Arkiv för Matematik

In this article, we study the axiomatic approach of Grivaux in [Gri10] for rational Bott–Chern cohomology, and use it in particular to define Chern classes of coherent sheaves in rational Bott–Chern cohomology. This method also allows us to derive a Riemann–Roch–Grothendieck formula for a projective morphism between smooth complex compact manifolds.

A semi-matching coloring of a finite simple graph $G=(V,E)$ is a mapping $\varphi$ from $V$ to $\lbrace 1,\dotsc,k \rbrace$ such that (i) every color class is an independent set, and (ii) the edge set of the graph induced by the union of any two consecutive color classes is a matching. A semi-matching coloring $\varphi$ is a local coloring if, in addition, (iii) the union of any three consecutive color classes induces a triangle-free subgraph of $G$. In this paper, we give two counterexamples and one positive solution to three problems arisen in recent papers of You, Cao, Wang. In particular, we show that the local and semi-matching coloring problems are NP-complete on the class of split graphs.

We consider a perturbation problem for embedded eigenvalues of a self-adjoint differential operator in $L^2 (\mathbb{R} ; \mathbb{R}^n)$. In particular, we study the set of all small perturbations in an appropriate Banach space for which the embedded eigenvalue remains embedded in the continuous spectrum. We show that this set of small perturbations forms a smooth manifold and we specify its co-dimension. Our methods involve the use of exponential dichotomies, their roughness property and Lyapunov–Schmidt reduction.

Suppose that $p \in (1,\infty], \nu \in [1/2,\infty), {S_{\nu }}=\left\{({x_{1}},{x_{2}})\in {\mathbb{R}^{2}}\setminus \{(0,0)\}:|\phi | \lt \frac{\pi }{2\nu }\right\}$, where $\phi$ is the polar angle of $(x_1, x_2)$. Let $R \gt 0$ and $\omega_p (x)$ be the $p$-harmonic measure of $\partial B(0,R) \cap \mathcal{S}_\nu$ at $x$ with respect to $B(0,R) \cap S_\nu$. We prove that there exists a constant $C$ such that whenever $x \in B(0,R) \cap \mathcal{S}_{2 \nu}$ and where the exponent $k(\nu, p)$ is given explicitly as a function of $\nu$ and $p$. Using this estimate we derive local growth estimates for $p$-sub- and $p$-superharmonic functions in planar domains which are locally approximable by sectors, e.g., we conclude bounds of the rate of convergence near the boundary where the domain has an inwardly or outwardly pointed cusp. Using the estimates of $p$-harmonic measure we also derive a sharp Phragmén–Lindelöf theorem for $p$-subharmonic functions in the unbounded sector $S_\nu$. Moreover, if $p=\infty$ then the above mentioned estimates extend from the setting of two-dimensional sectors to cones in $\mathbb{R}^n$. Finally, when $\nu \in (1/2,\infty)$ and $p \in (1,\infty)$ we prove uniqueness (modulo normalization) of positive $p$-harmonic functions in $\mathcal{S}_\nu$ vanishing on $\partial \mathcal{S}_\nu$.

For any $n \in \mathbb{Z}_{\geq 2}$, let $\mathfrak{m}_n$ be the subalgebra of $\mathfrak{sp}_{2n}$ spanned by all long negative root vectors $X_{-2 \varepsilon_i} , i=1, \dotsc , n$. Then ($\mathfrak{sp}_{2n}$, $\mathfrak{m}_n$) is a Whittaker pair in the sense of a definition given by Batra and Mazorchuk. In this paper, we use differential operators to study the category of $\mathfrak{sp}_{2n}$-modules that are locally finite over $\mathfrak{m}_n$. We show that when $\mathbf{a} \in (\mathbb{C}^\ast)^n$, each non-empty block $\mathcal{WH}^{\chi \mu}_{\mathbf{a}}$ with the central character $\chi \mu$ is equivalent to the Whittaker category $\mathcal{W}^{\mathbf{a}}$ of the even Weyl algebra $\mathcal{D}^{ev}_n$ which is semi-simple. Any module in $\mathcal{WH}^{\chi \mu}_{\mathbf{a}}$ has the minimal Gelfand–Kirillov dimension $n$. We also characterize all possible algebra homomorphisms from $U(\mathfrak{sp}_{2n}$) to the Weyl algebra $\mathcal{D}_n$ under a natural condition.

We study the regularity of symbolic powers of square-free monomial ideals. We prove that if $I = I_\Delta$ is the Stanley–Reisner ideal of a simplicial complex $\Delta$, then $\operatorname{reg}(I^{(n)}) \leqslant \delta (n-1)+b$ for all $n \geqslant 1$, where $\delta = \operatorname{lim}_{n \to \infty} \operatorname{reg}(I^{(n)}) / n$, $b=\max \lbrace \operatorname{reg}(I_\Gamma ) \vert \Gamma$ is a subcomplex of $\Delta$ with $\mathcal{F}(\Gamma) \subseteq F(\Delta ) \rbrace$, and $F(\Gamma)$ and $F(\Delta)$ are the set of facets of $\Gamma$ and $\Delta$, respectively. This bound is sharp for any $n$. When $I=I(G)$ is the edge ideal of a simple graph $G$, we obtain a general linear upper bound $\operatorname{reg}(I^{(n)}) \leqslant 2n+ \operatorname{ord-match} (G) - 1$, where $\operatorname{ord-match} (G)$ is the ordered matching number of $G$.

Using methods from complex analysis we obtain some qualitative results for certain streamline characteristics in a deep-water Stokes flow.

We study conditions on closed sets $C, F \subset \mathbb{R}$ making the product $C \times F$ removable or non-removable for $W^{1,p}$. The main results show that the Hausdorff-dimension of the smaller dimensional component $C$ determines a critical exponent above which the product is removable for some positive measure sets $F$, but below which the product is not removable for another collection of positive measure totally disconnected sets $F$. Moreover, if the set $C$ is Ahlfors-regular, the above removability holds for any totally disconnected $F$.

We consider a constrained version of the $\operatorname{HL}(0)$ Hastings–Levitov model of aggregation in the complex plane, in which particles can only attach to the part of the cluster that has already been grown. Although one might expect that this gives rise to a non-trivial limiting shape, we prove that the cluster grows explosively: in the upper half plane, the aggregate accumulates infinite diameter as soon as it reaches positive capacity. More precisely, we show that after $nt$ particles of (half-plane) capacity $1/(2n)$ have attached, the diameter of the shape is highly concentrated around $\sqrt{t \operatorname{log} n}$, uniformly in $t \in [0, T]$. This illustrates a new instability phenomenon for the growth of single trees/fjords in unconstrained $\operatorname{HL}(0)$.

We give sufficient conditions for a general Dirichlet series to be universal with respect to translations or rearrangements.