inproceedings { Gutman20130830, author = "Gutman A.E.", title = "Sequentially convergent mappings and fixed-point theorems", booktitle = "Geometry Days in Novosibirsk -- 2013. International conference (Novosibirsk, August 28--31, 2013): Proceedings", address = "Novosibirsk", publisher = "Institute of Mathematics", year = "2013", pages = "34--35", language = "russian", annote = "It is shown that the main results of 28 articles devoted to generalizations of fixed-point theorems for mappings in special spaces are direct consequences of a single general theorem which translates previously known facts to the spaces under consideration." } @article { Gutman20131013, author = "Gutman A.E.", title = "Positive lifting in a measurable bundle of Banach lattices", journal = "Vladikavk. Math. J.", year = "2013", volume = "15", number = "4", pages = "12--13", doi = "10.23671/VNC.2013.4.7329", language = "russian", annote = "It is shown that every positive lifting in a measurable bundle of Banach lattices is a lattice homomorphism.", keywords = "Banach lattice, lattice homomorphism, Banach bundle, lifting" } @article { Gutman20131015, author = "Gutman A.E. and Koptev A.V.", title = "Lattice-metric decomposition of a monotone operator", journal = "Math. Notes NEFU", year = "2013", volume = "20", number = "2", pages = "34--40", language = "russian", annote = "In the paper, we consider the natural notion of monotone linear operator acting from a vector lattice into a normed space, show that every monotone operator admits a ``lattice-metric'' representation as the composition of a lattice homomorphism and a linear isometry, and present several applications of the results to the study of continuous and measurable bundles of Banach lattices." } @inproceedings { Gutman20131114, author = "Gutman A.E.", howpublished = "Electronic", title = "The technique of definable terms in Boolean valued analysis", booktitle = "Mal'tsev Meeting. International Conference (Novosibirsk, November 11--15, 2013): Proceedings", address = "Novosibirsk", year = "2013", pages = "164", annote = "A syntax technique related to the notion of $\Delta_1$ term is demonstrated by means of its applications to Boolean-valued analysis. As an example, the following question is considered: Which of the classical approaches to the definition of the field $R$ of reals and which Boolean algebras provide the explicit inclusion of $R^{\land}$ in $R$ inside the Boolean-valued universe $V^{(B)}$?" } @article { Gutman20131119, author = "Gutman A.E. and Koptev A.V.", title = "Continuous-discrete sections of bundles of Banach lattices", journal = "Vladikavk. Math. J.", year = "2013", volume = "15", number = "4", pages = "14--18", doi = "10.23671/VNC.2013.4.7330", language = "russian", annote = "The notion of a continuous bundle of Banach lattices is clarified and the order properties are studied of the space of $CD_0$-sections of such a bundle.", keywords = "continuous Banach bundle, Banach lattice, bundle of Banach lattices, $CD_0$-section of a Banach bundle, Alexandroff duplicate" } @article { Gutman20131023, author = "Gutman A.E. and Koptev A.V.", title = "Distribution of finite-dimensional and separable stalks of an ample Banach bundle", journal = "Proc. Acad. Sci.", year = "2014", volume = "456", number = "4", pages = "387--388", doi = "10.7868/S0869565214160038", language = "russian", annote = "The topological characteristics are studied of the set of points at which the stalks of an ample Banach bundle are finite-dimensional or separable. A connection is established between the property of the stalks of a bundle to be finite-dimensional or separable with the analogous property of the stalks of the ample hull of the bundle. A new criterion is obtained for existence of the dual bundle." } @article { Gutman20131024, author = "Gutman A.E. and Koptev A.V.", title = "Distribution of finite-dimensional and separable stalks of an ample Banach bundle", journal = "Doklady Math.", year = "2014", volume = "89", number = "3", pages = "319--320", doi = "10.1134/S1064562414030168", annote = "The topological characteristics are studied of the set of points at which the stalks of an ample Banach bundle are finite-dimensional or separable. A connection is established between the property of the stalks of a bundle to be finite-dimensional or separable with the analogous property of the stalks of the ample hull of the bundle. A new criterion is obtained for existence of the dual bundle." } @article { Gutman20131112, author = "Gutman A.E. and Koptev A.V.", title = "Convergence-preserving maps and fixed-point theorems", journal = "Mat. zametki", year = "2014", volume = "95", number = "5", pages = "790--794", doi = "10.4213/mzm10446", language = "russian", annote = "It is shown that the main results of 28 articles devoted to generalizations of fixed-point theorems for mappings in special spaces are direct consequences of a single general theorem which translates previously known facts to the spaces under consideration." } @article { Gutman20131113, author = "Gutman A.E. and Koptev A.V.", title = "Convergence-preserving maps and fixed-point theorems", journal = "Math. Notes", year = "2014", volume = "95", number = "5", pages = "738--742", doi = "10.1134/S0001434614050150", annote = "It is shown that the main results of 28 articles devoted to generalizations of fixed-point theorems for mappings in special spaces are direct consequences of a single general theorem which translates previously known facts to the spaces under consideration.", keywords = "sequential convergence, (pre)topological convergence, single-valued convergence, sequential topological space, convergence-preserving map, (sub)sequentially convergent map, fixed-point theorem" } @article { Gutman20131217, author = "Gutman A.E. and Koptev A.V.", title = "Finite dimensionality and separability of the stalks of Banach bundles", journal = "Sib. Matem. Zh.", year = "2014", volume = "55", number = "2", pages = "304--314", language = "russian", annote = "Topological characteristics are studied of the set