Piotr Krasoń

• Professor (Full) at University of Szczecin

In the paper we consider a generalization of classical Lissajous curves to the situation where corresponding differential forms involve square roots of quartics. We give a new interesting parametrization of these curves and fully analyze their behaviour in terms of roots of the quartics. We indicate natural applications of our method to the analysi...

In their recent work the second and the third authors extended the methods of M. A. Papanikolas and N. Ramachandran and determined the t-module structure on Ext1(Φ, Ψ) where Φ and Ψ were Anderson t-modules over A = Fq[t] of some specific types. This approach involved the concept of biderivation and certain reduction algorithm. In this paper, we gen...

In \cite{kk04} the second and third author extended the methods of \cite{pr} and determined the \tm module structure on $\Ext^1(\Phi,\Psi )$ where $\Phi $ and $\Psi$ were Anderson \tm modules over $A={\mathbf F}_q[t]$ of some specific types. This approach involved the concept of biderivation and certain reduction algorithm. In this paper we general...

In the work of M. A. Papanikolas and N. Ramachandran [A Weil-Barsotti formula for Drinfeld modules, Journal of Number Theory 98, (2003), 407-431] the Weil-Barsotti formula for the function field case concerning $\Ext_{\tau}^1(E,C)$ where $E$ is a Drinfeld module and $C$ is the Carlitz module was proved. We generalize this formula to the case where...

Abstract. Let A = Fq[t] be the polynomial ring over a finite field Fq and let φ and ψ be A−Drinfeld modules. In this paper we con- sider the group Ext^1(φ,ψ) with the Baer addition. We show that if rankφ > rankψ then Ext^1(φ,ψ) has the structure of a t−module. We give complete algorithm describing this structure. We generalize this to the cases: Ex...

We consider the relative motion of the system of two masses connected by a spring. We analyze it in a range of the Hooke’s law and show that the equations of the relative motion of the system are nonlinear once the equilibrium length of the spring is nonzero. Although the way of deriving the equations of motion is standard in classical mechanics so...

In this paper, we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In [G. Banaszak and P. Krasoń, On a local to global principle in étale K-groups of curves, J. K-Theory Appl. Algebra Geom. Topol. 12 (2013) 183–201], G. Banaszak and the author obtained the suf...

In this paper we develop an algorithm for obtaining some new linear relations among the Lauricella F D functions. Relations we obtain, generalize those hinted in the work of B. C. Carlson [C63]. The coefficients of these relations are contained in the ring of polynomials in the variables x 1 ,. .. , x N or in some exceptional cases in the field of...

In this paper we treat certain elliptic and hyper-elliptic integrals in a unified way. We introduce a new basis of these integrals coming from certain basis φn(x) of polynomials and show that the transition matrix between this basis and the traditional monomial basis is certain upper triangular band matrix. This allows us to obtain explicit formula...

In this paper we study both analytical and combinatorial properties of the solutions of the eigenproblem for the Heisenberg s-1/2 model for two deviations. Our analysis uses Chebyshev polynomials, Inverse Bethe Ansatz, winding numbers and rigged string configurations. We show some combinatorial aspects of strings in geometric way. We discuss some e...

We consider an eigenvalue problem for an inverted one dimensional harmonic oscilla-tor. We find a complete description for the eigenproblem in C ∞ (R). The eigenfunc-tions are described in terms of the confluent hypergeometric functions, the spectrum is C. The spectrum of the differential operator − d dx 2 − ω 2 x 2 is continuous and has physical s...

We consider an eigenvalue problem for an inverted one dimensional harmonic oscillator. We find a complete description for the eigenproblem in $C^{\infty}(\mathbb R)$. The eigenfunctions are described in terms of the confluent hypergeometric functions, the spectrum is ${\mathbb C}$. The spectrum of the differential operator $-{\frac{d}{dx^2}}-{\omeg...

In this paper we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In \cite{bk13} G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle for {\'e}tale $K$-theory of a curve . This condition in fact has bee...

In this paper we compute in some new cases the cardinalities of the fibers of certain natural fibrations that appear in the analysis of the configuration space of the Heisenberg ring. This is done by means of certain cyclic group actions on some subsets of restricted partitions.

In this paper we compute in some new cases the cardinalities of the fibers of certain natural fibrations that appear in the analysis of the configuration space of the Heisenberg ring. This is done by means of certain cyclic group actions on some subsets of restricted partitions.

