Sublinear Higson corona and Lipschitz extensions

The purpose of the paper is to characterize the dimension of sublinear Higson corona $\nu_L(X)$ of $X$ in terms of Lipschitz extensions of functions: Theorem: Suppose $(X,d)$ is a proper metric space. The dimension of the sublinear Higson corona $\nu_L(X)$ of $X$ is the smallest integer $m\ge 0$ with the following property: Any norm-preserving asymptotically Lipschitz function $f'\colon A\to \R^{m+1}$, $A\subset X$, extends to a norm-preserving asymptotically Lipschitz function $g'\colon X\to \R^{m+1}$. One should compare it to the result of Dranishnikov \cite{Dr1} who characterized the dimension of the Higson corona $\nu(X)$ of $X$ is the smallest integer $n\ge 0$ such that $\R^{n+1}$ is an absolute extensor of $X$ in the asymptotic category \AAA (that means any proper asymptotically Lipschitz function $f\colon A\to \R^{n+1}$, $A$ closed in $X$, extends to a proper asymptotically Lipschitz function $f'\colon X\to \R^{n+1}$). \par In \cite{Dr1} Dranishnikov introduced the category \tilde \AAA whose objects are pointed proper metric spaces $X$ and morphisms are asymptotically Lipschitz functions $f\colon X\to Y$ such that there are constants $b,c > 0$ satisfying $|f(x)|\ge c\cdot |x|-b$ for all $x\in X$. We show $\dim(\nu_L(X))\leq n$ if and only if $\R^{n+1}$ is an absolute extensor of $X$ in the category \tilde\AAA. \par As an application we reprove the following result of Dranishnikov and Smith \cite{DRS}: Theorem: Suppose $(X,d)$ is a proper metric space of finite asymptotic Assouad-Nagata dimension \asdim_{AN}(X). If $X$ is cocompact and connected, then \asdim_{AN}(X) equals the dimension of the sublinear Higson corona $\nu_L(X)$ of $X$.Comment: 13 page

Residues of pesticides in soil and water

V lončnih poskusih na prostem, brez vegetacije smo zasledovali razkroj in kopičenje prometrina (herbicid - iz skupine triazinov) v dveh različnih talnih tipih, v rendzini in v rjavih tleh. Razkroj prometrina v gornjem sloju rendzine (do 5 cm) s pH = 7 in s povprečno 9-% humusa je potekal v poletnih mesecih štirikrat počasneje, kot na rjavih tleh s pH = 5 do 6 in z manjšo količino humusa – 6%. Bolj enakomerno je potekala migracija v spodnje plasti (do 15 cm). Obratno pa je bila po nadaljnjih 6 mesecih relativna razgradnja prometrina v rjavih tleh v gornjem sloju počasnejša. Za orientacijo o prisotnosti .kloriranih ostankov pesticidov v vodah smo porabili vzorce rib. Znana je, da se v nekaterih delih vodnih živali reziduji pesticidov posebno nakopičijo. Pri analizah vzorcev rib ulovljenih ob slovenski obali Jadranskega morja smo ugotovili prisotnost DDT-ja in njegovih metabolitov. Ponekad je bil v merljivih količinah tuđi dieldrin.In pot experiments in the open, without vegetation, the degradation and accumulation of prometrine (herbicide of the triazine group) in two different soil types, the rendzina and brown soil, were observed. Degradation of prometrine in the upper layer of rendzina (up to 5 cm) with pH = 7 and an average of 9-% humus was in the summer season by 4 times slower than on brown soil with pH = 5-6 and a lesser amount of humus (6%). More regular was migration to the lower layers (up to 15 cm). After six months, however, the relative degradation of prometrine in brown soil was in the upper layer slower. The presence of chlorinated pesticide residues in water was determined or fish samples, since these residues are especially accumulated in some parts of aquatic animals. By analyses of fish samples from the Slovenian coast of the Adriatic, the presence of DDT and its metabolites was established. In some instances also measurable quantities of dieldrin were found

Compact maps and quasi-finite complexes

The simplest condition characterizing quasi-finite CW complexes $K$ is the implication $X\tau_h K\implies \beta(X)\tau K$ for all paracompact spaces $X$. Here are the main results of the paper: Theorem: If $\{K_s\}_{s\in S}$ is a family of pointed quasi-finite complexes, then their wedge $\bigvee\limits_{s\in S}K_s$ is quasi-finite. Theorem: If $K_1$ and $K_2$ are quasi-finite countable complexes, then their join $K_1\ast K_2$ is quasi-finite. Theorem: For every quasi-finite CW complex $K$ there is a family $\{K_s\}_{s\in S}$ of countable CW complexes such that $\bigvee\limits_{s\in S} K_s$ is quasi-finite and is equivalent, over the class of paracompact spaces, to $K$. Theorem: Two quasi-finite CW complexes $K$ and $L$ are equivalent over the class of paracompact spaces if and only if they are equivalent over the class of compact metric spaces. Quasi-finite CW complexes lead naturally to the concept of $X\tau {\mathcal F}$, where ${\mathcal F}$ is a family of maps between CW complexes. We generalize some well-known results of extension theory using that concept.Comment: 20 page

A classification of smooth embeddings of 3-manifolds in 6-space

We work in the smooth category. If there are knotted embeddings S^n\to R^m, which often happens for 2m<3n+4, then no concrete complete description of embeddings of n-manifolds into R^m up to isotopy was known, except for disjoint unions of spheres. Let N be a closed connected orientable 3-manifold. Our main result is the following description of the set Emb^6(N) of embeddings N\to R^6 up to isotopy. The Whitney invariant W : Emb^6(N) \to H_1(N;Z) is surjective. For each u \in H_1(N;Z) the Kreck invariant \eta_u : W^{-1}u \to Z_{d(u)} is bijective, where d(u) is the divisibility of the projection of u to the free part of H_1(N;Z). The group Emb^6(S^3) is isomorphic to Z (Haefliger). This group acts on Emb^6(N) by embedded connected sum. It was proved that the orbit space of this action maps under W bijectively to H_1(N;Z) (by Vrabec and Haefliger's smoothing theory). The new part of our classification result is determination of the orbits of the action. E. g. for N=RP^3 the action is free, while for N=S^1\times S^2 we construct explicitly an embedding f : N \to R^6 such that for each knot l:S^3\to R^6 the embedding f#l is isotopic to f. Our proof uses new approaches involving the Kreck modified surgery theory or the Boechat-Haefliger formula for smoothing obstruction.Comment: 32 pages, a link to http://www.springerlink.com added, to appear in Math. Zei

Multiple solutions for a class of perturbed second-order differential equations with impulses

n/