of points at which the stalks of an ample Banach bundle over an extremally disconnected compact space are finite-dimensional or separable. A connection is established between finite dimensionality or separability of the stalks of a bundle and the analogous properties of the stalks of the ample hull of the bundle. A new criterion is obtained for existence of a dual bundle.", keywords = "continuous Banach bundle, ample hull, extremally disconnected compact space, $\sigma$-isolated point" } @article { Gutman20131218, author = "Gutman A.E. and Koptev A.V.", title = "Finite dimensionality and separability of the stalks of Banach bundles", journal = "Sib. Math. J.", year = "2014", volume = "55", number = "2", pages = "246--253", doi = "10.1134/S0037446614020074", annote = "Topological characteristics are studied of the set of points at which the stalks of an ample Banach bundle over an extremally disconnected compact space are finite-dimensional or separable. A connection is established between finite dimensionality or separability of the stalks of a bundle and the analogous properties of the stalks of the ample hull of the bundle. A new criterion is obtained for existence of a dual bundle.", keywords = "continuous Banach bundle, ample hull, extremally disconnected compact space, $\sigma$-isolated point" } @inproceedings { Gutman20140610, author = "Gutman A.E.", title = "Object-oriented data as prefix rewriting systems", booktitle = "Advanced mathematics, computations and applications -- 2014. International conference (Novosibirsk, June 8--11, 2014): Abstracts", address = "Novosibirsk", publisher = "Academizdat", year = "2014", isbn = "978-5-9904865-8-4", pages = "48", annote = "A new approach is suggested for representing and analysing object-oriented data by means of rewriting systems." } @article { Gutman20140620, author = "Vodopyanov S.K. and Gordon E.I. and Gutman A.E. and Koptev A.V. and Kutateladze S.S. and Malyugin S.A. and Reshetnyak Yu.G.", title = "Contribution of Anatoly Kusraev to subdifferential calculus and Boolean-valued analysis (on the occasion of his 60th birthday)", journal = "Mathematical Forum (Scientific Outcome. South of Russia). Vladikavkaz: SMI VSC RAS and RNO-A", year = "2014", volume = "8", booktitle = "Research in mathematical analysis", number = "1", pages = "11--22", language = "russian", annote = "An address to professor A.G.Kusraev on the occasion of his 60th birthday, which includes descriptions and statements of his main results in subdifferential calculus and Boolean-valued analysis: the method of general position, formulas for subdifferentials in the most wide statement, the method of Boolean-valued representation of algebraic systems, Boolean-valued representation of lattice-normed spaces, bounded operators, and Maharam operators, injective Banach lattices, etc." } @article { Gutman20140828, author = "Vodopyanov S.K. and Gordon E.I. and Gutman A.E. and Koptev A.V. and Kutateladze S.S. and Malyugin S.A. and Reshetnyak Yu.G.", title = "Anatoly Georgievich Kusraev is 60", journal = "Mathematical Forum (Scientific Outcome. South of Russia). Vladikavkaz: SMI VSC RAS and RNO-A", year = "2014", volume = "8", booktitle = "Research in differential equations, mathematical modeling, and problems of mathematical education", number = "2", pages = "11--12", language = "russian", annote = "A short address to professor A.G.Kusraev on the occasion of his 60th birthday." } @inproceedings { Gutman20140926, author = "Gutman A.E. and Kusraev A.G. and Kutateladze S.S.", title = "The growth points of Boolean valued analysis", booktitle = "Geometry Days in Novosibirsk -- 2014. International conference dedicated to 85th anniversary of academician Yu.G.Reshetnyak (Novosibirsk, September 24--27, 2014): Proceedings", address = "Novosibirsk", publisher = "Sobolev Institute of Mathematics SB RAS", year = "2014", pages = "102", annote = "Boolean valued analysis is a powerful method of extending the scope of mathematical theories by means of the special nonstandard models of set theory. This communication pays attention to the Continuum Hypothesis, Kantorovich spaces, and the machinery of Boolean valued analysis." } @inproceedings { Gutman20140927, author = "Koptev A.V. and Gutman A.E.", title = "Homomorphisms of Banach bundles and separated convergent sequences", booktitle = "Geometry Days in Novosibirsk -- 2014. International conference dedicated to 85th anniversary of academician Yu.G.Reshetnyak (Novosibirsk, September 24--27, 2014): Proceedings", address = "Novosibirsk", publisher = "Sobolev Institute of Mathematics SB RAS", year = "2014", pages = "39--40", language = "russian", annote = "In the theory of continuous Banach bundles, the question remains open on existence of nonzero homomorphism in nonzero bundles. In this connection there is a need in general methods of constructing homomorphisms possessing certain approximating properties. The above topic covers the new facts presented here on existence of homomorphisms which assume prescribed values in the points of a convergent sequence." } @article { Gutman20131017, author = "Gutman A.E.", title = "Object-oriented data as prefix rewriting systems", journal = "Vladikavk. Math. J.", year = "2015", volume = "17", number = "3", pages = "23--35", doi = "10.23671/VNC.2017.3.7260", annote = "A deterministic longest-prefix rewriting system is a rewriting system such that there are no rewriting rules $X\rightarrow Y$, $X\rightarrow Z$ with $Y\ne Z$, and only longest prefixes of words are subject to rewriting. Given such a system, analogs are defined and examined of some concepts related to object-oriented data systems: inheritance of classes and objects, instances of classes, class and instance attributes, conceptual dependence and consistency, conceptual scheme, types and subtypes, etc. A special attention is paid to the effective verification of various properties of the rewriting systems under consideration. In particular, algorithms are presented for answering the following questions: Are all words finitely rewritable? Do