On the Hall algebra of semigroup representations over \mathbb F _1

Let $\mathrm{A }$ be a finitely generated semigroup with 0. An $\mathrm{A }$ -module over $\mathbb F _1$ (also called an $\mathrm{A }$ -set), is a pointed set $(M,*)$ together with an action of $\mathrm{A }$ . We define and study the Hall algebra $\mathbb H _{\mathrm{A }}$ of the category $\mathcal C _{\mathrm{A }}$ of finite $\mathrm{A }$ -modules. $\mathbb H _{\mathrm{A }}$ is shown to be the universal enveloping algebra of a Lie algebra $\mathfrak n _{\mathrm{A }}$ , called the Hall Lie algebra of $\mathcal C _{\mathrm{A }}$ . In the case of $\langle t \rangle $ —the free monoid on one generator $\langle t \rangle $ , the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent $\langle t \rangle $ -modules) is isomorphic to Kreimer’s Hopf algebra of rooted forests. This perspective allows us to define two new commutative operations on rooted forests. We also consider the examples when $\mathrm{A }$ is a quotient of $\langle t \rangle $ by a congruence, and the monoid $G \cup \{ 0\}$ for a finite group $G$ .     

Module amenability of the second dual and module topological center of semigroup algebras

In this paper we define the module topological center of the second dual $\mathcal{A}^{**}$ of a Banach algebra $\mathcal{A}$ which is a Banach $\mathfrak{A}$ -module with compatible actions on another Banach algebra $\mathfrak{A}$ . We calculate the module topological center of ? ¹(S)^**, as an ? ¹(E)-module, for an inverse semigroup S with an upward directed set of idempotents E. We also prove that ? ¹(S)^** is ? ¹(E)-module amenable if and only if an appropriate group homomorphic image of S is finite.     

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S _* related to the Carleson?CHunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in ${L^\infty(\mathbb{R}, V(\mathbb{R})), PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ and ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ on the weighted Lebesgue spaces ${L^p(\mathbb{R},w)}$ , with 1?< p <? ?? and ${w\in A_p(\mathbb{R})}$ . The Banach algebras ${L^\infty(\mathbb{R}, V(\mathbb{R}))}$ and ${PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ consist, respectively, of all bounded measurable or piecewise continuous ${V(\mathbb{R})}$ -valued functions on ${\mathbb{R}}$ where ${V(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded total variation, and the Banach algebra ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ consists of all Lipschitz ${V_d(\mathbb{R})}$ -valued functions of exponent ${\gamma \in (0,1]}$ on ${\mathbb{R}}$ where ${V_d(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded variation on dyadic shells. Finally, for the Banach algebra ${\mathfrak{A}_{p,w}}$ generated by all pseudodifferential operators a(x, D) with symbols ${a(x, \lambda) \in PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ on the space ${L^p(\mathbb{R}, w)}$ , we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ .     

Large free sets in powers of universal algebras

We prove that for each universal algebra ${(A, \mathcal{A})}$ of cardinality ${|A| \geq 2}$ and infinite set X of cardinality ${|X| \geq | \mathcal{A}|}$ , the X-th power ${(A^{X}, \mathcal{A}^{X})}$ of the algebra ${(A, \mathcal{A})}$ contains a free subset ${\mathcal{F} \subset A^{X}}$ of cardinality ${|\mathcal{F}| = 2^{|X|}}$ . This generalizes the classical Fichtenholtz–Kantorovitch–Hausdorff result on the existence of an independent family ${\mathcal{I} \subset \mathcal{P}(X)}$ of cardinality ${|\mathcal{I}| = |\mathcal{P}(X)|}$ in the Boolean algebra ${\mathcal{P}(X)}$ of subsets of an infinite set X.     

