Recent zbMATH articles in MSC 43Ahttps://zbmath.org/atom/cc/43A2024-04-15T15:10:58.286558ZWerkzeugA characterisation of local \(N\)-rings and an application to abstract harmonic analysishttps://zbmath.org/1530.130162024-04-15T15:10:58.286558Z"Wilkens, Bettina"https://zbmath.org/authors/?q=ai:wilkens.bettinaSummary: A commutative ring with unity is called an \(N\)-ring if each of its ideals is contracted from a Noetherian extension ring. The chief result of this paper is a characterisation of local \(N\)-rings by their subdirectly irreducible quotients. The results are applied to characterise spectral synthesis on \(G\)-invariant subspaces of the space of complex valued functions on an abelian group \(G\).Two-sided configuration equivalence and isomorphismhttps://zbmath.org/1530.200972024-04-15T15:10:58.286558Z"Malekan, Meisam Soleimani"https://zbmath.org/authors/?q=ai:soleimani-malekan.meisam"Rejali, Ali"https://zbmath.org/authors/?q=ai:rejali.aliSummary: The concept of configuration was first introduced by \textit{J. M. Rosenblatt} and \textit{G. A. Willis} [Can. Math. Bull. 44, No. 2, 231--241 (2001; Zbl 0980.43001)] to give a characterization for the amenability of groups. Then \textit{A. Rejali} and \textit{A. Yousofzadeh} [Algebra Colloq. 17, No. 4, 583--594 (2010; Zbl 1203.43003)] introduced the notion of two-sided configuration to study the normal subsets of a group. In [\textit{A. Abdollahi} et al., Ill. J. Math. 48, No. 3, 861--873 (2004; Zbl 1067.43001)], the authors have asked that if two configuration equivalent groups are isomorphic? We show that if \(G\) and \(H\) have same two-sided configuration sets and \(N\) is a normal subgroup of \(G\) with polycyclic or FC quotient, then there is a normal subgroup \(\mathfrak{N}\) of \(H\) such that \(G/N\cong H/\mathfrak{N} \). Also, we show that if \(G\) and \(H\) are two-sided equivalent groups, and if one of them is polycyclic or FC, then they are isomorphic.Counting topologically invariant means on \(L_\infty(G)\) and \(VN(G)\) with ultrafiltershttps://zbmath.org/1530.200982024-04-15T15:10:58.286558Z"Hopfensperger, John"https://zbmath.org/authors/?q=ai:hopfensperger.johnSummary: In 1970, \textit{C. Chou} [Trans. Am. Math. Soc. 151, 443--456 (1970; Zbl 0202.14001)] showed there are \(|\mathbb{N}^\ast|=2^{2^{\mathbb{N}}}\) topologically invariant means on \(L_\infty (G)\) for any noncompact, \(\sigma\)-compact amenable group. Over the following 25 years, the sizes of the sets of topologically invariant means on \(L_\infty (G)\) and \(VN (G)\) were determined for any locally compact group. Each paper on a new case reached the same conclusion -- ``the cardinality is as large as possible'' -- but a unified proof never emerged. In this paper, I show \(L_1 (G)\) and \(A(G)\) always contain orthogonal nets converging to invariance. An orthogonal net indexed by \(\Gamma\) has \(| \Gamma^\ast |\) accumulation points, where \(|\Gamma^\ast |\) is determined by ultrafilter theory.
Among a smattering of other results, I prove \textit{A. L. T. Paterson}'s conjecture [Pac. J. Math. 84, 391--397 (1979; Zbl 0429.43001)] that left and right topologically invariant means on \(L_\infty (G)\) coincide if and only if \(G\) has precompact conjugacy classes.