there exist recurrent words? Is the system conceptually consistent? Given two words $X$ and $Y$, does $X$ conceptually depend on $Y$? Does the type of $X$ coincide with that of $Y$? Is the type of $X$ a subtype of that of $Y$?", keywords = "prefix rewriting, term rewriting, object-oriented data system, information system, consistency verification, ontology of a data model" } @inproceedings { Gutman20150828, author = "Gutman A.E.", title = "The problem of existence of nonclosed Archimedean cones", booktitle = "Geometry Days in Novosibirsk -- 2015. International conference (Novosibirsk, August 26--29, 2015): Proceedings", address = "Novosibirsk", publisher = "Sobolev Institute of Mathematics SB RAS", year = "2015", pages = "91--92", language = "russian", annote = "There are certain results that essentially reduce the class of locally convex spaces in which all Archimedean cones are closed, but, in general, the problem of describing such spaces remains open. We present the main results which are obtained in the course of solving the problem." } @article { Gutman20150908, author = "Gutman A.E. and Emelyanov E.Yu. and Matyukhin A.V.", title = "Nonclosed Archimedean cones in locally convex spaces", journal = "Vladikavk. Math. J.", year = "2015", volume = "17", number = "3", pages = "36--43", doi = "10.23671/VNC.2017.3.7262", language = "russian", annote = "The problem is stated of describing the class of locally convex spaces which include nonclosed Archimedean cones. Certain results are presented in the course of solving the problem.", keywords = "Archimedean ordered vector space, locally convex space, weak topology, cone, wedge" } @inproceedings { Gutman20150914, author = "Gutman A.E. and Matyukhin A.V.", howpublished = "Electronic", title = "The problem of describing the locally convex spaces which include nonclosed Archimedean cones", booktitle = "Actual Questions of Contemporary Science. International scientific conference (Moscow, September 14--15, 2015): Proceedings", address = "Moscow", publisher = "RusAlliance Owl", year = "2015", isbn = "5990722532, 9785990722538", pages = "8--12", language = "russian", annote = "The problem is stated of describing the class of locally convex spaces which include nonclosed Archimedean cones. Certain results are presented which are obtained in the course of solving the problem.", keywords = "locally convex space, ordered vector space, Archimedean cone" } @article { Gutman20150926, author = "Gutman A.E. and Matyukhin A.V.", howpublished = "Electronic", title = "Topological vector spaces with nonclosed Archimedean cones", journal = "Prospero", year = "2015", volume = "8", number = "20", pages = "62--64", language = "russian", annote = "The problem is stated of describing the class of topological vector spaces with nonclosed Archimedean cones. Some results are presented in the course of solving the problem.", keywords = "topological vector space, ordered vector space, axiom of Archimedes, cone" } @booklet { Gutman20160601, author = "Gutman A.E. and Katz M.G. and Kudryk T.S. and Kutateladze S.S.", note = "Electronic preprint / arXiv:1606.00160 [math.HO]", howpublished = "Electronic", title = "The Mathematical Intelligencer flunks the Olympics", address = "arXiv.org", year = "2016", pages = "25", annote = "The Mathematical Intelligencer recently published a note by Y. Sergeyev that challenges both mathematics and intelligence. We examine Sergeyev's claims concerning his purported Infinity computer. We compare his grossone system with the classical Levi--Civita fields and with the hyperreal framework of A. Robinson, and analyze the related algorithmic issues inevitably arising in any genuine computer implementation. We show that Sergeyev's grossone system is unnecessary and vague, and that whatever consistent subsystem could be salvaged is subsumed entirely within a stronger and clearer system (IST). Lou Kauffman, who published an article on a grossone, places it squarely outside the historical panorama of ideas dealing with infinity and infinitesimals." } @inproceedings { Gutman20160921, author = "Gutman A.E. and Matyukhin A.V.", title = "Nonclosed Archimedean cones", booktitle = "Geometry Days in Novosibirsk -- 2016. International conference (Novosibirsk, September 21--24, 2016): Proceedings", address = "Novosibirsk", publisher = "Institute of Mathematics", year = "2016", pages = "46--47", language = "russian", annote = "The notion of Archimedean convex set is introduced and clarified. The problem is considered of describing the class of topological vector spaces which include nonclosed Archimedean cones. The main results are presented which are obtained in the course of solving the problem and its variations with cones replaced by wedges and closedness, by sequential closedness." } @inproceedings { Gutman20161211, author = "Gutman A.E. and Kononenko L.I.", title = "Formalization of inverse problems and applications to systems of equations with parameters", booktitle = "Geometric Analysis and Control Theory. International conference (Novosibirsk, December, 8--12, 2016): Proceedings", address = "Novosibirsk", publisher = "Sobolev Institute of Mathematics SB RAS", year = "2016", pages = "40--42", annote = "We show how binary correspondences can be used for simple formalization of the notion of problem, definition of the basic components of problems, their properties, and constructions (the condition of a problem, its data and unknowns, solvability and unique solvability of a problem, inverse problem, composition and restriction of problems, etc.). As an illustration, we consider a system of differential equations which describe a process in chemical kinetics, as well as the inverse problem." } @inproceedings { Gutman20161212, author = "Gutman A.E. and Matyukhin A.V.", title = "Nonclosed Archimedean cones", booktitle = "Geometric Analysis and Control Theory. International conference (Novosibirsk, December, 8--12, 2016): Proceedings", address = "Novosibirsk", publisher = "Sobolev Institute of Mathematics SB RAS", year = "2016", pages = "42--44", annote = "The notion of Archimedean convex set is introduced and studied. The problem is considered of describing the class of topological vector spaces which include nonclosed Archimedean cones. The main results are presented which are obtained when solving the problem and its variations with cones replaced by wedges and closedness, by sequential closedness." } @article { Gutman20160315, author = "Gutman A.E. and Katz M.G. and Kudryk T.S. and Kutateladze S.S.", title = "The Mathematical Intelligencer flunks the Olympics", journal = "Found. Sci.", year = "2017", volume = "22", number = "3", pages = "539--555", doi = "10.1007/s10699-016-9485-8", annote = "The Mathematical Intelligencer recently published a note by Y. Sergeyev that challenges both mathematics and intelligence. We examine Sergeyev's claims concerning his purported Infinity computer. We compare his grossone system with the classical Levi--Civita fields and with the hyperreal framework of A. Robinson, and analyze the related algorithmic issues inevitably arising in any genuine computer implementation. We show that Sergeyev's grossone system is unnecessary and vague, and that whatever consistent subsystem could be salvaged is subsumed entirely within a stronger and clearer system (IST). Lou Kauffman, who published an article on a grossone, places it squarely outside the historical panorama of ideas dealing with infinity and infinitesimals." } @article { Gutman20161030, author = "Gutman A.E. and Kononenko L.I.", title = "Formalization of inverse problems and its applications", journal = "Sib. J. Pure. Appl. Math.", year = "2017", volume = "17", number = "4", pages = "49--56", doi = "10.17377/PAM.2017.17.5", language = "russian", annote = "We show how binary correspondences can be used for simple formalization of the notion of problem, definition of the basic components of problems, their properties, and constructions (the condition of a problem, its data and unknowns, solvability and unique solvability of a problem, inverse problem, composition and restriction of problems, etc.). We also consider topological problems and the related notions of stability and correctness. Particular attention is paid to problems with parameters. As an illustration, we consider a system of differential equations which describes a process in chemical kinetics, as well as the inverse problem.", keywords = "inverse problem, binary correspondence, solvability, composition, stability, correctness, differential equation, chemical kinetics" } @inbook { Gutman20170808, author = "Gutman A.E. and Kutateladze S.S.", title = "Laboratory of Functional Analysis", booktitle = "Sobolev Institute of Mathematics SB RAS. 60 years", address = "Novosibirsk", publisher = "Institute of Mathematics", year = "2017", pages = "151--158", language = "russian", annote = "A photo-illustrated story about the Laboratory of Functional Analysis confined to the 70th anniversary of the Sobolev Institute of Mathematics SB RAS." } @inproceedings { Gutman20170817, author = "Gutman A.E. and Kononenko L.I.", title = "Binary correspondences and multidimensional problems of chemical kinetics", booktitle = "Mathematics in the Modern World. International conference dedicated to the 60th anniversary of the Sobolev Institute of Mathematics (Novosibirsk, August 14--19, 2017): Proceedings", address = "Novosibirsk", publisher = "Institute of Mathematics", year = "2017", pages = "205", language = "russian", annote = "It is shown that binary correspondences provide a simple and adequate formalization for the main components of problems (the condition of a problem, its data and unknowns), their basic properties and constructions (solvability and unique solvability of a problem, inverse problem, composition and restriction of problems), make it possible to formalize topological problems and related properties (stability, correctness), and also speak of parametrizations of problems and dependence of solutions on parameters. As an example, a singularly perturbed system is considered of ordinary differential equations which describes a process of chemical kinetics and burning." } @inproceedings { Gutman20170818, author = "Gutman A.E. and Matyukhin A.V.", title = "Archimedean cones and incoming directions", booktitle = "Mathematics in the Modern World. International conference dedicated to the 60th anniversary of the Sobolev Institute of Mathematics (Novosibirsk, August 14--19, 2017): Proceedings", address = "Novosibirsk", publisher = "Institute of Mathematics", year = "2017", pages = "154", language = "russian", annote = "The notion is introduced and studied of incoming direction for a given convex set at a given point, a new criterion is provided for a wedge to be Archimedean, and the question is answered of when a wedge is included in a half-space." } @inproceedings { Gutman20170822, author = "Gutman A.E. and Kononenko L.I.", title = "Binary correspondences and problems of chemical kinetics with many-sheeted slow surface", booktitle = "Sobolev Readings. International School-Conference (Novosibirsk, August 20--23, 2017): Proceedings", address = "Novosibirsk", publisher = "Institute of Mathematics", year = "2017", pages = "122", annote = "It is shown that binary correspondences provide a simple and adequate formalization for components of problems, their properties and constructions, make it possible to formalize topological problems, their parametrizations, and dependence of solutions on parameters. As an example, a singularly perturbed system is considered of ordinary differential equations which describes a process of chemical kinetics with many-sheeted slow surface." } @inproceedings { Gutman20171115, author = "Gutman A.E. and Kononenko L.I.", howpublished = "Electronic", title = "Binary correspondences in problems of chemical kinetics", booktitle = "Lomonosov's Reading in Altai: Fundamental Problems of Science and Education, International conference (Barnaul, November 14--17, 2017): Collection of scientific articles", address = "Barnaul", publisher = "Altai State Univ.", year = "2017", issn = "2309-463X", pages = "424--430", language = "russian", annote = "Is it shown how binary correspondences can be used for simple formalization of the notion of problem, definition of the basic components of problems, their properties, and constructions (the condition of a problem, its data and unknowns, solvability and unique solvability of a problem, inverse problem, composition and restriction of problems, etc.). Topological problems are also considered as well as the related notions of stability and correctness. Particular attention is paid to problems with parameters. As an illustration, a system of differential equations is considered which describes a process in chemical kinetics, as well as the inverse problem." } @article { Gutman20171204, author = "Gutman A.E. and Kononenko L.I.", howpublished = "Electronic", title = "The inverse problem of chemical kinetics as a composition of binary correspondences", journal = "Sib. Electron. Math. Rep.", year = "2018", volume = "15", pages = "48--53", doi = "10.17377/semi.2018.15.006", language = "russian", annote = "Binary correspondences are employed for formalization of the notion of problem, definition of the basic components of problems, their properties, and constructions (the condition of a problem, its data and unknowns, solvability and unique solvability of a problem, inverse problem, and composition of problems). As an illustration, we consider a system of differential equations which describe a process in chemical kinetics. Within the study of the inverse problem, a criterion is established for linear independence of functions in terms of finite sets of their values.", keywords = "differential equation, chemical kinetics, inverse problem, linear independence, binary correspondence, solvability, composition" } @article { Gutman20180306, author = "Gutman A.E.", title = "On the structure of the Boolean-valued universe", journal = "Vladikavk. Math. J.", year = "2018", volume = "20", number = "2", pages = "38--48", doi = "10.23671/VNC.2018.2.14718", language = "russian", annote = "The logical machinery is clarified which justifies declaration of hypotheses. In particular, attention is paid to hypotheses and conclusions constituted by infinitely many formulas. Formal definitions are presented for Boolean-valued algebraic system and model of a theory, for the system of terms of Boolean-valued truth value of formulas, for ascent and mixing. Logical interrelations are described between the ascent, mixing, and maximum principles. It is shown that every ascent with arbitrary weights can be transformed into an ascent with constant weight. The notion of restriction of an element of a Boolean-valued algebraic system is introduced and studied. It is proven that every Boolean-valued model of Set theory which meets the ascent principle has multilevel structure analogous to von Neumann's cumulative hierarchy.", keywords = "Set theory, Boolean-valued model, universe, cumulative hierarchy" } @article { Gutman20180703, author = "Gutman A.E. and Kononenko L.I.", title = "Binary correspondences and the inverse problem of chemical kinetics", journal = "Vladikavk. Math. J.", year = "2018", volume = "20", number = "3", pages = "37--47", doi = "10.23671/VNC.2018.3.17981", annote = "We show how binary correspondences can be used for simple formalization of the notion of problem, definition of the basic components of problems, their properties, and constructions. In particular, formalization of the following notions is presented: condition, data, unknowns, and solutions of a problem, solvability and unique solvability, inverse problem, composition and restriction of problems, isomorphism between problems. We also consider topological problems and the related notions of stability and correctness. A connection is indicated between the stability and continuity of a uniquely solvable topological problem. The definition of parametrized set is given. The notions are introduced of parametrized problem, the problem of reconstruction of an object by the values of parameters, as well as the notions of locally free set of parameters and stability with respect to a set of parameters. As an illustration, we consider a singularly perturbed system of ordinary differential equations which describe a process in chemical kinetics and burning. Direct and inverse problems are stated for such a system. We extend the class of problems under study by considering polynomials of arbitrary degree as the right-hand sides of the differential equations. It is shown how the inverse problem of chemical kinetics can be corrected and made more practical by means of composition with a simple auxiliary problem which represents the relation between functions and finite sets of numerical characteristics being measured. For the corrected inverse problem, formulas for the solution are presented and the conditions of unique solvability are indicated. Within the study of solvability, a criterion is established for linear independence of functions in terms of finite sets of their values. With the help of the criterion, realizability is clarified of the condition for unique solvability of the inverse problem of chemical kinetics.", keywords = "binary correspondence, inverse problem, solvability, composition, stability, correctness, differential equation, chemical kinetics, linear independence" } @inproceedings { Gutman20180919, author = "Gutman A.E.", title = "Archimedean and directionally closed cones", booktitle = "Geometry Days in Novosibirsk -- 2018. International conference (Novosibirsk, September 19--22, 2018): Proceedings", address = "Novosibirsk", publisher = "Institute of Mathematics", year = "2018", isbn = "978-5-86134-220-9", pages = "15", annote = "A criterion is provided for a wedge to be Archimedean, which is based on the notion of closed set along a direction." } @inproceedings { Gutman20181210, author = "Gutman A.E. and Kononenko L.I.", title = "Binary correspondences and the inverse problem of chemical kinetics", booktitle = "Sobolev Readings. International