Donsker-type theorems for nonparametric maximum likelihood estimators

Let ${\mathcal{P}}$ be a nonparametric probability model consisting of smooth probability densities and let ${\hat{p}_{n}}$ be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law ${\mathbb{P}}$ . With $\hat{\mathbb{P}}_{n}$ denoting the measure induced by the density ${\hat{p}_{n}}$ , define the stochastic process ${\hat{\nu}}_{n}: f\longmapsto \sqrt{n} \int fd({\hat{\mathbb{P}}}_{n} -\mathbb{P})$ where f ranges over some function class ${\mathcal{F}}$ . We give a general condition for Donsker classes ${\mathcal{F}}$ implying that the stochastic process $\hat{\nu}_{n}$ is asymptotically equivalent to the empirical process in the space ${\ell ^{\infty }(\mathcal{F})}$ of bounded functions on ${ \mathcal{F}}$ . This implies in particular that $\hat{\nu}_{n}$ converges in law in ${\ell ^{\infty }(\mathcal{F})}$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes ${\mathcal{ F}}$ . We give a number of applications: convergence of the probability measure ${\hat{\mathbb{P}}_{n}}$ to ${\mathbb{P}}$ at rate ${\sqrt{n}}$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; ${\sqrt{n}}$ -efficient estimation of nonlinear functionals defined on ${\mathcal{P}}$ ; limit theorems at rate ${\sqrt{n}}$ for the maximum likelihood estimator of the convolution product ${\mathbb{P\ast P}}$ .     

Norm Transitive Semigroups on Operator Algebra

Let $\mathcal A $ be a semigroup of bounded linear operators on the Banach algebra $B(X)$ for a separable Banach space $X$ . We show the transitivity of $\mathcal A $ with the operator norm topology, implies hypercyclicity with the strong operator topology (SOT) while the converse may not be true. As a consequence, SOT-transitive semigroup of left multiplication operators on $B(X)$ is characterized.     

A commutant realization of Odake’s algebra

The bcβγ-system $ \mathcal{W} $ of rank 3 has an action of the affine vertex algebra $ {V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ , and the commutant vertex algebra $ \mathcal{C}=\mathrm{Com}\left( {{V_0}\left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right),\mathcal{W}} \right) $ contains copies of V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and Odake’s algebra $ \mathcal{O} $ . Odake’s algebra is an extension of the N = 2 super-conformal algebra with c = 9, and is generated by eight fields which close nonlinearly under operator product expansions. Our main result is that V _?3/2 $ \left( {\mathfrak{s}{{\mathfrak{l}}_2}} \right) $ and $ \mathcal{O} $ form a Howe pair (i.e., a pair of mutual commutants) inside $ \mathcal{C} $ . More generally, any finite-dimensional representation of a Lie algebra $ \mathfrak{g} $ gives rise to a similar Howe pair, and this example corresponds to the adjoint representation of $ \mathfrak{s}{{\mathfrak{l}}_2} $ .     

Yangians and quantum loop algebras

Let $\mathfrak{g }$ be a complex, semisimple Lie algebra. Drinfeld showed that the quantum loop algebra $U_\hbar (L\mathfrak g )$ of $\mathfrak{g }$ degenerates to the Yangian ${Y_\hbar (\mathfrak g )}$ . We strengthen this result by constructing an explicit algebra homomorphism $\Phi $ from $U_\hbar (L\mathfrak g )$ to the completion of ${Y_\hbar (\mathfrak g )}$ with respect to its grading. We show moreover that $\Phi $ becomes an isomorphism when ${U_\hbar (L\mathfrak g )}$ is completed with respect to its evaluation ideal. We construct a similar homomorphism for $\mathfrak{g }=\mathfrak{gl }_n$ and show that it intertwines the actions of $U_\hbar (L\mathfrak gl _{n})$ and $Y_\hbar (\mathfrak gl _{n})$ on the equivariant $K$ -theory and cohomology of the variety of $n$ -step flags in ${\mathbb{C }}^d$ constructed by Ginzburg–Vasserot.     

