We consider the Szegő reproducing kernel associated with the space of H-harmonic functions on the unit ball in n-dimensional space, i.e. functions that are characterized by being annihilated by the hyperbolic Laplacian. This paper derives an explicit series expansion for the reproducing kernel in terms of a triple hypergeometric function introduced of Exton. Moreover, we demonstrate that the Szegő kernel admits a representation as a finite sum of hypergeometric functions. We further show that the Szegő kernel, for linearly dependent arguments, can be expressed in terms of the first Appell hypergeometric function. In addition we provide a series expansion for the weighted Bergman kernels.
We study the class of squarefree principal vector-spread Borel ideals. We compute the minimal primary decomposition of these ideals and thereby we prove that they are sequentially Cohen-Macaulay. As the final conclusion of our results, we completely classify the ideals in our class having the property that their ordinary and symbolic powers coincide.
We show that a general nonlinearity a(x,u) is uniquely determined, possibly up to a gauge, in a neighborhood of a fixed solution from boundary measurements of the corresponding semilinear equation. The main theorems are low regularity counterparts of the results in our recent paper (Johansson, Nurminen, Salo; ArXiv preprint 2312.12196).
The Gorenstein property in local algebra admits several characterizations via its module category. The goal of this paper is to collect and generalize such characterizations to the relative setting, i.e., to Gorenstein morphisms as defined by [AF92]. We achieve this by proving these characterizations more generally for graded-commutative Gorenstein dg-algebras.
In our previous research, we constructed the affine varieties $\Sigma _{\mathbb {A}}^{13}$ and $\Pi _{\mathbb {A}}^{14}$ whose partial projectivizations admit $\mathbb {P}^{2}\times \mathbb {P}^{2}$-fibrations with relative Picard number one. In this paper, we produce prime quasi-smooth ℚ-Fano 3-folds which are anticanonically embedded of codimension four and belong to 23 (resp. 8) classes in the Graded Ring Database, as weighted complete intersections in weighted projectivizations of $\Sigma _{\mathbb {A}}^{13}$ (resp. $\Pi _{\mathbb {A}}^{14}$ or its cone). We also show that a general member of the anticanonical linear system of a general prime ℚ-Fano $3$-fold constructed in this way is a quasi-smooth $K3$ surface with at worst Du Val singularities.
We define essential Cartan pairs of $C^*$-algebras generalising the definition of Renault [Cartan subalgebras in $C^*$-algebras, Irish Math. Soc. Bull. (2008), no. 61, 29–63] and show that such pairs are given by essential twisted groupoid $C^*$-algebras as defined by Kwaśniewski and Meyer [Essential crossed products by inverse semigroup actions: Simplicity and pure infiniteness, Doc. Math. 26 (2021), 271–335]. We show that the underlying twisted groupoid is effective, and is unique up to isomorphism among twists over effective groupoids giving rise to the essential Cartan pair. We also show that for twists over effective groupoids giving rise to such pairs, the automorphism group of the twist is isomorphic to the automorphism group of the induced essential Cartan pair via explicit constructions.
The vertex cover ideal $J(G)$ of a finite graph $G$ is studied. We characterize when a Cohen–Macaulay vertex cover ideal $J(G)$ has a Scarf minimal free resolution. Furthermore, by using both combinatorial and topological techniques, the graded Betti number $\beta _{i,i+j}(J(G))$, where $i$ and $j$ are the projective dimension and the regularity of $J(G)$, is computed, when $G$ is either a path or a cycle.
We derive an interior estimate up to second orders in Hardy spaces for solutions to the parabolic problem $$ \begin {cases} u_t - \sum _{i, j=1}^n a_{i j} \partial ^2_{ij} u = f & \text {in $\Omega _T$}, \\ u \in h^p(0,T; h^{1,p}(\Omega )) \cap h^{1,p}_{loc }(0,T ; h^{2,p}_{loc }(\Omega )) \end {cases} $$ within an appropriate framework. In the course of proof, we also establish the boundedness results of parabolic singular integrals and their commutators on Hardy spaces which are of independent interest.
We explain how results comparing the homology of spaces of algebraic and continuous maps to projective spaces can be leveraged to compare moduli stacks of families of algebraic and continuous maps.
