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The higher order degrees are Alexander-type invariants of complements to an affine plane curve. In this paper, we characterize the vanishing of such invariants for a curve C given as a transversal union of plane curves C $C^{\prime }$ and C $C^{\prime \prime }$ in terms of the finiteness and the vanishing properties of the invariants of C $C^{\prime }$ and C $C^{\prime \prime }$ , and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial Δ C multi $\Delta ^{\operatorname{multi}}_C$ is a power of ( t 1 ) $(t-1)$ , and we characterize when Δ C multi = 1 $\Delta ^{\operatorname{multi}}_C=1$ in terms of the defining equations of C $C^{\prime }$ and C $C^{\prime \prime }$ . Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves.  相似文献   

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We show that U ( k ) $U(k)$ -invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in g l ( k , C ) ${\mathfrak {g} \mathfrak {l}}(k,{\mathbb {C}})$ correspond to algebraic curves C of genus ( k 1 ) 2 $(k-1)^2$ , equipped with a flat projection π : C P 1 $\pi :C\rightarrow {\mathbb {P}}^1$ of degree k, and an antiholomorphic involution σ : C C $\sigma :C\rightarrow C$ covering the antipodal map on P 1 ${\mathbb {P}}^1$ .  相似文献   

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Inspired by some iterative algorithms useful for proving the real analyticity (or the Gevrey regularity) of a solution of a linear partial differential equation with real-analytic coefficients, we consider the following question. Given a smooth function defined on [ a , b ] R $[a,b]\subset {\mathbb {R}}$ and given an increasing divergent sequence d n $d_n$ of positive integers such that the derivative of order d n $d_n$ of f has a growth of the type M d n $M_{d_n}$ , when can we deduce that f is a function in the Denjoy–Carleman class C M ( [ a , b ] ) $C^M([a,b])$ ? We provide a positive result and show that a suitable condition on the gaps between the terms of the sequence d n $d_n$ is needed.  相似文献   

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We show the existence of a solution for an equation where the nonlinearity is logarithmically singular at the origin, namely, Δ u = ( log u + f ( u ) ) χ { u > 0 } $-\Delta u =(\log u+f(u))\chi _{\lbrace u>0\rbrace }$ in Ω R 2 $\Omega \subset \mathbb {R}^{2}$ with Dirichlet boundary condition. The function f has exponential growth, which can be subcritical or critical with respect to the Trudinger–Moser inequality. We study the energy functional I ε $I_\epsilon$ corresponding to the perturbed equation  Δ u + g ε ( u ) = f ( u ) $-\Delta u + g_\epsilon (u) = f(u)$ , where g ε $g_\epsilon$ is well defined at 0 and approximates log u $ - \log u$ . We show that I ε $I_\epsilon$ has a critical point u ε $u_\epsilon$ in H 0 1 ( Ω ) $H_0^1(\Omega )$ , which converges to a legitimate nontrivial nonnegative solution of the original problem as ε 0 $\epsilon \rightarrow 0$ . We also investigate the problem with f ( u ) $f(u)$ replaced by λ f ( u ) $\lambda f(u)$ , when the parameter λ > 0 $\lambda >0$ is sufficiently large.  相似文献   

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In this paper, we study geometry of totally real minimal surfaces in the complex hyperquadric Q N 2 $Q_{N-2}$ , and obtain some characterizations of the harmonic sequence generated by these minimal immersions. For totally real flat surfaces that are minimal immersed in both Q N 2 $Q_{N-2}$ and C P N 1 $\mathbb {C}P^{N-1}$ , we determine them for N = 4 , 5 , 6 $N=4, 5, 6$ , and give a classification theorem when they are Clifford solutions.  相似文献   

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Let p ( · ) $p(\cdot )$ be a measurable function defined on R d ${\mathbb {R}}^d$ and p : = inf x R d p ( x ) $p_-:=\inf _{x\in {\mathbb {R}}^d}p(x)$ . In this paper, we generalize the Hardy–Littlewood maximal operator. In the definition, instead of cubes or balls, we take the supremum over all rectangles the side lengths of which are in a cone-like set defined by a given function ψ. Moreover, instead of the integral means, we consider the L q ( · ) $L_{q(\cdot )}$ -means. Let p ( · ) $p(\cdot )$ and q ( · ) $q(\cdot )$ satisfy the log-Hülder condition and p ( · ) = q ( · ) r ( · ) $p(\cdot )= q(\cdot ) r(\cdot )$ . Then, we prove that the maximal operator is bounded on L p ( · ) $L_{p(\cdot )}$ if 1 < r $1<r_- \le \infty$ and is bounded from L p ( · ) $L_{p(\cdot )}$ to the weak L p ( · ) $L_{p(\cdot )}$ if 1 r $1 \le r_- \le \infty$ . We generalize also the theorem about the Lebesgue points.  相似文献   

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We obtained order estimates for the entropy numbers of the Nikol'skii–Besov classes of functions B p , θ r ( T d ) $B^{\bm{r}}_{p,\theta }(\mathbb {T}^d)$ with mixed smoothness in the metric of the space of quasi-continuous functions Q C ( T d ) $QC(\mathbb {T}^d)$ . We also showed that for 2 p $2\le p \le \infty$ , 2 θ < $2\le \theta < \infty$ , r 1 > 1 2 $r_1>\frac{1}{2}$ , d 2 $d\ge 2$ , the estimate of the corresponding asymptotic characteristic is exact in order.  相似文献   

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This paper will establish the asymptotic behavior of an anisotropic area-preserving flow which shows the existence of smooth solutions of the planar L p $L_p$ Minkowski problem for p 0 $p\ne 0$ .  相似文献   

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In this paper, we study anabelian geometry of curves over algebraically closed fields of positive characteristic. Let X = ( X , D X ) $X^{\bullet }=(X, D_{X})$ be a pointed stable curve over an algebraically closed field of characteristic p > 0 $p>0$ and Π X $\Pi _{X^{\bullet }}$ the admissible fundamental group of X $X^{\bullet }$ . We prove that there exists a group-theoretical algorithm, whose input datum is the admissible fundamental group Π X $\Pi _{X^{\bullet }}$ , and whose output data are the topological and the combinatorial structures associated with X $X^{\bullet }$ . This result can be regarded as a mono-anabelian version of the combinatorial Grothendieck conjecture in the positive characteristic. Moreover, by applying this result, we construct clutching maps for moduli spaces of admissible fundamental groups.  相似文献   

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In order to improve the classical Bohr inequality, we explain some refined versions for a quasi-subordination family of functions in this paper, one of which is key to build our results. Using these investigations, we establish an improved Bohr inequality with refined Bohr radius under particular conditions for a family of harmonic mappings defined in the unit disk D ${\mathbb {D}}$ . Along the line of extremal problems concerning the refined Bohr radius, we derive a series of results. Here, the family of harmonic mappings has the form f = h + g ¯ $f=h+\overline{g}$ , where g ( 0 ) = 0 $g(0)=0$ , the analytic part h is bounded by 1 and that | g ( z ) | k | h ( z ) | $|g^{\prime }(z)|\le k|h^{\prime }(z)|$ in D ${\mathbb {D}}$ and for some k [ 0 , 1 ] $k\in [0,1]$ .  相似文献   

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