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1.
In this paper we give the complete answer to a question posed by A. Arhangel?skii and prove that the sphere SnSn is diagonal resolvable if and only if SnSn is an H  -space if and only if n∈{0,1,3,7}n{0,1,3,7}. Moreover, we prove that any upper half even dimensional QQ-sphere cannot be diagonal resolvable.  相似文献   

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The period annuli of the planar vector field x=−yF(x,y)x=yF(x,y), y=xF(x,y)y=xF(x,y), where the set {F(x,y)=0}{F(x,y)=0} consists of k   different isolated points, is defined by k+1k+1 concentric annuli. In this paper we perturb it with polynomials of degree n and we study how many limit cycles bifurcate, up to a first order analysis, from all the period annuli simultaneously in terms of k and n  . Additionally, we prove that the associated Abelian integral is piecewise rational and, when k=1k=1, the provided upper bound is reached. Finally, the case k=2k=2 is also treated.  相似文献   

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A weak selection on an infinite set X   is a function σ:[X]2→Xσ:[X]2X such that σ({x,y})∈{x,y}σ({x,y}){x,y} for each {x,y}∈[X]2{x,y}[X]2. A weak selection on a space is said to be continuous if it is a continuous function with respect to the Vietoris topology on [X]2[X]2 and the topology on X  . We study some topological consequences from the existence of a continuous weak selection on the product X×YX×Y for the following particular cases:
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Both X and Y are spaces with one non-isolated point.  相似文献   

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In this paper, we give some necessary and sufficient conditions for the existence of Re-nnd and nonnegative definite {1,3}{1,3}- and {1,4}{1,4}-inverses of a matrix A∈Cn×nACn×n and completely described these sets. Moreover, we prove that the existence of nonnegative definite {1,3}{1,3}-inverse of a matrix A   is equivalent with the existence of its nonnegative definite {1,2,3}{1,2,3}-inverse and present the necessary and sufficient conditions for the existence of Re-nnd {1,3,4}{1,3,4}-inverse of A.  相似文献   

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The dimension of a point x   in Euclidean space (meaning the constructive Hausdorff dimension of the singleton set {x}{x}) is the algorithmic information density of x  . Roughly speaking, this is the least real number dim(x)dim(x) such that r×dim(x)r×dim(x) bits suffice to specify x   on a general-purpose computer with arbitrarily high precision 2−r2r. The dimension spectrum of a set X   in Euclidean space is the subset of [0,n][0,n] consisting of the dimensions of all points in X.  相似文献   

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We prove various Hardy-type and uncertainty inequalities on a stratified Lie group G  . In particular, we show that the operators Tα:f?|⋅|−αL−α/2fTα:f?||αLα/2f, where |⋅||| is a homogeneous norm, 0<α<Q/p0<α<Q/p, and L   is the sub-Laplacian, are bounded on the Lebesgue space Lp(G)Lp(G). As consequences, we estimate the norms of these operators sufficiently precisely to be able to differentiate and prove a logarithmic uncertainty inequality. We also deduce a general version of the Heisenberg–Pauli–Weyl inequality, relating the LpLp norm of a function f   to the LqLq norm of |⋅|βf||βf and the LrLr norm of Lδ/2fLδ/2f.  相似文献   

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We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a dd-parameter family of such semigroups satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the Julia set is the minimum of 2 and the zero of the pressure function. Moreover, the Hausdorff dimension of the exceptional set of parameters is estimated. We also show that if the zero of the pressure function is greater than 22, then typically the 2-dimensional Lebesgue measure of the Julia set is positive. Some sufficient conditions for a family to satisfy the transversality conditions are given. We give non-trivial examples of families of semigroups of non-linear polynomials with the transversality condition for which the Hausdorff dimension of the Julia set is typically equal to the zero of the pressure function and is less than 22. We also show that a family of small perturbations of the Sierpinski gasket system satisfies that for a typical parameter value, the Hausdorff dimension of the Julia set (limit set) is equal to the zero of the pressure function, which is equal to the similarity dimension. Combining the arguments on the transversality condition, thermodynamical formalisms and potential theory, we show that for each a∈CaC with |a|≠0,1|a|0,1, the family of small perturbations of the semigroup generated by {z2,az2}{z2,az2} satisfies that for a typical parameter value, the 2-dimensional Lebesgue measure of the Julia set is positive.  相似文献   

10.
We show that if T:X→XT:XX is a continuous linear operator on an FF-space X≠{0}X{0}, then the set of frequently hypercyclic vectors of TT is of first category in XX, and this answers a question of A. Bonilla and K.-G. Grosse-Erdmann. We also show that if T:X→XT:XX is a bounded linear operator on a Banach space X≠{0}X{0} and if TT is frequently hypercyclic (or, more generally, syndetically transitive), then the TT-orbit of every non-zero element of XX is bounded away from 0, and in particular TT is not hypercyclic.  相似文献   

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A new proof is given for the statement: For an irreducible, infinite Coxeter group (W,S)(W,S) and w∈WwW, if wSw-1=SwSw-1=S, then w=1w=1 (the identity element of W).  相似文献   

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A quasiplane f(V)f(V) is the image of an n-dimensional Euclidean subspace V   of RNRN (1≤n≤N−11nN1) under a quasiconformal map f:RN→RNf:RNRN. We give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi-Lipschitz n  -manifold and for a quasiplane to have big pieces of bi-Lipschitz images of RnRn. One main novelty of these results is that we analyze quasiplanes in arbitrary codimension N−nNn. To establish the big pieces criterion, we prove new extension theorems for “almost affine” maps, which are of independent interest. This work is related to investigations by Tukia and Väisälä on extensions of quasisymmetric maps with small distortion.  相似文献   

20.
The oscillation of solutions of f+Af=0f+Af=0 is discussed by focusing on four separate situations. In the complex case AA is assumed to be either analytic in the unit disc DD or entire, while in the real case AA is continuous either on (−1,1)(1,1) or on (0,∞)(0,). In all situations AA is expected to grow beyond bounds that ensure finite oscillation for all (non-trivial) solutions, and the separation between distinct zeros of solutions is considered.  相似文献   

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