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1.
For open discrete mappings f:D\{ b } ? \mathbbR3 f:D\backslash \left\{ b \right\} \to {\mathbb{R}^3} of a domain D ì \mathbbR3 D \subset {\mathbb{R}^3} satisfying relatively general geometric conditions in D \ {b} and having an essential singularity at a point b ? \mathbbR3 b \in {\mathbb{R}^3} , we prove the following statement: Let a point y 0 belong to [`(\mathbbR3)] \f( D\{ b } ) \overline {{\mathbb{R}^3}} \backslash f\left( {D\backslash \left\{ b \right\}} \right) and let the inner dilatation K I (x, f) and outer dilatation K O (x, f) of the mapping f at the point x satisfy certain conditions. Let B f denote the set of branch points of the mapping f. Then, for an arbitrary neighborhood V of the point y 0, the set Vf(B f ) cannot be contained in a set A such that g(A) = I, where I = { t ? \mathbbR:| t | < 1 } I = \left\{ {t \in \mathbb{R}:\left| t \right| < 1} \right\} and g:U ? \mathbbRn g:U \to {\mathbb{R}^n} is a quasiconformal mapping of a domain U ì \mathbbRn U \subset {\mathbb{R}^n} such that A ⊂ U.  相似文献   

2.
We study space mappings more general than the mappings with bounded distortion in the sense of Reshetnyak. We consider questions related to the local behavior of mappings differentiable almost everywhere, possessing Properties N, N ?1, ACP, and ACP ?1, and such that quasiconformality coefficient satisfies a certain restriction on growth. We show that the value of a mapping satisfying these requirements on an arbitrary neighborhood of an essential singularity can be greater in absolute value than the logarithm of the inverse radius of the ball raised to an arbitrary positive power.  相似文献   

3.
For the mappings f:D ? D¢,  D,  D¢ ì \mathbbRn f:D \to D',\,\,D,\,\,D' \subset {\mathbb{R}^n} , n ≥ 2, satisfying certain geometric conditions in the fixed domain D, we have proved estimates of the form K I (x, f) ≤ Q(x) almost everywhere, where K I (x, f) is the inner dilatation of f at a point x, and Q(x) is a fixed real-valued function responsible for the “control” over a distortion of the families of curves in D at a mapping f.  相似文献   

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In this paper, we prove some positive results on discreteness and openness of mappings of finite distortion under some integrability condition on the distortion and the multiplicity function of the mapping. We also show that in some sense, our results are sharp.  相似文献   

6.
We consider a family of open discrete mappings f:D ?[`(\mathbb Rn)] f:D \to \overline {{{\mathbb R}^n}} that distort, in a special way, the p-modulus of a family of curves that connect the plates of a spherical condenser in a domain D in \mathbb Rn {{\mathbb R}^n} ; p > n-1; p < n; and bypass a set of positive p-capacity. We establish that this family is normal if a certain real-valued function that controls the considered distortion of the family of curves has finite mean oscillation at every point or only logarithmic singularities of order not higher than n - 1: We show that, under these conditions, an isolated singularity x 0D of a mapping f:D\{ x0 } ?[`(\mathbb Rn)] f:D\backslash \left\{ {{x_0}} \right\} \to \overline {{{\mathbb R}^n}} is removable, and, moreover, the extended mapping is open and discrete. As applications, we obtain analogs of the known Liouville and Sokhotskii–Weierstrass theorems.  相似文献   

7.
Translated from Sibirski Matematicheski, Vol. 36, No. 2, pp. 404–408, March–April, 1995.  相似文献   

8.
The article is devoted to the study of the boundary properties of homeomorphisms f: DD′, D,D′ ? ? n , satisfying some geometric conditions responsible for the control of the measure of distortion of families of curves in D. Under additional requirements on the boundaries ?D and ?D′ of the domains, we prove that the family of all such homeomorphisms is equicontinuous in $\bar D$ .  相似文献   