In this paper we treat certain elliptic and hyper-elliptic integrals in a unified way. We introduce a new basis of these integrals coming from certain basis ${\phi}_n(x)$ of polynomials and show that the transition matrix between this basis and the traditional monomial basis is certain upper triangular band matrix. This allows us to obtain explicit...

In this paper we investigate a local to global principle for Mordell-Weil group defined over a ring of integers ${\cal O}_K$ of $t$-modules that are products of the Drinfeld modules ${\widehat\varphi}={\phi}_{1}^{e_1}\times \dots \times {\phi}_{t}^{e_{t}}.$ Here $K$ is a finite extension of the field of fractions of $A={\mathbb F}_{q}[t].$ We assum...

We consider a new generalization of Hermite polynomials to the case of several variables. Our construction is based on an analysis of the generalized eigenvalue problem for the operator ∂Ax+D∂Ax+D, acting on a linear space of polynomials of N variables, where A is an endomorphism of the Euclidean space RNRN and D is a second order differential oper...

Let X be a smooth, proper and geometrically irreducible curve X defined over a number field F and let χ be a regular and proper model of X over OF,Sl . In this paper we address the problem of detecting the linear dependence over ℤl of elements in the étale K-theory of χ. To be more specific, let P ∊ K et 2n (χ) and let ⋀̂ ⊂ K et 2n (χ) be a...

In this paper we investigate the image of the $l$-adic representation attached to the Tate module of an abelian variety defined over a number field. We consider simple abelian varieties of type III in the Albert classification. We compute the image of the $l$-adic and mod $l$ Galois representations and we prove the Mumford-Tate and Lang conjectures...

In this paper we investigate linear dependence of points in Mordell-Weil groups of abelian varieties via reduction maps. In particular we try to determine the conditions for detecting linear dependence in Mordell-Weil groups via finite number of reductions. Comment: 20 pages. Revision of introduction and bibliography

In this paper we investigate reduction of nontorsion elements in the étale K-theory of a curve X over a global field F. We show that the reduction map can be well understood in terms of Galois cohomology of l-adic representations.

In this paper we investigate the image of the $l$-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert classification. We compute the image explicitly and verify the classical conjectures of Mumford-Tate, Hodge, Lang and Tate, for a large family of abelian...

We consider the local to global principle for detecting linear dependence of points in groups of the Mordell-Weil type. As applications of our general setting we obtain corresponding statements for Mordell-Weil groups of non-CM elliptic curves and some higher dimensional abelian varieties defined over number fields, and also for odd dimensional K-g...

We consider the local to global principle for detecting linear dependence of points in groups of the Mordell-Weil type. As applications of our general setting we obtain corresponding statements for Mordell-Weil groups of non{-}CM elliptic curves and some higher dimensional abelian varieties defined over number fields, and also for odd dimensional K...

In this paper we study the image of l-adic representations coming from Tate module of an abelian variety defined over a number field. We treat abelian varieties with complex and real multiplications. We verify the Mumford-Tate conjecture for a new class of abelian varieties with real multiplication. 1. Introduction. Let l be an odd prime number, F...

We consider the support problem of Erdös in the context of l-adic representations of the absolute Galois group of a number field. Main applications of the results of the paper concern Galois cohomology of the Tate module of abelian varieties with real and complex multiplications, the algebraic K-theory groups of number fields and the integral homol...

. In this paper we consider a support problem for the reduction map on the odd dimensional algebraic K-theory of number fields. 1. Introduction. Pal Erdos stated the following problem for the integers in 1988. Support Problem. For two positive integers x and y the following two statements are equivalent. (1) For every n # N and every prime p we hav...

In this paper we consider the Quillen–Lichtenbaum conjecture for number fields using description of the higher K-theory in terms of SK 1 groups.

. In the present work, we investigate the reduction map on the 'etale K- theory of a curve defined over a global field. We prove that on the even-dimensional K-groups the map has finite kernel and reduce the odd-dimensional case to a conjecture of Jannsen. 1. Introduction The algebraic K-groups of arithmetic schemes are expected to carry deep arith...

In this note we prove that the integral homology of the special linear group of the henselization of some local rings imbeds into the homology of the special linear group of the completion. We define henselian adele ring and prove that the integral homology of its special linear group injects into the homology of Sl of the finite adele ring. The me...

