Showing papers in "Revista Matematica Complutense in 2021"

Reconstruction of compacta by finite approximations and inverse persistence

Abstract: The aim of this paper is to show how the homeomorphism and homotopy types of compact metric spaces can be reconstructed by the inverse limit of an inverse sequence of finite approximations of the space. This recovering allows us to propose an alternative way to construct persistence modules from a point cloud.

Normalized ground states for the critical fractional NLS equation with a perturbation

Abstract: In this paper, we study normalized ground states for the following critical fractional NLS equation with prescribed mass: $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^{s}u=\lambda u +\mu |u|^{q-2}u+|u|^{2_{s}^{*}-2}u,&{}x\in \mathbb {R}^{N}, \\ \int _{\mathbb {R}^{N}}u^{2}dx=a^{2},\\ \end{array}\right. } \end{aligned}$$ where $$(-\Delta )^{s}$$ is the fractional Laplacian, $$02s$$ , $$20$$ , $$\mu \in \mathbb {R}$$ . By using Jeanjean’s trick in Jeanjean (Nonlinear Anal 28:1633–1659, 1997), and the standard method which can be found in Brezis and Nirenberg (Commun Pure Appl Math 36:437–477, 1983) to overcome the lack of compactness, we first prove several existence and nonexistence results for a $$L^{2}$$ -subcritical (or $$L^{2}$$ -critical or $$L^{2}$$ -supercritical) perturbation $$\mu |u|^{q-2}u$$ , then we give some results about the behavior of the ground state obtained above as $$\mu \rightarrow 0^{+}$$ . Our results extend and improve the existing ones in several directions.

A Hölder Infinity Laplacian obtained as limit of Orlicz Fractional Laplacians

Abstract: This paper concerns the study of the asymptotic behavior of the solutions to a family of fractional type problems on a bounded domain, satisfying homogeneous Dirichlet boundary conditions. The family of differential operators includes the fractional $$p_n$$ -Laplacian when $$p_n\rightarrow \infty $$ as a particular case, tough it could be extended to a function of the Holder quotient of order s, whose primitive is an Orlicz function satisfying appropriated growth conditions. The limit equation involves the Holder infinity Laplacian.

Some asymptotic properties of kernel regression estimators of the mode for stationary and ergodic continuous time processes.

Abstract: In the present paper, we consider the nonparametric regression model with random design based on $$(\mathbf{X}_\mathrm{t},\mathbf{Y}_\mathrm{t})_{\mathrm{t}\ge 0}$$ a $$\mathbb {R}^{d}\times \mathbb {R}^{q}$$ -valued strictly stationary and ergodic continuous time process, where the regression function is given by $$m(\mathbf{x},\psi ) = \mathbb {E}(\psi (\mathbf{Y}) \mid \mathbf{X} = \mathbf{x}))$$ , for a measurable function $$\psi : \mathbb {R}^{q} \rightarrow \mathbb {R}$$ We focus on the estimation of the location $${\varvec{\Theta }}$$ (mode) of a unique maximum of $$m(\cdot , \psi )$$ by the location $$ \widehat{{\varvec{\Theta }}}_\mathrm{T}$$ of a maximum of the Nadaraya–Watson kernel estimator $$\widehat{m}_\mathrm{T}(\cdot , \psi )$$ for the curve $$m(\cdot , \psi )$$ Within this context, we obtain the consistency with rate and the asymptotic normality results for $$ \widehat{{\varvec{\Theta }}}_\mathrm{T}$$ under mild local smoothness assumptions on $$m(\cdot , \psi )$$ and the design density $$f(\cdot )$$ of $$\mathbf{X}$$ Beyond ergodicity, any other assumption is imposed on the data This paper extends the scope of some previous results established under the mixing condition The usefulness of our results will be illustrated in the construction of confidence regions

Calculus of variations on locally finite graphs

Abstract: Let $$G=(V,E)$$ be a locally finite graph. Firstly, using calculus of variations, including a direct method of variation and the mountain-pass theory, we get sequences of solutions to several local equations on G (the Schrodinger equation, the mean field equation, and the Yamabe equation). Secondly, we derive uniform estimates for those local solution sequences. Finally, we obtain global solutions by extracting convergent sequence of solutions. Our method can be described as a variational method from local to global.