School-Conference dedicated to the 110th anniversary of the birthday of S.L.Sobolev (Novosibirsk, December 10--16, 2018): Proceedings", address = "Novosibirsk", publisher = "Institute of Mathematics", year = "2018", isbn = "978-5-86134-222-3", pages = "81", language = "russian", annote = "By treating the notion of problem as binary correspondence, we provide a simple and adequate formalization for the main components of problems, their basic properties, and constructions. In particular, we arrive at natural formal notions of inverse problem and composition of problems. As an illustration, the inverse problem is considered to a singularly perturbed system of ordinary differential equations which describe a process in chemical kinetics and burning. The inverse problem is corrected and made more practical by means of composition with a simple auxiliary problem which represents the relation between functions and finite sets of numers. For the corrected inverse problem, formulas for the solution are presented and the conditions of unique solvability are indicated." } @article { Gutman20190710, author = "Gutman A.E. and Emelianenkov I.A.", title = "Lexicographic structures on vector spaces", journal = "Vladikavk. Math. J.", year = "2019", volume = "21", number = "4", pages = "42--55", doi = "10.23671/VNC.2019.21.44621", language = "russian", annote = "Basic properties are described of the Archimedean equivalence and dominance in a totally ordered vector space. The notion of (pre)lexicographic structure on a vector space is introduced and studied. A lexicographic structure is a duality between vectors and points which makes it possible to represent an abstract ordered vector space as an isomorphic space of real-valued functions endowed with a lexicographic order. The notions of function and basic lexicographic structures are introduced. Interrelations are clarified between an ordered vector space and its function lexicographic representation. A new proof is presented for the theorem on isomorphic embedding of a totally ordered vector space into a lexicographically ordered space of real-valued functions with well-ordered supports. A criterion is obtained for denseness of a maximal cone with respect to the strongest locally convex topology. Basic maximal cones are described in terms of sets constituted by pairwise nonequivalent vectors. The class is characterized of vector spaces in which there exist nonbasic maximal cones.", keywords = "maximal cone, dense cone, totally ordered vector space, Archimedean equivalence, Archimedean dominance, lexicographic order, Hamel basis, locally convex space" } @article { Gutman20190720, author = "Gutman A.E.", title = "Boolean-valued universe as an algebraic system. I: Basic principles", journal = "Sib. Matem. Zh.", year = "2019", volume = "60", number = "5", pages = "1041--1062", doi = "10.33048/smzh.2019.60.505", language = "russian", annote = "The paper is devoted to the study of Boolean-valued algebraic systems of set-theoretic signature. The technique of partial elements of these systems is developed. Some formal apparatus is presented for using partial elements and Boolean-valued classes in the truth values of formulas. The predicative Boolean-valued classes are studied that admit quantification. Logical interrelations are described between the basic properties of Boolean-valued systems: the transfer, mixing, and maximum principles.", keywords = "Boolean-valued algebraic system, set theory, Boolean-valued analysis" } @article { Gutman20190721, author = "Gutman A.E.", title = "Boolean-valued universe as an algebraic system. I: Basic principles", journal = "Sib. Math. J.", year = "2019", volume = "60", number = "5", pages = "810--827", doi = "10.1134/S0037446619050057", annote = "The paper is devoted to the study of Boolean-valued algebraic systems of set-theoretic signature. The technique of partial elements of these systems is developed. Some formal apparatus is presented for using partial elements and Boolean-valued classes in the truth values of formulas. The predicative Boolean-valued classes are studied that admit quantification. Logical interrelations are described between the basic properties of Boolean-valued systems: the transfer, mixing, and maximum principles.", keywords = "Boolean-valued algebraic system, set theory, Boolean-valued analysis" } @inproceedings { Gutman20190922, author = "Gutman A.E. and Kononenko L.I.", title = "Binary correspondences and an algorithm for solving an inverse problem of chemical kinetics", booktitle = "International conference on Geometric Analysis in honor of the 90th anniversary of academician Yu.G.Reshetnyak (Novosibirsk, September, 22--28, 2019): Proceedings", address = "Novosibirsk", publisher = "Sobolev Institute of Mathematics SB RAS", year = "2019", isbn = "978-5-4437-0949-9", pages = "67", annote = "Binary correspondences are used for formalization of problems, their basic components, properties, and constructions. A singularly perturbed system of ordinary differential equations is considered which describes a process in chemical kinetics. An iteration algorithm is proposed for finding an approximate solution to the inverse problem." } @inproceedings { Gutman20190923, author = "Gutman A.E.", title = "Cumulative structure of a Boolean-valued model of set theory", booktitle = "International conference on Geometric Analysis in honor of the 90th anniversary of academician Yu.G.Reshetnyak (Novosibirsk, September, 22--28, 2019): Proceedings", address = "Novosibirsk", publisher = "Sobolev Institute of Mathematics SB RAS", year = "2019", isbn = "978-5-4437-0949-9", pages = "64--66", annote = "We show that every Bolean-valued universe has a multilevel structure analogous to the von Neumann cumulative hierarchy, in which, at each level, the ascents are added of the Boolean-valued functions defined on subsets of the previous levels. Another cumulative structure is obtained if we consider the ascents of constant functions only and add mixings at the limit steps. Such cumulative hierarchies clarify the