Categories of partial algebras for critical points between varieties of algebras

We denote by Con_c A the ${(\vee, 0)}$ -semilattice of all finitely generated congruences of an algebra A. A lifting of a ${(\vee, 0)}$ -semilattice S is an algebra A such that ${S \cong {\rm Con}_{\rm c} A}$ . The assignment Conc can be extended to a functor. The notion of lifting is generalized to diagrams of ${(\vee, 0)}$ -semilattices. A gamp is a partial algebra endowed with a partial subalgebra together with a semilattice-valued distance; gamps form a category that lends itself to a universal algebraic-type study. The raison d’être of gamps is that any algebra can be approximated by its finite subgamps, even in case it is not locally finite. Let ${\mathcal{V}}$ and ${\mathcal{W}}$ be varieties of algebras (on finite, possibly distinct, similarity types). Let P be a finite lattice. We assume the existence of a combinatorial object, called an ${\aleph_0}$ -lifter of P, of infinite cardinality ${\lambda}$ . Let ${\vec{A}}$ be a P-indexed diagram of finite algebras in ${\mathcal{V}}$ . If ${{\rm Con}_{\rm c} \circ \vec{A}}$ has no partial lifting in the category of gamps of ${\mathcal{W}}$ , then there is an algebra ${A \in \mathcal{V}}$ of cardinality ${\lambda}$ such that Con_c A is not isomorphic to Con_c B for any ${B \in \mathcal{W}}$ . This makes it possible to generalize several known results. In particular, we prove the following theorem, without assuming that ${\mathcal{W}}$ is locally finite. Let ${\mathcal{V}}$ be locally finite and let ${\mathcal{W}}$ be congruence-proper (i.e., congruence lattices of infinite members of ${\mathcal{W}}$ are infinite). The following equivalence holds. Every countable ${(\vee, 0)}$ -semilattice with a lifting in ${\mathcal{V}}$ has a lifting in ${\mathcal{W}}$ if and only if every ${\omega}$ -indexed diagram of finite ${(\vee, 0)}$ -semilattices with a lifting in ${\mathcal{V}}$ has a lifting in ${\mathcal{W}}$ . Gamps are also applied to the study of congruence-preserving extensions. Let ${\mathcal{V}}$ be a non-semidistributive variety of lattices and let n ≥ 2 be an integer. There is a bounded lattice ${A \in \mathcal{V}}$ of cardinality ${\aleph_1}$ with no congruence n-permutable, congruence-preserving extension. The lattice A is constructed as a condensate of a square-indexed diagram of lattices.     

Some results on metric n-Lie algebras

We study the structure of a metric n-Lie algebra G over the complex field C. Let G = SR be the Levi decomposition, where R is the radical of G and S is a strong semisimple subalgebra of G. Denote by m(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R ⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R ⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to m(G) + 1. The centralizer of R in G is equal to the sum of all minimal ideals; it is the direct sum of R ⊥ and the center of G. Finally, G has no strong semisimple ideals if and only if R⊥■R.     

On uniformly bounded spherical functions in Hilbert space

Let G be a commutative group, written additively, with a neutral element 0, and let K be a finite group. Suppose that K acts on G via group automorphisms ${G \ni a \mapsto ka \in G}$ , ${k \in K}$ . Let ${{\mathfrak{H}}}$ be a complex Hilbert space and let ${{\mathcal L}({\mathfrak{H}})}$ be the algebra of all bounded linear operators on ${{\mathfrak{H}}}$ . A mapping ${u \colon G \to {\mathcal L}({\mathfrak{H}})}$ is termed a K-spherical function if it satisfies (1) ${|K|^{-1} \sum_{k\in K} u (a+kb)=u (a) u (b)}$ for any ${a,b\in G}$ , where |K| denotes the cardinality of K, and (2) ${u (0) = {\rm id}_{\mathfrak {H}},}$ where ${{\rm id}_{\mathfrak {H}}}$ designates the identity operator on ${{\mathfrak{H}}}$ . The main result of the paper is that for each K-spherical function ${u \colon G \to {\mathcal {L}}({\mathfrak {H}})}$ such that ${\| u \|_{\infty} = \sup_{a\in G} \| u (a)\|_{{\mathcal L}({\mathfrak{H}})} < \infty,}$ there is an invertible operator S in ${{\mathcal L}({\mathfrak{H}})}$ with ${\| S \| \, \| S^{-1}\| \leq |K| \, \| u \|_{\infty}^2}$ such that the K-spherical function ${{\tilde{u}} \colon G \to {\mathcal L}({\mathfrak{H}})}$ defined by ${{\tilde{u}}(a) = S u (a) S^{-1},\,a \in G,}$ satisfies ${{\tilde{u}}(-a) = {\tilde{u}}(a)^*}$ for each ${a \in G}$ . It is shown that this last condition is equivalent to insisting that ${{\tilde{u}}(a)}$ be normal for each ${a \in G}$ .     