The fourth power mean of Dirichlet $L$-functions in $\mathbb{F}_q [T]$

Abstract: We prove results on moments of L-functions in the function field setting, where the moment averages are taken over primitive characters of modulus R, where R is a polynomial in $${\mathbb {F}}_{q}[T]$$ . We consider the behaviour as $${{\,\mathrm{deg}\,}}R \rightarrow \infty $$ and the cardinality of the finite field is fixed. Specifically, we obtain an exact formula for the second moment provided that R is square-full, an asymptotic formula for the second moment for any R, and an asymptotic formula for the fourth moment for any R. The fourth moment result is a function field analogue of Soundararajan’s result in the number field setting that improved upon a previous result by Heath-Brown. Both the second and fourth moment results extend work done by Tamam in the function field setting who focused on the case where R is prime. As a prerequisite for the fourth moment result, we obtain, for the special case of the divisor function, the function field analogue of Shiu’s generalised Brun–Titchmarsh theorem.

Characterization of Ulrich bundles on Hirzebruch surfaces

Abstract: In this work we characterize Ulrich bundles of any rank on polarized rational ruled surfaces over $${\mathbb {P}^1}$$ . We show that every Ulrich bundle admits a resolution in terms of line bundles. Conversely, given an injective map between suitable totally decomposed vector bundles, we show that its cokernel is Ulrich if it satisfies a vanishing in cohomology. As a consequence we obtain, once we fix a polarization, the existence of Ulrich bundles for any admissible rank and first Chern class. Moreover we show the existence of stable Ulrich bundles for certain pairs $$({\text {rk}}(E),c_1(E))$$ and with respect to a family of polarizations. Finally we construct examples of indecomposable Ulrich bundles for several different polarizations and ranks.

The geometric classification of 2-step nilpotent algebras and applications

Abstract: We give a geometric classification of complex $n$-dimensional $2$-step nilpotent (all, commutative and anticommutative) algebras. Namely, has been found the number of irreducible components and their dimensions. As a corollary, we have a geometric classification of complex $5$-dimensional nilpotent associative algebras. In particular, it has been proven that this variety has $14$ irreducible components and $9$ rigid algebras.

Regularizing effects concerning elliptic equations with a superlinear gradient term

Abstract: We consider the homogeneous Dirichlet problem for an elliptic equation driven by a linear operator with discontinuous coefficients and having a subquadratic gradient term. This gradient term behaves as $$g(u)| abla u|^q$$ , where $$1

Fano 3-folds from homogeneous vector bundles over Grassmannians

Abstract: We rework the Mori–Mukai classification of Fano 3-folds, by describing each of the 105 families via biregular models as zero loci of general global sections of homogeneous vector bundles over products of Grassmannians.

$$\omega ^\omega $$ ω ω -Base and infinite-dimensional compact sets in locally convex spaces

Abstract: A locally convex space (lcs) E is said to have an $$\omega ^{\omega }$$ -base if E has a neighborhood base $$\{U_{\alpha }:\alpha \in \omega ^\omega \}$$ at zero such that $$U_{\beta }\subseteq U_{\alpha }$$ for all $$\alpha \le \beta $$ . The class of lcs with an $$\omega ^{\omega }$$ -base is large, among others contains all (LM)-spaces (hence (LF)-spaces), strong duals of distinguished Frechet lcs (hence spaces of distributions $$D^{\prime }(\Omega )$$ ). A remarkable result of Cascales-Orihuela states that every compact set in an lcs with an $$\omega ^{\omega }$$ -base is metrizable. Our main result shows that every uncountable-dimensional lcs with an $$\omega ^{\omega }$$ -base contains an infinite-dimensional metrizable compact subset. On the other hand, the countable-dimensional vector space $$\varphi $$ endowed with the finest locally convex topology has an $$\omega ^\omega $$ -base but contains no infinite-dimensional compact subsets. It turns out that $$\varphi $$ is a unique infinite-dimensional locally convex space which is a $$k_{\mathbb {R}}$$ -space containing no infinite-dimensional compact subsets. Applications to spaces $$C_{p}(X)$$ are provided.