structure of Boolean-valued systems and, in particular, make it possible to easily prove the uniqueness of a Boolean-valued universe up to isomorphism. We also present a general tool for adding ascents to Boolean-valued systems which builds the cumulative hierarchy starting from an arbitrary extensional system. This makes it possible to construct examples of Boolean-valued systems with unusual properties. By means of the tool, we show that each of the five conditions listed in the definition of a Boolean-valued universe, is essential and does not follow from the other conditions." } @article { Gutman20200308, author = "Gutman A.E.", title = "Boolean-valued universe as an algebraic system. II: Intensional hierarchies", journal = "Sib. Matem. Zh.", year = "2020", volume = "61", number = "3", pages = "539--571", doi = "10.33048/smzh.2020.61.305", language = "russian", annote = "For Boolean-valued algebraic systems of set-theoretic signature, the notions of transitivity, regularity, and $\sigma$-regularity are studied. The notion of a universe over an arbitrary extensional Boolean-valued system is introduced. A description is proposed of the structure of the universe by means of various hierarchies. The results are used for proving the uniqueness of a Boolean-valued universe up to a unique isomorphism.", keywords = "Boolean-valued algebraic system, set theory, Boolean-valued analysis, universe, cumulative hierarchy" } @article { Gutman20200309, author = "Gutman A.E.", title = "Boolean-valued universe as an algebraic system. II: Intensional hierarchies", journal = "Sib. Math. J.", year = "2020", volume = "61", number = "3", pages = "426--452", doi = "10.1134/S0037446620030052", annote = "For Boolean-valued algebraic systems of set-theoretic signature, the notions of transitivity, regularity, and $\sigma$-regularity are studied. The notion of a universe over an arbitrary extensional Boolean-valued system is introduced. A description is proposed of the structure of the universe by means of various hierarchies. The results are used for proving the uniqueness of a Boolean-valued universe up to a unique isomorphism.", keywords = "Boolean-valued algebraic system, set theory, Boolean-valued analysis, universe, cumulative hierarchy" } @inproceedings { Gutman20200914, author = "Gutman A.E. and Kononenko L.I.", title = "Binary correspondences and an algorithm for solving an inverse problem of chemical kinetics in a nondegenerate case", booktitle = "Geometry in the Large. Conference dedicated to the 90th birthday of Victor Toponogov (St. Petersburg, 2021): Abstracts", address = "St. Petersburg", publisher = "Euler International Mathematical Institute", year = "2021", pages = "12", annote = "Binary correspondences are used for formalization of problems, their basic components, properties, and constructions. A singularly perturbed system of ordinary differential equations is considered which describes a process in chemical kinetics in a nondegenerate case. An iteration algorithm is proposed for finding an approximate solution to the inverse problem." } @article { Gutman20210404, author = "Gutman A.E.", title = "Boolean-valued set-theoretic systems: General formalism and basic technique", journal = "Mathematics", year = "2021", volume = "9", number = "9", article = "1056", pages = "78", doi = "10.3390/math9091056", annote = "The article is devoted to the study of the Boolean-valued universe as an algebraic system. We start with the logical backgrounds of the notion and present the formalism of extending the syntax of Boolean truth values by the use of definable symbols, internal classes, outer terms, and external Boolean-valued classes. Next, we enrich the collection of Boolean-valued research tools with the technique of partial elements and the corresponding joins, mixings, and ascents. Passing on to the set-theoretic signature, we prove that bounded formulas are absolute for transitive Boolean-valued subsystems. We also introduce and study intensional, predicative, cyclic, and regular Boolean-valued systems, examine the maximum principle, and analyze its relationship with the ascent and mixing principles. The main applications relate to the universe over an arbitrary extensional Boolean-valued system. A close interrelation is established between such a universe and the intensional hierarchy. We~prove the existence and uniqueness of the Boolean-valued universe up to a unique isomorphism and show that the conditions in the corresponding axiomatic characterization are logically independent. We also describe the structure of the universe by means of several cumulative hierarchies. Another application, based on the quantifier hierarchy of formulas, improves the transfer principle for the canonical embedding in the Boolean-valued universe.", keywords = "Boolean-valued universe, algebraic system, set theory, cumulative hierarchy" } @inproceedings { Gutman20210921, author = "Gutman A.E.", title = "Boolean-Valued Analysis: See the Simple in the Complex", booktitle = "International Conference «Order Analysis and Related Problems of Mathematical Modeling, XVI. Operator Theory and Differential Equations» (Vladikavkaz, September 20--25, 2021): Proceedings", address = "Vladikavkaz", publisher = "SMI VSC RAS and RNO-A", year = "2021", pages = "1", language = "russian", annote = "The audience is offered a story about Boolean-valued models of set theory. These are very unusual models with very unusual logic. Assertions in such models do not have to be true or false, and their truth can take intermediate values. This was the idea that helped to successfully solve the most famous mathematical problem of the 20th century, the Continuum Problem." } @article { Gutman20230503, author = "Gutman A.E. and Emelianenkov I.A.", title = "Locally convex spaces with all Archimedean cones closed", journal = "Sib. Matem. Zh.", year = "2023", volume = "64", number = "5", pages = "945--970", doi = "10.33048/smzh.2023.64.505", language = "russian", annote = "We provide an exhaustive description of the class of locally convex spaces