Multidegrees of tame automorphisms of C^n

Let F=(F_1,...,F_n):C^n --> C^n be a polynomial mapping. By the multidegree of the mapping F we mean mdeg F=(deg F_1,...,deg F_n), an element of N^n. The aim of this paper is to study the following problem (especially for n=3): for which sequence (d_1,...,d_n) in N^n there is a tame automorphism F of C^n such that mdeg F=(d_1,...,d_n). In other words we investigate the set mdeg(Tame(C^n)), where Tame(C^n) denotes the group of tame automorphisms of C^n and mdeg denotes the mapping from the set of polynomial endomorphisms of C^n into the set N^n. Since for all permutation s of {1,...,n} we have (d_1,...,d_n) is in mdeg(Tame(C^n)) if and only if (d_s(1),...,d_s(n)) is in mdeg(Tame(C^n)) we may focus on the set mdeg(Tame(C^n)) intersected with {(d_1,...,d_n) : d_1<=...<=d_n}. In the paper, among other things, we give complete description of the sets: mdeg(Tame(C^n)) intersected with {(3,d_2,d_3):3<=d_2<=d_3}}, mdeg(Tame(C^n)) intersected with {(5,d_2,d_3):5<=d_2<=d_3}}, In the examination of the last set the most difficult part is to prove that (5,6,9) is not in mdeg(Tame(C^n)). As a surprising consequence of the method used in proving that (5,6,9) is not in mdeg(Tame(C^n)), we obtain the result saying that the existence of tame automorphism F of C^3 with mdeg F=(37,70,105) implies that two dimensional Jacobian Conjecture is not true. Also, we give the complete description of the following sets: mdeg(Tame(C^n)) intersected with {(p_1,p_2,d_3):3<=p_1<p_2<=d_3}}, where p_1 and p_2 are prime numbers, mdeg(Tame(C^n)) intersected with {(d_1,d_2,d_3):d_1<p_2<=d_3}}, where d_1 and d_2 are odd numbers such that gcd(d_1,d_2)=1. Using description of the last set we show that the set mdeg(Aute(C^n))\mdeg(Tame(C^n)) is infinite

Extension of polynomial mappings with a given Łojasiewicz exponent

Abstract. Let V ⊂ Cn be an affine subspace. We prove that for a poly-nomial mapping f: V → Cm, n ≤ m, there is an extension F: Cm → Cm with the same Lojasiewicz exponent at infinity. Let V be an infinite algebraic subset of Cn and let f: V → Cm be a polynomial mapping. By the Lojasiewicz exponent at infinity (or the global Lojasiewicz exponent) of f, we mean the number L∞(f) = sup{ν ∈ R: ∃A,B&gt; 0 ∀z ∈ V |z |&gt; B ⇒ A |z|ν ≤ |f(z)|}. The number L∞(f) does not depend on the choice of norms and on linear change of coordinates (see e.g. [1]). Thus in the sequel we will use the maximum norm. In this note we prove the following: Theorem 1. For every affine subspace V ⊂ Cn and for every polynomial mapping f: V → Cm, with n ≤ m, there exists a polynomial mapping F: Cn → Cm with F |V = f and L∞(F) = L∞(f). The above result is the very first step in a research into the following question: For what kind of sets V ⊂ Cn and what kind of mappings f: V → Cm, n ≤ m, does there exist a polynomial extension F: Cn → Cm of f with L∞(F) = L∞(f)? or into a more general one: What can we say about max{L∞(F) : F |V = f}

A note on triangular automorphisms

Abstract. In this short note we propose a new very easy and elementary proof of the known fact that every triangular automorphism of kn is the ex-ponent of a suitably chosen locally nilpotent k-derivation on k[x1,..., xn]. Two other, different proofs of this fact can be found in [2] and [3]. 1. Introduction. Le