Set-theoretic solutions of the Yang–Baxter equation, associated quadratic algebras and the minimality condition

Abstract: Given a finite non-degenerate set-theoretic solution (X, r) of the Yang–Baxter equation and a field K, the structure K-algebra of (X, r) is $$A=A(K,X,r)=K\langle X\mid xy=uv \text{ whenever } r(x,y)=(u,v)\rangle $$ . Note that $$A=\oplus _{n\ge 0} A_n$$ is a graded algebra, where $$A_n$$ is the linear span of all the elements $$x_1\cdots x_n$$ , for $$x_1,\dots ,x_n\in X$$ . One of the known results asserts that the maximal possible value of $$\dim (A_2)$$ corresponds to involutive solutions and implies several deep and important properties of A(K, X, r). Following recent ideas of Gateva-Ivanova (A combinatorial approach to noninvolutive set-theoretic solutions of the Yang–Baxter equation. arXiv:1808.03938v3 [math.QA], 2018), we focus on the minimal possible values of the dimension of $$A_2$$ . We determine lower bounds and completely classify solutions (X, r) for which these bounds are attained in the general case and also in the square-free case. This is done in terms of the so called derived solution, introduced by Soloviev and closely related with racks and quandles. Several problems posed by Gateva-Ivanova (2018) are solved.

Local atomic decompositions for multidimensional Hardy spaces

Abstract: We consider a nonnegative self-adjoint operator L on $$L^2(X)$$ , where $$X\subseteq {{\mathbb {R}}}^d$$ . Under certain assumptions, we prove atomic characterizations of the Hardy space $$\begin{aligned} H^1(L) = \left\{ f\in L^1(X) \ : \ \left\| \sup _{t>0} \left| \exp (-tL)f \right| \right\| _{L^1(X)}<\infty \right\} . \end{aligned}$$ We state simple conditions, such that $$H^1(L)$$ is characterized by atoms being either the classical atoms on $$X\subseteq {\mathbb {R}^d}$$ or local atoms of the form $$|Q|^{-1}\chi _Q$$ , where $$Q\subseteq X$$ is a cube (or cuboid). One of our main motivation is to study multidimensional operators related to orthogonal expansions. We prove that if two operators $$L_1, L_2$$ satisfy the assumptions of our theorem, then the sum $$L_1 + L_2$$ also does. As a consequence, we give atomic characterizations for multidimensional Bessel, Laguerre, and Schrodinger operators. As a by-product, under the same assumptions, we characterize $$H^1(L)$$ also by the maximal operator related to the subordinate semigroup $$\exp (-tL^ u )$$ , where $$ u \in (0,1)$$ .

New constructions of unexpected hypersurfaces in $\mathbb{P}^n$

Abstract: In the paper we present new examples of unexpected varieties. The research on unexpected varieties started with a paper of Cook II, Harbourne, Migliore and Nagel and was continued in the paper of Harbourne, Migliore, Nagel and Teitler. Here we present three ways of producing unexpected varieties that expand on what was previously known. In the paper of Harbourne, Migliore, Nagel and Teitler, cones on varieties of codimension 2 were used to produce unexpected hypersurfaces. Here we show that cones on positive dimensional varieties of codimension 2 or more almost always give unexpected hypersurfaces. For non-cones, almost all previous work has been for unexpected hypersurfaces coming from finite sets of points. Here we construct unexpected surfaces coming from lines in $$\mathbb {P}^3$$ , and we generalize the construction using birational transformations to obtain unexpected hypersurfaces in higher dimensions.

Pointwise Convergence along non-tangential direction for the Schrödinger equation with Complex Time

Abstract: We study the pointwise convergence to the initial data in a cone region for the fractional Schrodinger operator $$P^{t}_{a,\gamma }$$ with complex time. By stationary phase analysis, we establish the maximal estimate for $$P^{t}_{a,\gamma }$$ in a cone region. As a consequence of the maximal estimate, the pointwise convergence holds through a standard argument. Our results extend those obtained by Cho–Lee–Vargas (J Fourier Anal Appl 18:972–994, 2012) and Shiraki ( arXiv:1903.02356v1 ) from the real value time to the complex value time.