Finally, I discuss some open problems arising from the study of the sizes of sets of invariant means on groups and semigroups.Isometric actions on \(L_p\)-spaces: dependence on the value of \(p\)https://zbmath.org/1530.220052024-04-15T15:10:58.286558Z"Marrakchi, Amine"https://zbmath.org/authors/?q=ai:marrakchi.amine"de la Salle, Mikael"https://zbmath.org/authors/?q=ai:de-la-salle.mikaelSummary: Answering a question by Chatterji-Druţu-Haglund, we prove that, for every locally compact group \(G\), there exists a critical constant \(p_G \in [0,\infty]\) such that \(G\) admits a continuous affine isometric action on an \(L_p\) space \((0< p<\infty)\) with unbounded orbits if and only if \(p \geq p_G\). A similar result holds for the existence of proper continuous affine isometric actions on \(L_p\) spaces. Using a representation of cohomology by harmonic cocycles, we also show that such unbounded orbits cannot occur when the linear part comes from a measure-preserving action, or more generally a state-preserving action on a von Neumann algebra and \(p>2\). We also prove the stability of this critical constant \(p_G\) under \(L_p\) measure equivalence, answering a question of Fisher.Spectral multipliers on 2-step stratified groups. I.https://zbmath.org/1530.220072024-04-15T15:10:58.286558Z"Calzi, Mattia"https://zbmath.org/authors/?q=ai:calzi.mattiaSummary: Given a 2-step stratified group which does not satisfy a slight strengthening of the Moore-Wolf condition, a sub-Laplacian \({\mathcal{L}}\) and a family \({\mathcal{T}}\) of elements of the derived algebra, we study the convolution kernels associated with the operators of the form \(m({\mathcal{L}}, -\,i {\mathcal{T}})\). Under suitable conditions, we prove that: (i) if the convolution kernel of the operator \(m(\mathcal{L},- \, i\mathcal{T})\) belongs to \(L^1\), then \(m\) equals almost everywhere a continuous function vanishing at \(\infty \) (`Riemann-Lebesgue lemma'); (ii) if the convolution kernel of the operator \(m(\mathcal{L}, - \, i\mathcal{T})\) is a Schwartz function, then \(m\) equals almost everywhere a Schwartz function.Isotypic multiharmonic polynomials and Gelbart-Helgason reciprocityhttps://zbmath.org/1530.220102024-04-15T15:10:58.286558Z"Kable, Anthony C."https://zbmath.org/authors/?q=ai:kable.anthony-cSummary: A result of \textit{S. S. Gelbart} [Trans. Am. Math. Soc. 192, 29--50 (1974; Zbl 0292.22016)] and \textit{S. Helgason} [Adv. Math. 5, 1--154 (1970; Zbl 0209.25403)] that describes the harmonic analysis of the space of functions on certain Stiefel manifolds and a result of Folland that describes the harmonic analysis of the de Rham complex on a sphere are generalized to describe the harmonic analysis of the de Rham complex on certain Stiefel manifolds. Equivalently, the cobranching law from a fundamental representation of an orthogonal group to a larger orthogonal group is determined.The Dunkl-Laplace transform and Macdonald's hypergeometric serieshttps://zbmath.org/1530.330132024-04-15T15:10:58.286558Z"Brennecken, Dominik"https://zbmath.org/authors/?q=ai:brennecken.dominik"Rösler, Margit"https://zbmath.org/authors/?q=ai:rosler.margitSummary: We continue a program generalizing classical results from the analysis on symmetric cones to the Dunkl setting for root systems of type \(A\). In particular, we prove a Dunkl-Laplace transform identity for Heckman-Opdam hypergeometric functions of type \(A\) and more generally, for the associated Opdam-Cherednik kernel. This is achieved by