Algebraic curve in the unit ball in ℂ2 that passes through the origin and all of whose boundary components are arbitrarily short

A negative answer is given to the following question of A.G. Vitushkin: Does there exist a nontrivial lower bound for the length of the maximal component of intersection of the unit sphere and an algebraic curve passing through the origin. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 253, pp. 135–157. To the memory of Anatolii Georgievich Vitushkin     

Proof schemata in Hilbert-type axiomatic theories

A derivation schema in an axiomatic theory is defined as a finite sequence of analysis of applications of rules and axioms. A derivation by a schema U is any derivation whose list of analyses of applications of rules and axioms is precisely U. A derivation schema is admissible if a corresponding derivation can be constructed. Let G be a Hilbert-type axiomatic theory. The following problems are considered: a) to decide whether a given derivation scheme is admissible in G; b) to decide whether a formula is derivable by a given derivation schema in G. In the usual formulations of the predicate calculus without equality, the first problem is shown to be decidable, the second undecidable.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 174, pp. 132–146, 1988.     

An example in connection with the Jacobian conjecture

Translated from Matematicheskie Zametki, Vol. 47, No. 1, pp. 127–136, January, 1990.     

Remark on Kreisel's conjecture

At the end of the 60's, Kreisel conjectured that formal arithmetics admit infinite induction rules if the lengths of proofs of their premises are uniformly bounded by the same number. By the length of a proof we mean the number of applications of axioms and rules of inference. In this article we construct a theory R ^* with a finite number of specific axioms. The language of R ^* contains a constant 0, a unary function symbol, equality, and ternary predicates for addition and multiplication. It is proved that for any consistent axiomatizable extensionR^* it is possible to find a formula A(a) that satisfies the following conditions: a) xA(x) cannot be derived in; b) for any n the length of the proof of the formula A(0⁽ⁿ⁾) is no greater than c₁[log₂ (n+1)]+c₂, where the constants c₁ and c₂ are independent of n. Here the expression 0⁽ⁿ⁾ indicates 0 with n primes.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institua im. V. A. Steklova Akademii Nauk SSSR, Vol. 176, pp. 118–126, 1989.     

On rational cuspidal curves

Let be a rational curve of degree d which has only one analytic branch at each point. Denote by m the maximal multiplicity of singularities of C. It is proven in [MS] that . We show that where is the square of the “golden section”. We also construct examples which show that this estimate is asymptotically sharp. When , we show that and this estimate is sharp. The main tool used here, is the logarithmic version of the Bogomolov-Miyaoka-Yau inequality. For curves as above we give an interpretation of this inequality in terms of the number of parameters describing curves of a given degree and the number of conditions imposed by singularity types. Received: 11 February 2000 / Published online: 8 November 2002 RID="*" ID="*" Partially supported by Grants RFFI-96-01-01218 and DGICYT SAB95-0502     

A new decidable Horn fragment of predicate calculus

We describe an extension of the class ∀∃ of Horn formulas in predicate calculus. We prove the decidability of this class. We describe complexity characteristics such that fixing them splits this extended class into polynomially decidable subclasses. Fixing the maximum arity of predicates splits our class into subclasses belonging to NP. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 316, 2004, pp. 147–162.     

Rudolph diagrams and an analytic realization of the Vitushkin covering

An example of a three-sheeted covering over the ball in ² is constructed. The covering is the union of a four-dimensional ball with an analytic disk. The projection of the ball is a local homeomorphism ramified along the disk, whose projection is a part of an algebraic curve.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 206–224, August, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-011-225.     

Three ways of recognizing inessential formulas in sequents

The first method is based on the familiar method of lowering thinnings downwards and is a further development of the lemmas on weeding of [1]. The second method is based on the use of sufficiently wide classes of sequents, for which derivability in the intuitionistic predicate calculus coincides with derivability in the classical predicate calculus and the familiar property of disjunction is true. By this method one can get, e.g., a syntactic proof of the following assertion. If the positive formula A is derivable in the theory of groups under additional assumptions of the form then A is also derivable in the theory of groups without these assumptions. As the third method there is proposed a syntactically formulated test for the conservativeness of extensions of intuitionistic axiomatic theories. With the help of this test one can get, for example, a syntactic proof of the hereditary undecidability of the intuitionistic theory of equality, with additional axioms which are the formula, all formulas of the form and all negations of formulas derivable in the classical predicate calculus.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 163–175, 1979.