Extending structures for Lie algebras

Let $\mathfrak{g }$ be a Lie algebra, $E$ a vector space containing $\mathfrak{g }$ as a subspace. The paper is devoted to the extending structures problem which asks for the classification of all Lie algebra structures on $E$ such that $\mathfrak{g }$ is a Lie subalgebra of $E$ . A general product, called the unified product, is introduced as a tool for our approach. Let $V$ be a complement of $\mathfrak{g }$ in $E$ : the unified product $\mathfrak{g } \,\natural \, V$ is associated to a system $(\triangleleft , \, \triangleright , \, f, \{-, \, -\})$ consisting of two actions $\triangleleft $ and $\triangleright $ , a generalized cocycle $f$ and a twisted Jacobi bracket $\{-, \, -\}$ on $V$ . There exists a Lie algebra structure $[-,-]$ on $E$ containing $\mathfrak{g }$ as a Lie subalgebra if and only if there exists an isomorphism of Lie algebras $(E, [-,-]) \cong \mathfrak{g } \,\natural \, V$ . All such Lie algebra structures on $E$ are classified by two cohomological type objects which are explicitly constructed. The first one $\mathcal{H }^{2}_{\mathfrak{g }} (V, \mathfrak{g })$ will classify all Lie algebra structures on $E$ up to an isomorphism that stabilizes $\mathfrak{g }$ while the second object $\mathcal{H }^{2} (V, \mathfrak{g })$ provides the classification from the view point of the extension problem. Several examples that compute both classifying objects $\mathcal{H }^{2}_{\mathfrak{g }} (V, \mathfrak{g })$ and $\mathcal{H }^{2} (V, \mathfrak{g })$ are worked out in detail in the case of flag extending structures.     

Euclidean hypersurfaces with genuine deformations in codimension two

We classify hypersurfaces of rank two of Euclidean space ${\mathbb{R}^{n+1}}$ that admit genuine isometric deformations in ${\mathbb{R}^{n+2}}$ . That an isometric immersion ${\hat{f}\colon M^n \to \mathbb{R}^{n+2}}$ is a genuine isometric deformation of a hypersurface ${f\colon M^n\to\mathbb{R}^{n+1}}$ means that ${\hat f}$ is nowhere a composition ${\hat f=\hat F\circ f}$ , where ${\hat{F} \colon V\subset \mathbb{R}^{n+1} \to\mathbb{R}^{n+2}}$ is an isometric immersion of an open subset V containing the hypersurface.     