analytic continuation from a Laplace transform identity for non-symmetric Jack polynomials which was stated, for the symmetric case, as a key conjecture by \textit{I. G. Macdonald} [``Hypergeometric functions I'', Preprint, \url{arXiv:1309.4568v1}]. Our proof for the Jack polynomials is based on Dunkl operator techniques and the raising operator of \textit{F. Knop} and \textit{S. Sahi} [Invent. Math. 128, No. 1, 9--22 (1997; Zbl 0870.05076)]. Moreover, we use these results to establish Laplace transform identities between hypergeometric series in terms of Jack polynomials. Finally, we conclude with a Post-Widder inversion formula for the Dunkl-Laplace transform.Sharp estimates for \(W\)-invariant Dunkl and heat kernels in the \(A_n\) casehttps://zbmath.org/1530.351192024-04-15T15:10:58.286558Z"Graczyk, Piotr"https://zbmath.org/authors/?q=ai:graczyk.piotr"Sawyer, Patrice"https://zbmath.org/authors/?q=ai:sawyer.patriceSummary: In this article, we prove exact estimates for the \(W\)-invariant Dunkl kernel and heat kernel, for the root system of type \(A\) with arbitrary positive multiplicities. We apply the estimates of the \(W\)-invariant Dunkl heat kernel to compute sharp estimates for the Newton kernel and for the \(s\)-stable semigroups generated by a fractional power of the \(W\)-invariant Dunkl Laplacian.Modulus of smoothness and approximation theorems in Clifford analysishttps://zbmath.org/1530.420192024-04-15T15:10:58.286558Z"Tyr, Othman"https://zbmath.org/authors/?q=ai:tyr.othmanSummary: This paper uses some basic results on Clifford analysis introduced by \textit{E. Hitzer} [Clifford Anal. Clifford Algebr. Appl. 2, No. 3, 223--235 (2013; Zbl 1297.43006)], to study some problems in the theory of approximation of functions in the space of square integral functions in the Clifford algebra. The equivalence between the moduli of smoothness of all orders constructed by the Steklov function and the K-functionals constructed from the Sobolev-type space is proved. A consequence of this equivalence theorem is given at the end of this work.Multilinear spectral multipliers on Besov and Triebel-Lizorkin spaces on Lie groups of polynomial growthhttps://zbmath.org/1530.420222024-04-15T15:10:58.286558Z"Fang, Jingxuan"https://zbmath.org/authors/?q=ai:fang.jingxuan"Li, Hongbo"https://zbmath.org/authors/?q=ai:li.hongbo"Zhao, Jiman"https://zbmath.org/authors/?q=ai:zhao.jimanSummary: In this paper, on Lie groups of polynomial growth \(G\), we prove the boundedness of multilinear spectral multipliers from the product of Besov spaces \(B_{p_1,q_1}^{s_1}(G)\times B_{p_2,q_2}^{s_2}(G) \times \cdots \times B_{p_N,q_N}^{s_N}(G)\) to Lebesgue spaces \(L^p (G)\) with \(p_1, \ldots, p_N\), \(q_1, \ldots, q_N\), \(p\geqslant 1\) and \(s_1, \ldots, s_N \in\mathbb{R}\). Then we prove the boundedness from the product of Triebel-Lizorkin spaces \(T_{p_1,q_1}^{s_1}(G)\times T_{p_2,q_2}^{s_2}(G) \times \cdots \times T_{p_N,q_N}^{s_N}(G)\) to Lebesgue spaces \(L^p (G)\) with \(p_1, \ldots, p_N\), \(q_1, \ldots, q_N >1\), \(p\geqslant 1\), \(s_1, \ldots, s_N \in\mathbb{R}\).Dilational symmetries of decomposition and coorbit spaceshttps://zbmath.org/1530.420562024-04-15T15:10:58.286558Z"Führ, Hartmut"https://zbmath.org/authors/?q=ai:fuhr.hartmut"Raisi-Tousi, Reihaneh"https://zbmath.org/authors/?q=ai:tousi.reihaneh-raisiSummary: We investigate the invariance properties of general wavelet coorbit spaces and Besov-type decomposition spaces under dilations by matrices. We show that these