Let $(\mathscr {E}, \langle \cdot , \cdot \rangle )$ be a Hilbert $C^*$-module over a $C^*$-algebra $\mathfrak {A}$. The space of adjointable operators on $\mathscr {E}$ is denoted by $\mathcal {L}(\mathscr {E})$. The sets of all states and pure states on $\mathfrak {A}$ are denoted by $\mathcal {S}(\mathfrak {A})$ and $\mathcal {P}( \mathfrak {A})$, respectively. For $\tau \in \mathcal {S}( \mathfrak {A})$, let us define $\mathcal {N}^{\mathscr {E}}_{\tau }:= \lbrace x\in \mathscr {E}:\tau ( \langle x,x\rangle )=0 \rbrace $. The Hilbert completion of $\mathscr {E}/{\mathcal {N}^{\mathscr {E}}_{\tau }}$ is denoted by $\mathscr {E}_{\tau }$. For $T\in \mathcal {L}(\mathscr {E})$, the operator $T_{\mathscr {E}_\tau }\in \mathbb {B}( \mathscr {E}_\tau )$, is defined by $ T_{\mathscr {E}_\tau }(x+\mathcal {N}^{\mathscr {E}}_{\tau })=Tx+\mathcal {N}^{\mathscr {E}}_{\tau }$. In this paper, we show that $\mathscr {E}_{\tau }={\mathscr {E}}/{\mathcal {N}^{\mathscr {E}}_{\tau }}$ when $\mathfrak {A}$ either is a $C^*$-algebra of compact operators or is commutative. We introduce a quantity in the context of Hilbert $C^*$-modules, denoted by $\pi ^{\mathscr {E}}_k(\cdot )$ for $k\geq 1$. We prove that $\pi ^{\mathscr {E}}_k(T)\leq \sup _{\tau \in \mathcal {P}( \mathfrak {A})}\|T_{\mathscr {E}_\tau }\|_{(k)}\leq {\pi }^{\mathscr {E}^{\sharp }}_k(T_{\mathscr {E}^{\sharp }})$ for every $T\in \mathcal {L}(\mathscr {E})$, where the space $\mathscr {E}^{\sharp }$ is constructed as the extension of $\mathscr {E}$ by the embedding of $\mathfrak {A}$ into its enveloping von Neumann algebra $\mathfrak {A}^{**}$. Thus, the norm $\pi ^{\mathscr {E}}_k(\cdot )$ preserves the Schatten properties, as in Hilbert spaces, particularly when $\pi ^{\mathscr {E}}_k(T) = {\pi }^{\mathscr {E}^{\sharp }}_k(T_{\mathscr {E}^{\sharp }})$ for all $T \in \mathcal {L}(\mathscr {E})$. This equality holds for $\mathscr {E}$ over commutative $C^*$-algebras with a frame. Moreover, it holds for $\mathscr {E}$ over $C^*$-algebras of compact operators.
Let u : Omega subset of C-n -> C-m, for n >= 2 and m >= 1. Let 1 <= p <= 2, and 2(2n)(2 )- 1 <= q < infinity such that 1/p + 1/p ' = 1 and 1/p - 1/p ' = 1/q. Suppose |partial derivativeu| <= V|u|, where V is an element of L-loc(q)(Omega). Then u has a unique continuation property in the following sense: if u is an element of W-loc(1,p)(Omega) and for some z(0) is an element of Omega, parallel to u parallel to(Lp '(B(z0,r))) decays faster than any powers of r as r -> 0, then u equivalent to 0. The same result holds for q = infinity if u is scalar-valued (m = 1).
Let $u\colon \Omega \subset \mathbb {C}^n \to \mathbb {C}^m$, for $n \geq 2$ and $m \geq 1$. Let $1 \leq p \leq 2$, and $2(2n)^2 -1 \leq q < \infty $ such that $\frac {1}{p} + \frac {1}{p'} = 1$ and $\frac {1}{p} - \frac {1}{p'} = \frac {1}{q}$. Suppose $\lvert \overline \partial u\rvert \leq V \lvert u\rvert $, where $V \in L^q_{\textrm {loc}}(\Omega )$. Then $u$ has a unique continuation property in the following sense: if $u \in W^{1,p}_{\textrm {loc}}(\Omega )$ and for some $z_0 \in \Omega $, $\lVert u \rVert _{L^{p'}(B(z_0,r))} $ decays faster than any powers of $r$ as $r \to 0$, then $u \equiv 0$. The same result holds for $q=\infty $ if $u$ is scalar-valued ($m=1$)
Utilizing the concept of Reeb Jacobi operator of Codazzi type, we investigate Hopf real hypersurfaces in the complex hyperbolic two-plane Grassmannian $G^{*}_2(\mathbb {C}^{m+2})$ which admit a constant Reeb function α along the Reeb direction of ξ. If the Reeb function α is constant along the Reeb direction, then the Reeb vector field $\xi =-JN$ either belongs to the distribution ${\mathfrak D}$ or the distribution ${\mathfrak D}^{\bot }$. By virtue of this fact, we have proved a new result about Reeb Jacobi operator of Codazzi type according to the Reeb vector field $\xi \in \mathfrak {D}^{\bot }$ or $\xi \in \mathfrak {D}$, respectively.