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Suppose a closed unbounded set F Rn is a union of a finite number p of closed unbounded sets Fi that are pairwise disjoint, and suppose f is a continuous mapping of F into the metric space r(2). With each set Fi there is associated a point at infinity i, at which it is assumed that f has a finite limit Ai R(2), i=1, 2, ..., p. It is proved that: 1) f is bounded on F; 2) if f is a real functional, then the set contains a smallest and a largest value; 3) if the distance between Fi and Fj is greater than zero whenever i j, then f is uniformly continuous on F.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 422–427, March, 1991.  相似文献   

11.
Suppose a closed unbounded set F Rn is a union of a finite number p of closed unbounded sets Fi that are pairwise disjoint, and suppose f is a continuous mapping of F into the metric space R(2). With each set Fi there is associated a point at infinity i, at which it is assumed that f has a finite limit Ai R(2), i=1, 2, ..., p. It is proved that: 1) f is bounded on F; 2) if f is a real functional, then the set contains a smallest and a largest value; 3) if the distance between Fi and Fj is greater than zero whenever i j, then f is uniformly continuous on F.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 3, pp. 422–427, March, 1991.  相似文献   

12.
The principle aim of this essay is to illustrate how different phenomena is captured by different discretizations of the Hopf equation and general hyperbolic conservation laws. This includes dispersive schemes, shock capturing schemes as well as schemes for computing multi-valued solutions of the underlying equation. We introduce some model equations which describe the behavior of the discrete equation more accurate than the original equation. These model equations can either be conveniently discretized for producing novel numerical schemes or further analyzed to enrich the theory of nonlinear partial differential equations.  相似文献   

13.
It follows from Banach’s fixed point theorem that every nonexpansive self-mapping of a bounded, closed and convex set in a Banach space has approximate fixed points. This is no longer true, in general, if the set is unbounded. Nevertheless, as we show in the present paper, there exists an open and everywhere dense set in the space of all nonexpansive self-mappings of any closed and convex (not necessarily bounded) set in a Banach space (endowed with the natural metric of uniform convergence on bounded subsets) such that all its elements have approximate fixed points.  相似文献   

14.
We prove discreteness and openness of the mappings with finite distortion defined on ℍ-type Carnot groups whose distortion functions belong to certain Orlicz spaces. Similar results in Euclidean spaces were established bymany authors under different assumptions about the mappings and their distortion functions. The text was submitted by the authors in English.  相似文献   

15.
The Fock–Bargmann–Hartogs domain Dn,m(μ)Dn,m(μ) (μ>0μ>0) in Cn+mCn+m is defined by the inequality ‖w‖2<e−μ‖z‖2w2<eμz2, where (z,w)∈Cn×Cm(z,w)Cn×Cm, which is an unbounded non-hyperbolic domain in Cn+mCn+m. Recently, Yamamori gave an explicit formula for the Bergman kernel of the Fock–Bargmann–Hartogs domains in terms of the polylogarithm functions and Kim–Ninh–Yamamori determined the automorphism group of the domain Dn,m(μ)Dn,m(μ). In this article, we obtain rigidity results on proper holomorphic mappings between two equidimensional Fock–Bargmann–Hartogs domains. Our rigidity result implies that any proper holomorphic self-mapping on the Fock–Bargmann–Hartogs domain Dn,m(μ)Dn,m(μ) with m≥2m2 must be an automorphism.  相似文献   

16.
A metrizable compact convex set K is stable if and only if continuous averaging on K is open.  相似文献   

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We prove newa priori estimates for the resolvent of a minimal quantum dynamical semigroup. These estimates simplify well-known conditions sufficient for conservativity and impose continuity conditions on the time-dependent operator coefficients ensuring the existence of conservative solutions of the Markov evolution equations. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 125–140, January, 1997. Translated by A. M. Chebotarev  相似文献   

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