In this note we prove that the integral homology of the special linear group of the henselization of some local rings imbeds into the homology of the special linear group of the completion. We define hens&an ad& ring and prove that the integral homology of its special linear group injects into the homology of Sl of the finite adele ring. The method...

Contents In the paper a generalization of the Global Element Method developed in [3] for elliptic boundary value problems to complex Helmholtz is regarded. The method in this case relies upon minimization of two functionals apart with respect to real and imaginary parts. The structure of the resulting system of simultaneous linear equations has als...

In the paper an application of the Global Element Method in static field problems is considered. — The method is based on variational principle for elliptic boundary value problems and preserves the advantages of the global variational techniques such as rapid convergence and ease of increasing the accuracy of solution. A system consisting of an el...

In the paper the distributions of magnetic and electric fields in one-dimensional nonlinear electromagnetic field problems have been regarded. An analysis have been performed numerically by means of the finite difference method. The distributions of the electromagnetic field in a shield iron and in a hollowed iron cylinder have been calculated to i...

An analysis of the force acting on the core of the coil in shape of a rotation ellipsoid has been performed. Formulas which determine the value of force related to geometrical dimensions have been obtained. Regarding certain limit, one can examine a particular case of the ball core. A numerical analysis of corresponding formulas has been performed...

A system which appears in eddy current defectoscopy has been considered. The aim of the paper was an analysis of the distribution of eddy current induced by alternating magnetic field. Possibilities of the Galerkin and least square method have been investigated and compared with an analysis by means of the integral equation method.

The Ritz variational method is suggested for use in the determination of the equivalent time constant of diffusion processes. General formulas are obtained and illustrated with examples.RésuméLa méthode variationnelle de Ritz est suggérée pour la détermination de la constante de temps équivalente du processus de diffusion. Des formules générales on...

Contents In the paper the eddy-current distribution and the coefficient of screening for axial-symmetric problems are regarded.—The algorithm used for calculations is based on the analysis of the Fredholm integral equation of the second kind which corresponds to these problems.

The paper deals with the integral equation to the eddy current problems. The various methods which are possible to apply for solving the appropriate Fredholm integral equation are compared. Numerical calculations have been performed for several examplary systems.

The distribution of the magnetic field for a cable with a strand having an elliptic cross section has been performed. The exact formulas describing the inductance of the cable have been obtained. An analysis has proved that there is an essential influence of deformation over the inductance. Results have been compared with the changes of capacitance...

The object of this work is the development of algorithms which allow one to describe properties of electromagnetic screen in shape of a drilled cylinder and also the efficiency of a screen protecting the ferromagnetic tank placed in a window of the power transformer. The method used for computation is based on the Fredholm integral equation of the...

Übersicht In der vorliegenden Arbeit wird die Stromverteilung in einer ebenen Anordnung zweier rechteckiger Leiter mit Hilfe der Fredholmschen Integralgleichung analysiert. Durch Einführung der nicht leitenden hochpermeablen Abschirmung wird die Integralgleichung als eine Integralgleichung mit degeneriertem Kern beschrieben und als Lösung eines lin...

A theoretical study of the uniform magnetic field generation at the midplane of a thin-walled solenoid is presented. The field synthesis problem is posed and an appropriate mathematical method proposed to solve it. Some numerical results are presented.

Contents The paper deals with the synthesis of magnetic field in the plane perpendicular to the axis of a solenoid.

Übersicht In der vorliegenden Arbeit wird die Stromverteilung in einem Bandleiter mit Hilfe der Fredholmschen Integralgleichung analysiert. Es wird der Vergleich zwischen den numerisch ausgewerteten und den in der Literatur mit Hilfe von anderen Verfahren berechneten Ergebnisse durchgeführt.

The synthesis of the magnetic field at the mid-plane of a solenoid is considered. The problem is formulated by means of the integral Fredholm equation of the first kind. The regularization method proposed by A. N. Tichonov is applied to solving the problem. Results of computations are included and analyzed.

In the present work, we investigate the reduction map on the etale K -theory of a curve deened over a global eld. We prove that on the even-dimensional K -groups the map has nite kernel and reduce the odd-dimensional case to a con-jecture of Jannsen.

Paper from a conference held Sept. 1995 in Poznan, Poland Incluye bibliografía