Compact embeddings in Besov-type and Triebel–Lizorkin-type spaces on bounded domains

Abstract: We study embeddings of Besov-type and Triebel–Lizorkin-type spaces, $${\text {id}}_\tau {:}\,{B}_{p_1,q_1}^{s_1,\tau _1}(\varOmega )\,\hookrightarrow \,{B}_{p_2,q_2}^{s_2,\tau _2}(\varOmega )$$ and $${\text {id}}_\tau {:}\,{F}_{p_1,q_1}^{s_1,\tau _1}(\varOmega ) \hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}(\varOmega ) $$ , where $$\varOmega \subset {{\mathbb R}^d}$$ is a bounded domain, and obtain necessary and sufficient conditions for the compactness of $${\text {id}}_\tau $$ . Moreover, we characterize its entropy and approximation numbers. Surprisingly, these results are completely obtained via embeddings and the application of the corresponding results for classical Besov and Triebel–Lizorkin spaces as well as for Besov–Morrey and Triebel–Lizorkin–Morrey spaces.

Moderate deviation principles for nonparametric recursive distribution estimators using Bernstein polynomials

Abstract: In this paper we prove moderate deviations principles for the recursive estimators of a distribution function defined by the stochastic approximation algorithm based on Bernstein polynomials introduced by Jmaei el al. (J Nonparametr Stat 29:792–805, 2017). We show that the considered estimator gives the same pointwise moderate deviations principle (MDP) as the recursive kernel distribution estimator proposed in Slaoui (Math Methods Stat 23(4):306–325, 2014b) and whose large and moderate deviation principles were established by Slaoui (Stat Interface 12(3):439–455, 2009).

Fractional semilinear heat equations with singular and nondecaying initial data

Abstract: We study integrability conditions for existence and nonexistence of a local-in-time integral solution of fractional semilinear heat equations with rather general growing nonlinearities in uniformly local $$L^p$$ spaces. Our main results about this matter consist of Theorems 1.4, 1.6, 5.1 and 5.3. We introduce a supersolution of an integral equation which can be applied to a nonlocal parabolic equation. When the nonlinear term is $$u^p$$ or $$e^u$$ , a local-in-time solution can be constructed in the critical case, and integrability conditions for the existence and nonexistence are completely classified. Our analysis is based on the comparison principle, Jensen’s inequality and $$L^p$$ - $$L^q$$ type estimates.

On weighted compactness of commutator of semi-group maximal function and fractional integrals associated to Schrödinger operators

Abstract: Let $$\mathcal {T}^*$$ and $$\mathcal {I}_\alpha $$ be the semi-group maximal function and fractional integrals associated to the Schrodinger operator $$-\Delta +V(x)$$ , respectively, with V satisfying an appropriate reverse Holder inequality. In this paper, we show that the commutator of $$\mathcal {T}^*$$ is a compact operator on $$L^p(w)$$ for $$1

The group generated by Riordan involutions

Abstract: We prove that any element in the group generated by the Riordan involutions is the product of at most four of them. We also give a description of this subgroup as a semidirect product of a special subgroup of the commutator subgroup and the Klein four-group.

Spectra and ergodic properties of multiplication and convolution operators on the space $${\mathcal S}({\mathbb R})$$ S ( R )

Abstract: In this paper we investigate the spectra and the ergodic properties of the multiplication operators and the convolution operators acting on the Schwartz space $${\mathcal S}({\mathbb R})$$ of rapidly decreasing functions, i.e., operators of the form $$M_h: {\mathcal S}({\mathbb R})\rightarrow {\mathcal S}({\mathbb R})$$ , $$f \mapsto h f $$ , and $$C_T:{\mathcal S}({\mathbb R})\rightarrow {\mathcal S}({\mathbb R})$$ , $$f\mapsto T\star f$$ . Precisely, we determine their spectra and characterize when those operators are power bounded and mean ergodic.

Removability of zero modular capacity sets

Abstract: We introduce a notion of modular with a corresponding modular function space in order to build a modular capacity theory We give two different definitions of capacity, one of them of variational type, the other one through either the modular of the test functions, or the modular of their gradients We study, in both cases, the removability of sets of zero capacity in fairly general abstract Sobolev spaces with zero boundary values As a key tool, we establish a modular Poincare inequality With the notion of modular function space in hands, we find a way to introduce a Banach function space, which allows to compare the zero capacity sets with respect to both notions Thanks to this comparison, we characterize the compact sets of zero variational type capacity as removable sets The paper is enriched with several examples, extending and unifying many results already known in literature in the settings of Musielak–Orlicz–Sobolev spaces, Lorentz–Sobolev spaces, variable exponent Sobolev spaces

Complete convergence for weighted sums of widely orthant-dependent random variables and its statistical application

Abstract: In this paper, we investigate the complete convergence for weighted sums of widely orthant-dependent (WOD, for short) random variables. Our results extend the corresponding ones of Chen and Sung (Stat Probab Lett 154, 2019) to a much more general type of complete convergence. As an application of our main results, we establish the complete consistency for the estimator in the nonparametric regression models and provide a simulation study to assess the finite sample performance of the theoretical results.

