The number of trees half of whose vertices are leaves and asymptotic enumeration of plane real algebraic curves

The number of topologically different plane real algebraic curves of a given degree $d$ has the form $\exp(C d^2 + o(d^2))$. We determine the best available upper bound for the constant $C$. This bound follows from Arnold inequalities on the number of empty ovals. To evaluate its rate we show its equivalence with the rate of growth of the number of trees half of whose vertices are leaves and evaluate the latter rate.Comment: 13 pages, 3 figure

Integrating a Global Induction Mechanism into a Sequent Calculus

Most interesting proofs in mathematics contain an inductive argument which requires an extension of the LK-calculus to formalize. The most commonly used calculi for induction contain a separate rule or axiom which reduces the valid proof theoretic properties of the calculus. To the best of our knowledge, there are no such calculi which allow cut-elimination to a normal form with the subformula property, i.e. every formula occurring in the proof is a subformula of the end sequent. Proof schemata are a variant of LK-proofs able to simulate induction by linking proofs together. There exists a schematic normal form which has comparable proof theoretic behaviour to normal forms with the subformula property. However, a calculus for the construction of proof schemata does not exist. In this paper, we introduce a calculus for proof schemata and prove soundness and completeness with respect to a fragment of the inductive arguments formalizable in Peano arithmetic.Comment: 16 page

The Epsilon Calculus and Herbrand Complexity

Hilbert's epsilon-calculus is based on an extension of the language of predicate logic by a term-forming operator $\epsilon_{x}$. Two fundamental results about the epsilon-calculus, the first and second epsilon theorem, play a role similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand's Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions of existential theorems obtained by this elimination procedure.Comment: 23 p

Classification of singular Q-homology planes. I. Structure and singularities

A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We obtain a structure theorem for Q-homology planes with smooth locus of non-general type. We show that if a Q-homology plane contains a non-quotient singularity then it is a quotient of an affine cone over a projective curve by an action of a finite group respecting the set of lines through the vertex. In particular, it is contractible, has negative Kodaira dimension and only one singular point. We describe minimal normal completions of such planes.Comment: improved results from Ph.D. thesis (University of Warsaw, 2009), 25 pages, to appear in Israel J. Mat

Treatment and diagnosis of arteriovenous malformation of the gastrointestinal tract using endoscopic methods

The analysis of 21 medical records of patients with arteriovenous malformation was performed. The features of the clinical course were revealed, and various types of endoscopic hemostasis were analyzed in patients with this syndromeВыполнен анализ 21 истории болезни пациентов с артериовенозной мальформацией. Выявлены особенности клинического течения, проанализированы различные виды эндоскопического гемостаза у пациентов с данным синдромо

The method of crossing the pancreas during resection

As a result of the study, a unique method of crossing the pancreas was createdв результате исследования был создан собственный уникальный метод пересечения поджелудочной желез