Let $C$ be a smooth curve of genus $g$ and let $d$,$r$ be integers with $1 \leq r \leq g-2$ and $2r\leq d \leq g-2+r$. H. Martens' Theorem states that $\dim (W^r_d(C))=d-2r$ implies $C$ is hyperelliptic. It is known that for a metric graph Γ of genus $g$ such statement using $\dim (W^r_d(\Gamma ))$ does not hold. However replacing $\dim (W^r_d(\Gamma ))$ by the so-called Brill-Noether rank $w^r_d(\Gamma )$ it was stated as a conjecture. Using a similar definition in the case of curves one has $\dim (W^r_d(C))=w^r_d(C)$. Let $\Gamma$ be a chain of cycles of genus $g$ and let $r,d$ be integers with $1 \leq r \leq g-2$ and $2r\leq d \leq g-3+r$. If $w^r_d(\Gamma )=d-2r$ then we prove Γ is hyperelliptic. In case $g \geq 2r+3$ then we prove there exist non-hyperelliptic chains of cycles satisfying $w^r_{g-2+r}(\Gamma )=g-2-r$, contradicting the conjecture. We give a complete description of all counterexamples within the set of chains of cycles to the statement of H. Martens' Theorem. Those counterexamples also give rise to chains of cycles such that $w^r_{g-2+r}(\Gamma ) \neq w^1_{g-r}(\Gamma )$. This shows that the Riemann-Roch duality does not hold for the Brill-Noether ranks of metric graphs.
A Schrödinger operator on a metric graph is isospectral to the Laplacian if and only if the potential is zero. This result is extended to the case of magnetic Schrödinger operators. We show that if a magnetic Schrödinger operator is isospectral to the Laplacian then the magnetic potential can be eliminated. Even the case of delta couplings at the vertices is considered.
The paper “Minimal piecewise linear cones in $\mathbb {R}^4$” (Math. Scand. 130 (2024), 149–160) contains two errors. The result is weakened from a full classification of the five minimal piecewise linear cones in $\mathbb {R}^4$ to a classification of five minimizers and two candidates that we could not determine whether were minimizing or not.
This work discusses the existence and uniqueness of the weak solution to a spatial fractional diffusion equation, which can be applied in image processing. The proposed model combines the advantages of both second- and fourth-order diffusion equations along with Gaussian filtering by employing spatial fractional derivatives and a Gaussian filter. This approach enhance edges preservation and robustness to noise. The existence and uniqueness of the weak solution for the model are proved by applying Schauder's fixed-point theorem.
Let $\mathfrak {a}$ be an ideal of a Noetherian ring $R$ such that the category of $\mathfrak {a}$-cofinite modules is an Abelian subcategory of the category of $R$-modules. Let $M$ be a finitely generated $R$-module such that the $R$-modules $H^i_{\mathfrak {a}}(M)$ are $\mathfrak {a}$-cofinite for all integers $i\in \mathbb{N}_0$. In this paper it is shown that $\dim H^i_{\mathfrak {a}}(M)\leq 1$ for all integers $i\geq 2$.
It follows from Grothendieck's little inequality that to any complex $m \times n $ matrix $X$ of column norm at most 1, and an $0 <\varepsilon <1,$ there exist a natural number $l, $ a matrix $C$ in $M_{(m,l)}(\mathbb {C})$ with $(1-\varepsilon)^2I_m \leq CC^* \leq (4/\pi ) (1 + \varepsilon)^2 I_m $ and an $l \times n$ matrix $Z$ with entries in the torus $\mathbb {T},$ such that $X= l^{-(1/2)}CZ.$ Grothendieck's little inequality with the constant $k_G^\mathbb {C} = 4/\pi $ follows from this factorization. Grothendieck's big inequality may be reproduced with the estimate $K_G^\mathbb {C} \leq k_G^\mathbb {C}/(2- k_G^\mathbb {C}).$