Composition operators on spaces of double Dirichlet series

Abstract: We study composition operators on spaces of double Dirichlet series, focusing our interest on the characterization of the composition operators of the space of bounded double Dirichlet series $${\mathcal {H}}^\infty ({\mathbb {C}}_+^2)$$ . We also show how the composition operators of this space of Dirichlet series are related to the composition operators of the corresponding spaces of holomorphic functions. Finally, we give a characterization of the superposition operators in $${\mathcal {H}}^\infty ({\mathbb {C}}_+)$$ and in the spaces $${\mathcal {H}}^p$$ .

Different notions of Sierpiński–Zygmund functions

Abstract: A function $$f:{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}$$ is Sierpinski–Zygmund, $$f\in {{\,\mathrm{SZ}\,}}(\mathrm {C})$$ , provided its restriction $$f{\restriction }M$$ is discontinuous for any $$M\subset {{\mathbb {R}}}$$ of cardinality continuum. Often, it is slightly easier to construct a function $$f:{{\mathbb {R}}}\rightarrow {{\mathbb {R}}}$$ , denoted as $$f\in {{\,\mathrm{SZ}\,}}(\mathrm {Bor})$$ , with a seemingly stronger property that $$f{\restriction }M$$ is not Borel for any $$M\subset {{\mathbb {R}}}$$ of cardinality continuum. It has been recently noticed that the properness of the inclusion $${{\,\mathrm{SZ}\,}}(\mathrm {Bor})\subseteq {{\,\mathrm{SZ}\,}}(\mathrm {C})$$ is independent of ZFC. In this paper we explore the classes $${{\,\mathrm{SZ}\,}}(\Phi )$$ for arbitrary families $$\Phi $$ of partial functions from $${{\mathbb {R}}}$$ to $${{\mathbb {R}}}$$ . We investigate additivity and lineability coefficients of the class $${{\mathbb {S}}}:={{\,\mathrm{SZ}\,}}(\mathrm {C}){\setminus } {{\,\mathrm{SZ}\,}}(\mathrm {Bor})$$ . In particular we show that if $${{\mathfrak {c}}}=\kappa ^+$$ and $${{\mathbb {S}}} e \emptyset $$ , then the additivity of $${{\mathbb {S}}}$$ is $$\kappa $$ , that $${{\mathbb {S}}}$$ is $${{\mathfrak {c}}}^+$$ -lineable, and it is consistent with ZFC that $${{\mathbb {S}}}$$ is $${{\mathfrak {c}}}^{++}$$ -lineable. We also construct several examples of functions from $${{\,\mathrm{SZ}\,}}(\mathrm {C}){\setminus } {{\,\mathrm{SZ}\,}}(\mathrm {Bor})$$ that belong also to other important classes of real functions.

The compact approximation property for spaces of holomorphic mappings on Fréchet spaces

Abstract: In this paper, the compact approximation property on Frechet spaces is characterized in terms of holomorphic mappings. We show that a Frechet space E has the compact approximation property if and only if every holomorphic mapping on a balanced open subset $$U\subset E$$ with values in a Frechet space can be approximated uniformly on compact subsets of U by compact holomorphic mappings. This extends the well-known linear characterization to the holomorphic setting. We also give characterizations of the compact approximation property in terms of bounded holomorphic mappings on Banach spaces.

When is $${\mathfrak {m}}:{\mathfrak {m}}$$ an almost Gorenstein ring?

