Latest articles of Arkiv för Matematik
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https://content.intlpress.com/https://content.intlpress.com/Latest articles of Arkiv för Matematik
http://content.intlpress.com/journal/ARKIV/feeds/latest
enSat, 18 May 2024 16:12:35 +0300<![CDATA[Explosive growth for a constrained Hastings–Levitov aggregation model]]>
https://content.intlpress.com/journal/ARKIV/article/11621
https://content.intlpress.com/journal/ARKIV/article/11621We 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 $\operatorname{HL}(0)$.
PDFXML]]>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 $\operatorname{HL}(0)$.
PDFXML]]>Nathanaël Berestycki,Vittoria SilvestriWed, 26 Apr 2023 00:00:00 +0300<![CDATA[Estimates of $p$-harmonic functions in planar sectors]]>
https://content.intlpress.com/journal/ARKIV/article/11622
https://content.intlpress.com/journal/ARKIV/article/11622Suppose 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

\[
C^{-1} {\left( \dfrac{\lvert x \rvert}{R} \right)}^{k (\nu, p)} \leq \omega_p (x) \leq C {\left( \dfrac{\lvert x \rvert}{R} \right)}^{k (\nu, p)}
\]

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 $\partial \mathcal{S}_\nu$.
PDFXML]]>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

\[
C^{-1} {\left( \dfrac{\lvert x \rvert}{R} \right)}^{k (\nu, p)} \leq \omega_p (x) \leq C {\left( \dfrac{\lvert x \rvert}{R} \right)}^{k (\nu, p)}
\]

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 $\partial \mathcal{S}_\nu$.
PDFXML]]>Niklas L. P. Lundström,Jesper SinghWed, 26 Apr 2023 00:00:00 +0300<![CDATA[Regularity of symbolic powers of square-free monomial ideals]]>
https://content.intlpress.com/journal/ARKIV/article/11623
https://content.intlpress.com/journal/ARKIV/article/11623We 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 $G$.
PDFXML]]>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 $G$.
PDFXML]]>Thi Hien Truong,Nam Trung TranWed, 26 Apr 2023 00:00:00 +0300<![CDATA[Universality of general Dirichlet series with respect to translations and rearrangements]]>
https://content.intlpress.com/journal/ARKIV/article/11624
https://content.intlpress.com/journal/ARKIV/article/11624We give sufficient conditions for a general Dirichlet series to be universal with respect to translations or rearrangements.
PDFXML]]>We give sufficient conditions for a general Dirichlet series to be universal with respect to translations or rearrangements.
PDFXML]]>Frédéric BayartWed, 26 Apr 2023 00:00:00 +0300<![CDATA[Perturbations of embedded eigenvalues for self-adjoint ODE systems]]>
https://content.intlpress.com/journal/ARKIV/article/11625
https://content.intlpress.com/journal/ARKIV/article/11625We 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.
PDFXML]]>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.
PDFXML]]>Sara Maad Sasane,Alexia PapalazarouWed, 26 Apr 2023 00:00:00 +0300<![CDATA[On local and semi-matching colorings of split graphs]]>
https://content.intlpress.com/journal/ARKIV/article/11626
https://content.intlpress.com/journal/ARKIV/article/11626A 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.
PDFXML]]>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.
PDFXML]]>Yaroslav ShitovWed, 26 Apr 2023 00:00:00 +0300<![CDATA[Intersection theory and Chern classes in Bott–Chern cohomology]]>
https://content.intlpress.com/journal/ARKIV/article/11627
https://content.intlpress.com/journal/ARKIV/article/11627In 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.
PDFXML]]>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.
PDFXML]]>Xiaojun WuWed, 26 Apr 2023 00:00:00 +0300<![CDATA[A Whittaker category for the symplectic Lie algebra and differential operators]]>
https://content.intlpress.com/journal/ARKIV/article/11628
https://content.intlpress.com/journal/ARKIV/article/11628For 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.
PDFXML]]>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.
PDFXML]]>Yang Li,Jun Zhao,Yuanyuan Zhang,Genqiang LiuWed, 26 Apr 2023 00:00:00 +0300<![CDATA[Yagita’s counter-examples and beyond]]>
https://content.intlpress.com/journal/ARKIV/article/11629
https://content.intlpress.com/journal/ARKIV/article/11629A conjecture on a relationship between the Chow and Grothendieck rings for the generically twisted variety of Borel subgroups in a split semisimple group $G$, stated by the second author, has been disproved by Nobuaki Yagita in characteristic $0$ for $G=\operatorname{Spin}(2n+1)$ with $n=8$ and $n=9$. For $n=8$, the second author provided an alternative simpler proof of Yagita’s result, working in any characteristic, but failed to do so for $n=9$. This gap is filled here by involving a new ingredient—Pieri type $K$-theoretic formulas for highest orthogonal grassmannians. Besides, a similar counter-example for $n=10$ is produced. Note that the conjecture on $\operatorname{Spin}(2n+1)$ holds for $n$ up to $5$; it remains open for $n=6$, $n=7$, and every $n \geq 11$.
PDFXML]]>A conjecture on a relationship between the Chow and Grothendieck rings for the generically twisted variety of Borel subgroups in a split semisimple group $G$, stated by the second author, has been disproved by Nobuaki Yagita in characteristic $0$ for $G=\operatorname{Spin}(2n+1)$ with $n=8$ and $n=9$. For $n=8$, the second author provided an alternative simpler proof of Yagita’s result, working in any characteristic, but failed to do so for $n=9$. This gap is filled here by involving a new ingredient—Pieri type $K$-theoretic formulas for highest orthogonal grassmannians. Besides, a similar counter-example for $n=10$ is produced. Note that the conjecture on $\operatorname{Spin}(2n+1)$ holds for $n$ up to $5$; it remains open for $n=6$, $n=7$, and every $n \geq 11$.
PDFXML]]>Sanghoon Baek,Nikita A. KarpenkoWed, 26 Apr 2023 00:00:00 +0300<![CDATA[A complex-analytic approach to streamline properties of deep-water Stokes waves]]>
https://content.intlpress.com/journal/ARKIV/article/11630
https://content.intlpress.com/journal/ARKIV/article/11630Using methods from complex analysis we obtain some qualitative results for certain streamline characteristics in a deep-water Stokes flow.
PDFXML]]>Using methods from complex analysis we obtain some qualitative results for certain streamline characteristics in a deep-water Stokes flow.
PDFXML]]>Olivia ConstantinWed, 26 Apr 2023 00:00:00 +0300