Założenia „Normatywnego słownika polskich nazw miejscowych z obszaru Litwy“

The aim of the article is to elaborate on the principles of the online edition of The Normative Dictionary of the Polish Place Names from the area of Lithuania. This is a normative dictionary. It contains Polish equivalents of contemporary Lithuanian place names from the area of Lithuania, their declination, adjectives and names of residents. The dictionary contains names of districts, towns and villages, counting over 20 residents from Vilnius, Šalčininkai, Trakai, Švenčionys and Širvintos regions. The dictionary includes names from the Vilnius area inhabited by numerous Polish minority, and selected names from different parts of Lithuania, important in terms of history, culture and society (e.g. from the area of Lauda and the vicinity of Kaunas). The article outlines the structure of the dictionary, looks into difficulties encountered in its development and supplements this study with a brief description of the Dictionary and of the present state of work on it.Lietuvos teritorijos lenkiškų vietovardžių normatyvinio žodyno prielaidos Straipsnio tikslas – aptarti internetinio Lietuvos teritorijos lenkiškų vietovardžių normatyvinio žodyno prielaidas. Minėtas žodynas yra norminis, jame pateikiami sprendimai dėl lenkiškų atitikmenų dabartinių lietuviškų vietovardžių Lietuvos teritorijoje, jų linksniavimo, darinių, ypač būdvardžių ir gyventojų pavadinimų taisyklingumo. Žodyne yra savivaldybių ir rajonų miestų ir miestelių pavadinimai, taip pat Vilniaus, Šalčininkų, Trakų, Švenčionių ir Širvintų rajonų, t.y. Vilniaus krašto, kuriame tankiausiai gyvena lenkų mažuma Lietuvoje, vietovių, turinčių daugiau negu 20 gyventojų pavadinimai. Taip pat atrinkti vietovardžiai iš kitų Lietuvos vietovių, svarbių istorine, kultūrine ir socialine prasme (pvz., iš Kauno apylinkių). Straipsnyje aptariamos žodyno prielaidos, jo kūrimo sunkumai, pateikiamas trumpas aprašymas ir darbų jį rengiant būklė

Pure torsion problem in tensor notation

The paper examines the application of the tensor calculus to the classic problem of the pure torsion of prismatic rods. The introduction contains a short description of the reference frames, base vectors, contravariant and covariant vector coordinates when applying the Einstein summation convention. Torsion formulas were derived according to Coulomb’s and Saint-Venant’s theories, while, as a link between the theories, so-called Navier’s error was discussed. Groups of the elasticity theory equations were used

Hanks-Type Serine/Threonine Protein Kinases and Phosphatases in Bacteria: Roles in Signaling and Adaptation to Various Environments

Reversible phosphorylation is a key mechanism that regulates many cellular processes in prokaryotes and eukaryotes. In prokaryotes, signal transduction includes two-component signaling systems, which involve a membrane sensor histidine kinase and a cognate DNA-binding response regulator. Several recent studies indicate that alternative regulatory pathways controlled by Hanks-type serine/threonine kinases (STKs) and serine/threonine phosphatases (STPs) also play an essential role in regulation of many different processes in bacteria, such as growth and cell division, cell wall biosynthesis, sporulation, biofilm formation, stress response, metabolic and developmental processes, as well as interactions (either pathogenic or symbiotic) with higher host organisms. Since these enzymes are not DNA-binding proteins, they exert the regulatory role via post-translational modifications of their protein targets. In this review, we summarize the current knowledge of STKs and STPs, and discuss how these enzymes mediate gene expression in prokaryotes. Many studies indicate that regulatory systems based on Hanks-type STKs and STPs play an essential role in the regulation of various cellular processes, by reversibly phosphorylating many protein targets, among them several regulatory proteins of other signaling cascades. These data show high complexity of bacterial regulatory network, in which the crosstalk between STK/STP signaling enzymes, components of TCSs, and the translational machinery occurs. In this regulation, the STK/STP systems have been proved to play important roles

The simple method of dynamic visco-elastic analysis of road structure. on rheological foundation

In contrast to computationally advanced methods of road pavement dynamic analysis, the one-dimensional, simple method is derived on the basis of visco-elastic simple beam lying on generalized Winkler visco-elastic foundation. By virtue of least square method the visco-elastic constants could be estimated with technically admissible accuracy. The introduced method is useful enough to predict any pavement deformation process in the range of linear visco-elasticity.In contrast to computationally advanced methods of road pavement dynamic analysis, the one-dimensional, simple method is derived on the basis of visco-elastic simple beam lying on generalized Winkler visco-elastic foundation. By virtue of least square method the visco-elastic constants could be estimated with technically admissible accuracy. The introduced method is useful enough to predict any pavement deformation process in the range of linear visco-elasticity