Abstract: Given a one-dimensional Cohen-Macaulay local ring $(R,\mathfrak{m},k)$, we prove that it is almost Gorenstein if and only if $\mathfrak{m}$ is a canonical module of the ring $\mathfrak{m}:\mathfrak{m}$. Then, we generalize this result by introducing the notions of almost canonical ideal and gAGL ring and by proving that $R$ is gAGL if and only if $\mathfrak{m}$ is an almost canonical ideal of $\mathfrak{m}:\mathfrak{m}$. We use this fact to characterize when the ring $\mathfrak{m}:\mathfrak{m}$ is almost Gorenstein, provided that $R$ has minimal multiplicity. This is a generalization of a result proved by Chau, Goto, Kumashiro, and Matsuoka in the case in which $\mathfrak{m}:\mathfrak{m}$ is local and its residue field is isomorphic to $k$.

Lipschitz spaces adapted to Schrödinger operators and regularity properties

Abstract: Consider the Schrodinger operator $$\mathcal {{L}}=-\Delta +V$$ in $$\mathbb {{R}}^n, n\ge 3,$$ where V is a nonnegative potential satisfying a reverse Holder condition of the type $$\begin{aligned} \left( \frac{1}{|B|}\int _B V(y)^qdy\right) ^{1/q}\le \frac{C}{|B|}\int _B V(y)dy, \, \text {{ for some }}q>n/2. \end{aligned}$$ We define $$\Lambda ^\alpha _\mathcal {{L}},\, 0<\alpha <2,$$ the class of measurable functions such that $$\begin{aligned} \Vert \rho (\cdot )^{-\alpha }f(\cdot )\Vert _\infty 0}\frac{\Vert f(\cdot +z)+f(\cdot -z)-2f(\cdot )\Vert _\infty }{|z|^\alpha }<\infty , \end{aligned}$$ where $$\rho $$ is the critical radius function associated to $$\mathcal {L}$$ . Let $$W_y f = e^{-y\mathcal {{L}}}f$$ be the heat semigroup of $$\mathcal {{L}}$$ . Given $$\alpha >0,$$ we denote by $$\Lambda _{\alpha /2}^{{W}}$$ the set of functions f which satisfy $$\begin{aligned} \Vert \rho (\cdot )^{-\alpha }f(\cdot )\Vert _\infty 0. \end{aligned}$$ We prove that for $$0<\alpha \le 2-n/q$$ , $$\Lambda ^\alpha _\mathcal {{L}}= \Lambda _{\alpha /2}^{{W}}.$$ As application, we obtain regularity properties of fractional powers (positive and negative) of the operator $$\mathcal {{L}}$$ , Schrodinger Riesz transforms, Bessel potentials and multipliers of Laplace transforms type. The proofs of these results need in an essential way the language of semigroups. Parallel results are obtained for the classes defined through the Poisson semigroup, $$P_yf= e^{-y\sqrt{\mathcal {{L}}}}f.$$

Kreiss bounded and uniformly Kreiss bounded operators

Abstract: If T is a Kreiss bounded operator on a Banach space, then $$\Vert T^n\Vert =O(n)$$ . Forty years ago Shields conjectured that in Hilbert spaces, $$\Vert T^n\Vert = O(\sqrt{n})$$ . A negative answer to this conjecture was given by Spijker, Tracogna and Welfert in 2003. We improve their result and show that this conjecture is not true even for uniformly Kreiss bounded operators. More precisely, for every $$\varepsilon >0$$ there exists a uniformly Kreiss bounded operator T on a Hilbert space such that $$\Vert T^n\Vert \sim (n+1)^{1-\varepsilon }$$ for all $$n\in \mathbb {N}$$ . On the other hand, any Kreiss bounded operator on Hilbert spaces satisfies $$\Vert T^n\Vert =O(\frac{n}{\sqrt{\log n}})$$ . We also prove that the residual spectrum of a Kreiss bounded operator on a reflexive Banach space is contained in the open unit disc, extending known results for power bounded operators. As a consequence we obtain examples of mean ergodic Hilbert space operators which are not Kreiss bounded.

Nodal curves and polarizations with good properties

Abstract: In this paper we deal with polarizations on a nodal curve C with smooth components. Our aim is to study and characterize a class of polarizations, which we call “good”, for which depth one sheaves on C reflect some properties that hold for vector bundles on smooth curves. We will concentrate, in particular, on the relation between the $${{\underline{w}}}$$ -stability of $${\mathcal {O}}_C$$ and the goodness of $${{\underline{w}}}$$ . We prove that these two concepts agree when C is of compact type and we conjecture that the same should hold for all nodal curves.