in which all Archimedean cones are closed. We introduce the notion of quasidense set and prove that the above class consists of all finite-dimensional and countable-dimensional spaces $X$ whose topological dual $X'$ is quasidense in the algebraic dual $X^\#$ of $X$.", keywords = "Archimedean ordered vector space, locally convex space, weak topology, cone, wedge" } @article { Gutman20230504, author = "Gutman A.E. and Emelianenkov I.A.", title = "Locally convex spaces with all Archimedean cones closed", journal = "Sib. Math. J.", year = "2023", volume = "64", number = "5", pages = "1117--1136", doi = "10.1134/S0037446623050051", annote = "We provide an exhaustive description of the class of locally convex spaces in which all Archimedean cones are closed. We introduce the notion of quasidense set and prove that the above class consists of all finite-dimensional and countable-dimensional spaces $X$ whose topological dual $X'$ is quasidense in the algebraic dual $X^\#$ of $X$.", keywords = "Archimedean ordered vector space, locally convex space, weak topology, cone, wedge" } @article { Gutman20230731, author = "Gutman A.E.", title = "A sentence preservation theorem for Boolean algebras", journal = "J. Math. Sci.", year = "2023", pages = "8", doi = "10.1007/s10958-023-06599-4", annote = "At the initial stages of studying the theory of Boolean algebras, before trying to prove or disprove any simple sentence, students are usually asked to test their intuition using Venn diagrams or truth tables. A natural question arises: is it necessary to invent a proof after a positive check of this kind? Isn't such a check itself a rigorous proof of the verified sentence? And if this is not true in the general case, for which sentences is this true? We answer the question and prove an analog of the Jech Theorem for arbitrary (not necessarily complete) Boolean algebras.", keywords = "Boolean algebra, Venn diagram, truth table, Horn formula" } @inbook { Gutman20231012, author = "Gutman A.E. and Kusraev A.G.", title = "Boolean valued analysis and the Wickstead problem", booktitle = "Mathematical Forum. Vol. 14. Modern Mathematics. Introductory Lectures (Project OTDE-Workshop)", address = "Vladikavkaz", publisher = "SMI VSC RAS", year = "2023", isbn = "978-5-904695-46-0", pages = "11--48", language = "russian", annote = "The purpose of this mini-course, consisting of four lectures, is to sketch Boolean valued analysis and its application to one problem from the theory of linear operators in vector lattices.", keywords = "Kantorovich space, Wickstead problem, Cauchy functional equation, field extension, Boolean valued model, descent and ascent, Boolean valued reals" } @article { Gutman20231224, author = "Gutman A.E. and Emelianenkov I.A.", title = "Quasidenseness in $R^N$ and projective parallelotopes", journal = "Sib. Matem. Zh.", year = "2024", volume = "65", number = "2", pages = "258--276", doi = "10.33048/smzh.2024.65.203", language = "russian", annote = "We establish two new criteria for the closedness of Archimedean cones in countable-dimensional locally convex spaces in terms of projective parallelotopes and projective automorphisms. We also answer some open questions about quasidenseness and quasi-interior.", keywords = "Archimedean ordered vector space, locally convex space, weak topology, cone, quasi-interior, quasidense set" } @article { Gutman20231225, author = "Gutman A.E. and Emelianenkov I.A.", title = "Quasidenseness in $R^N$ and projective parallelotopes", journal = "Sib. Math. J.", year = "2024", volume = "65", number = "2", pages = "265--278", doi = "10.1134/S0037446624020034", annote = "We establish two new criteria for the closedness of Archimedean cones in countable-dimensional locally convex spaces in terms of projective parallelotopes and projective automorphisms. We also answer some open questions about quasidenseness and quasi-interior.", keywords = "Archimedean ordered vector space, locally convex space, weak topology, cone, quasi-interior, quasidense set" } @inproceedings { Gutman20240924, author = "Gutman A.E.", title = "Archimedean and closed cones", booktitle = "Conference on geometric analysis dedicated to the 95th anniversary of the birth of academician Yu.G.Reshetnyak (Novosibirsk, September 22--28, 2024): Proceedings", address = "Novosibirsk", year = "2024", pages = "40--42", doi = "10.5281/zenodo.13830148", language = "russian", annote = "A review of the results obtained towards describing the class of locally convex spaces in which all Archimedean cones are closed." } @article { Gutman20250114, author = "Gutman A.E. and Koptev A.V.", title = "Lateral convergence and homomorphisms of Banach bundles", journal = "Sib. Matem. Zh.", year = "2025", volume = "66", number = "2", pages = "188--203", doi = "10.33048/smzh.2025.66.205", language = "russian", annote = "We introduce and study the concepts of injective and lateral convergence in a topological space and obtain some results on the existence of homomorphisms of continuous Banach bundles, as well as continuous and weakly continuous vector-valued functions and sections that take preassigned values at the points of injectively and laterally convergent sequences.", keywords = "topological space, separation, convergent sequence, continuous Banach bundle, homomorphism, section" } @article { Gutman20250115, author = "Gutman A.E. and Emelianenkov I.A.", title = "Lateral convergence and homomorphisms of Banach bundles", journal = "Sib. Math. J.", year = "2025", volume = "66", number = "2", pages = "279--290", doi = "10.1134/S0037446625020053", annote = "We introduce and study the concepts of injective and lateral convergence in a topological space and obtain some results on the existence of homomorphisms of continuous Banach bundles, as well as continuous and weakly continuous vector-valued functions and sections that take preassigned values at the points of injectively and laterally convergent sequences.", keywords = "topological space, separation, convergent sequence, continuous Banach bundle, homomorphism, section" }