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1
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Two remarks about Picard-Vessiot extensions and elementary functions

100%
Żołądek H.
Colloquium Mathematicum
|
2000
|
tom 84/85
|
nr 1
173-183
EN
We present a simple proof of the theorem which says that for a series of extensions of differential fields K ⊂ L ⊂ M, where K ⊂ M is Picard-Vessiot, the extension K ⊂ L is Picard-Vessiot iff the differential Galois group $Gal_{L} M$ is a normal subgroup of $Gal_{K} M$. We also present a proof that the probability function Erf(x) is not an elementary function.
2
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On first integrals for polynomial differential equations on the line

100%
Żołądek H.
Studia Mathematica
|
1993
|
tom 107
|
nr 2
205-211
EN
We show that any equation dy/dx = P(x,y) with P a polynomial has a global (on ℝ²) smooth first integral nonconstant on any open domain. We also present an example of an equation without an analytic primitive first integral.
3
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Polynomial Riccati equations with algebraic solutions

100%
Żołądek H.
Banach Center Publications
|
2002
|
tom 58
|
nr 1
219-231
EN
We consider the equations of the form dy/dx = y²-P(x) where P are polynomials. We characterize the possible algebraic solutions and the class of equations having such solutions. We present formulas for first integrals of rational Riccati equations with an algebraic solution. We also present a relation between the problem of algebraic solutions and the theory of random matrices.
4
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On algebraic solutions of algebraic Pfaff equations

100%
Żołądek H.
Studia Mathematica
|
1995
|
tom 114
|
nr 2
117-126
EN
We give a new proof of Jouanolou's theorem about non-existence of algebraic solutions to the system $ẋ=z^{s}, ẏ=x^{s}, ż=y^{s}$. We also present some generalizations of the results of Darboux and Jouanolou about algebraic Pfaff forms with algebraic solutions.
5
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The XVI-th Hilbert problem about limit cycles

100%
Żołądek H.
Banach Center Publications
|
1995
|
tom 34
|
nr 1
167-174
EN
1. Introduction. The XVI-th Hilbert problem consists of two parts. The first part concerns the real algebraic geometry and asks about the topological properties of real algebraic curves and surfaces. The second part deals with polynomial planar vector fields and asks for the number and position of limit cycles. The progress in the solution of the first part of the problem is significant. The classification of algebraic curves in the projective plane was solved for degrees less than 8. Among general results we notice the inequalities of Harnack and Petrovski and Rohlin's theorem. Other results were obtained by Newton, Klein, Clebsch, Hilbert, Nikulin, Kharlamov, Gudkov, Arnold, Viro, Fidler. There are multidimensional generalizations: the theory of Khovansky, the inequalities of Petrovski and Oleinik, and others. In contrast to the algebraic part of the problem, the progress in the solution of the second part is small. In the present article we concentrate on the second part of the XVI-th Hilbert problem.
6
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Twierdzenie o powracaniu i pewne zagadki nierównowagowej mechaniki statystycznej

100%
Żołądek H.
Matematyka Poglądowa
|
2015
|
nr 2
16-21
PL
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7
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A simple proof of the non-integrability of the first and the second Painlevé equations

100%
Żołądek H.
Banach Center Publications
|
2011
|
tom 94
|
nr 1
295-302
EN
The first and the second Painlevé equations are explicitly Hamiltonian with time dependent Hamilton function. By a natural extension of the phase space one gets corresponding autonomous Hamiltonian systems in ℂ⁴. We prove that the latter systems do not have any additional algebraic first integral. In the proof equations in variations with respect to a parameter are used.
8
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New examples of holomorphic foliations without algebraic leaves

100%
Żołądek H.
Studia Mathematica
|
1998
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tom 131
|
nr 2
137-142
EN
We present a series of polynomial planar vector fields without algebraic invariant curves in $CP^2$.
9
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Linear differential equations and multiple zeta values. I. Zeta(2)

64%
Zakrzewski M., Żołądek H.
Fundamenta Mathematicae
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2010
|
tom 210
|
nr 3
207-242
EN
Certain generating fuctions for multiple zeta values are expressed as values at some point of solutions of linear meromorphic differential equations. We apply asymptotic expansion methods (like the WKB method and the Stokes operators) to solutions of these equations. In this way we give a new proof of the Euler formula ζ(2) = π²/6. In further papers we plan to apply this method to study some third order hypergeometric equation related to ζ(3).
10
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Geometry of Puiseux expansions

64%
Borodzik M., Żołądek H.
Annales Polonici Mathematici
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2008
|
tom 93
|
nr 3
263-280
EN
We consider the space Curv of complex affine lines t ↦ (x,y) = (ϕ(t),ψ(t)) with monic polynomials ϕ, ψ of fixed degrees and a map Expan from Curv to a complex affine space Puis with dim Curv = dim Puis, which is defined by initial Puiseux coefficients of the Puiseux expansion of the curve at infinity. We present some unexpected relations between geometrical properties of the curves (ϕ,ψ) and singularities of the map Expan. For example, the curve (ϕ,ψ) has a cuspidal singularity iff it is a critical point of Expan. We calculate the geometric degree of Expan in the cases gcd(degϕ,degψ) ≤ 2 and describe the non-properness set of Expan.
11
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Solution of the 1 : −2 resonant center problem in the quadratic case

52%
Fronville A., Sadovski A. P., Żołądek H.
Fundamenta Mathematicae
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1998
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tom 157
|
nr 2-3
191-207
EN
The 1:-2 resonant center problem in the quadratic case is to find necessary and sufficient conditions (on the coefficients) for the existence of a local analytic first integral for the vector field $(x + A_1x^2 + B_1xy + Cy^2) ∂_x+(-2y + Dx^2 + A_2xy + B_2y^2)∂_y$. There are twenty cases of center. Their necessity was proved in [4] using factorization of polynomials with integer coefficients modulo prime numbers. Here we show that, in each of the twenty cases found in [4], there is an analytic first integral. We develop a new method of investigation of analytic properties of polynomial vector fields.
12
Content available remote

Book reviews: ''Limit cycles of differential equations'' by Colin Christopher and Chengzhi Li

51%
Żołądek H.
Control and Cybernetics
|
2008
|
tom 37
|
nr 1
231-236
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