Extension of a theorem of Shi and Tam

In this note, we prove the following generalization of a theorem of Shi and Tam (J Differ Geom 62:79–125, 2002): Let (Ω, g) be an n-dimensional (n ≥ 3) compact Riemannian manifold, spin when n?>?7, with non-negative scalar curvature and mean convex boundary. If every boundary component Σ _i has positive scalar curvature and embeds isometrically as a mean convex star-shaped hypersurface ${{\hat \Sigma}_i \subset \mathbb{R}^n}$ , then $$ \int\limits_{\Sigma_i} H \ d \sigma \le \int\limits_{{\hat \Sigma}_i} \hat{H} \ d {\hat \sigma} $$ where H is the mean curvature of Σ _i in (Ω, g), ${\hat{H}}$ is the Euclidean mean curvature of ${{\hat \Sigma}_i}$ in ${\mathbb{R}^n}$ , and where d σ and ${d {\hat \sigma}}$ denote the respective volume forms. Moreover, equality holds for some boundary component Σ _i if, and only if, (Ω, g) is isometric to a domain in ${\mathbb{R}^n}$ . In the proof, we make use of a foliation of the exterior of the ${\hat \Sigma_i}$ ’s in ${\mathbb{R}^n}$ by the ${\frac{H}{R}}$ -flow studied by Gerhardt (J Differ Geom 32:299–314, 1990) and Urbas (Math Z 205(3):355–372, 1990). We also carefully establish the rigidity statement in low dimensions without the spin assumption that was used in Shi and Tam (J Differ Geom 62:79–125, 2002).     

1D Schrödinger Operators with Short Range Interactions: Two-Scale Regularization of Distributional Potentials

For real ${L_\infty(\mathbb{R})}$ -functions ${\Phi}$ and ${\Psi}$ of compact support, we prove the norm resolvent convergence, as ${\varepsilon}$ and ${\nu}$ tend to 0, of a family ${S_{\varepsilon \nu}}$ of one-dimensional Schrödinger operators on the line of the form $$S_{\varepsilon \nu} = -\frac{d^2}{dx^2} + \frac{\alpha}{\varepsilon^2} \Phi \left( \frac{x}{\varepsilon} \right) + \frac{\beta}{\nu} \Psi \left(\frac{x}{\nu} \right),$$ provided the ratio ${\nu/\varepsilon}$ has a finite or infinite limit. The limit operator S ₀ depends on the shape of ${\Phi}$ and ${\Psi}$ as well as on the limit of ratio ${\nu/\varepsilon}$ . If the potential ${\alpha\Phi}$ possesses a zero-energy resonance, then S ₀ describes a non trivial point interaction at the origin. Otherwise S ₀ is the direct sum of the Dirichlet half-line Schrödinger operators.     

Supports of (\mathfrak{g},\mathfrak{k})-modules of finite type

Let $\mathfrak{g}$ be a semisimple Lie algebra and $\mathfrak{k}$ be a reductive subalgebra in $\mathfrak{g}$ . We say that a $\mathfrak{g}$ -module M is a $(\mathfrak{g},\mathfrak{k})$ -module if M, considered as a $\mathfrak{k}$ -module, is a direct sum of finite-dimensional $\mathfrak{k}$ -modules. We say that a $(\mathfrak{g},\mathfrak{k})$ -module M is of finite type if all $\mathfrak{k}$ -isotopic components of M are finite-dimensional. In this paper we prove that any simple $(\mathfrak{g},\mathfrak{k})$ -module of finite type is holonomic. A simple $\mathfrak{g}$ -module M is associated with the invariants V(M), V(LocM), and L(M) reflecting the ??directions of growth of M.?? We also prove that for a given pair $(\mathfrak{g},\mathfrak{k})$ the set of possible invariants is finite.     

On Suborbital Graphs for the Extended Modular Group {\hat{\Gamma}}

In this paper, we show that the extended modular group ${\hat{\Gamma}}$ acts on ${\hat{\mathbb{Q}}}$ transitively and imprimitively. Then the number of orbits of ${\hat{\Gamma} _{0}(N)}$ on ${\hat{\mathbb{Q}}}$ is calculated and compared with the number of orbits of ${\Gamma _{0}(N)}$ on ${\hat{\mathbb{Q}}}$ . Especially, we obtain the graphs ${\hat{G}_{u, N}}$ of ${\hat{\Gamma}_{0}(N)}$ on ${\hat{\mathbb{Q}}}$ , for each ${N\in\mathbb{N}}$ and each unit ${u \in U_{N} }$ , then we determine the suborbital graph ${\hat{F}_{u,N}}$ . We also give the edge conditions in ${\hat{G}_{u, N}}$ and the necessary and sufficient conditions for a circuit to be triangle in ${\hat{F}_{u, N}.}$      

A Few Comments on the Clifford Algebra {C\ell_2} of the Standard Euclidean Plane

This paper is a continuation of the author’s plenary lecture given at ICCA 9 which was held in Weimar at the Bauhaus University, 15–20 July, 2011. We want to study on both the mathematical and the epistemological levels the thought of the brilliant geometer W. K. Clifford by presenting a few comments on the structure of the Clifford algebra ${C\ell_2}$ associated with the standard Euclidean plane ${\mathbb{R}^2}$ . Miquel’s theorem will be given in the algebraic context of the even Clifford algebra ${C\ell^+_2}$ isomorphic to the real algebra ${\mathbb{C}}$ . The proof of this theorem will be based on the cross ratio (the anharmonic ratio) of four complex numbers. It will lead to a group of homographies of the standard projective line ${\mathbb{C}P^1 = P(\mathbb{C}^2)}$ which appeared so attractive to W. K. Clifford in his overview of a general theory of anharmonics. In conclusion it will be shown how the classical Clifford-Hopf fibration S ¹ → S ³ → S ² leads to the space of spinors ${\mathbb{C}^2}$ of the Euclidean space ${\mathbb{R}^3}$ and to the isomorphism ${{\rm {PU}(1) = \rm {SU}(2)/\{I,-I\} \simeq SO(3)}}$ .     

Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. II: Local Spectra and Fredholmness

Let ${\mathcal{B}_{p,w}}$ be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space ${L^p(\mathbb{R},w)}$ , where ${p\in(1,\infty)}$ and w is a Muckenhoupt weight. We study the Banach subalgebra ${\mathfrak{U}_{p,w}}$ of ${\mathcal{B}_{p,w}}$ generated by all multiplication operators aI ( ${a\in PSO^\diamond}$ ) and all convolution operators W ⁰(b) ( ${b\in PSO_{p,w}^\diamond}$ ), where ${PSO^\diamond\subset L^\infty(\mathbb{R})}$ and ${PSO_{p,w}^\diamond\subset M_{p,w}}$ are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of ${\mathbb{R}\cup\{\infty\}}$ , and M _p,w is the Banach algebra of Fourier multipliers on ${L^p(\mathbb{R},w)}$ . Under some conditions on the Muckenhoupt weight w, using results of the local study of ${\mathfrak{U}_{p,w}}$ obtained in the first part of the paper and applying the theory of Mellin pseudodifferential operators and the two idempotents theorem, we now construct a Fredholm symbol calculus for the Banach algebra ${\mathfrak{U}_{p,w}}$ and establish a Fredholm criterion for the operators ${A\in\mathfrak{U}_{p,w}}$ in terms of their Fredholm symbols. In four partial cases we obtain for ${\mathfrak{U}_{p,w}}$ more effective results.     

Stability results for the N-dimensional Schiffer conjecture via a perturbation method

Given a eigenvalue $\mu _{0m}^2$ of $-\Delta $ in the unit ball $B_1$ , with Neumann boundary conditions, we prove that there exists a class $\mathcal{D}$ of $C^{0,1}$ -domains, depending on $\mu _{0m} $ , such that if $u$ is a no trivial solution to the following problem $ \Delta u+\mu u=0$ in $\Omega , u=0$ on $\partial \Omega $ , and $ \int \nolimits _{\partial \Omega }\partial _{\mathbf{n}}u=0$ , with $\Omega \in \mathcal{D}$ , and $\mu =\mu _{0m}^2+o(1)$ , then $\Omega $ is a ball. Here $\mu $ is a eigenvalue of $-\Delta $ in $\Omega $ , with Neumann boundary conditions.