matrices can be characterized by quasi-isometry properties with respect to a certain metric in frequency domain. We formulate versions of this phenomenon both for the decomposition and coorbit space settings. We then apply the general results to a particular class of dilation groups, the so-called shearlet dilation groups. We present a general, algebraic characterization of matrices that are coorbit compatible with a given shearlet dilation group. We explicitly determine the groups of compatible dilations, for a variety of concrete examples.Wiener amalgam spaces with respect to Orlicz spaces on the affine grouphttps://zbmath.org/1530.430012024-04-15T15:10:58.286558Z"Arıs, Büşra"https://zbmath.org/authors/?q=ai:aris.busra"Öztop, Serap"https://zbmath.org/authors/?q=ai:oztop.serapSummary: Let \(\mathbb{A}\) be the affine group, \(\Phi\) be a Young function and \(L^{\Phi}(\mathbb{A})\) be the corresponding Orlicz space. We study the Orlicz amalgam spaces \(W(L^{\Phi} (\mathbb{A}), L^1 (\mathbb{A}))\) and \(W(L^{\infty} (\mathbb{A}), L^{\Phi} (\mathbb{A}))\), where the local components are \(L^{\Phi}(\mathbb{A}), L^{\infty} (\mathbb{A})\) and the global components are \(L^1(\mathbb{A})\), \(L^{\Phi} (\mathbb{A})\), respectively. We obtain an equivalent discrete-type norm on the spaces \(W(L^{\Phi} (\mathbb{A}), L^1 (\mathbb{A}))\) and \(W(L^{\infty} (\mathbb{A}), L^{\Phi} (\mathbb{A}))\). This allows us to prove new convolution relations.Correction to: ``On harmonic Hilbert spaces on compact abelian groups''https://zbmath.org/1530.430022024-04-15T15:10:58.286558Z"Das, Suddhasattwa"https://zbmath.org/authors/?q=ai:das.suddhasattwa"Giannakis, Dimitrios"https://zbmath.org/authors/?q=ai:giannakis.dimitrios"Montgomery, Michael R."https://zbmath.org/authors/?q=ai:montgomery.michael-rFrom the text: We correct an error in Theorem 6 of the first and second author's paper [ibid. 29, No. 1, Paper No. 12, 26 p. (2023; Zbl 1515.43002)]. That theorem claimed that given a suitable (e.g., absolutely summable, symmetric, and subconvolutive) function \(\lambda : \hat{G}\to\mathbb{C}\) on the dual group \(\hat{G}\), there is an associated harmonic Hilbert space \(\mathcal{H}_\lambda\) of complex-valued functions on \(G\), which is a Banach \(^\ast\)-algebra
with respect to pointwise function multiplication and complex conjugation, and the Gelfand spectrum \(\sigma (\mathcal{H}_\lambda)\) is homeomorphic to \(G\). However, the stated assumptions on \(\lambda\) in that theorem are actually not sufficient to deduce that \(G\cong \sigma (\mathcal{H}_\lambda)\). Here, we show that [loc. cit., Theorem 6] remains valid if and only if \(\lambda\) satisfies the Gelfand-Raikov-Shilov condition. Aside from this modification on the assumptions of Theorem 6, the results of [loc. cit.] remain unchanged.Endomorphisms and derivations of the measure algebra of commutative hypergroupshttps://zbmath.org/1530.430032024-04-15T15:10:58.286558Z"Fechner, Żywilla"https://zbmath.org/authors/?q=ai:fechner.zywilla"Gselmann, Eszter"https://zbmath.org/authors/?q=ai:gselmann.eszter"Székelyhidi, László"https://zbmath.org/authors/?q=ai:szekelyhidi.laszloA hypergroup is defined by associating a convolution to the vector space of Radon measures on a locally compact space $X$, to make $M(X)$ a measure algebra. In classical Euclidean spaces derivatives are defined using the ring associated to the vector space. In the case of a hypergroup, a natural algebraic operation is not required on $X$. The authors use the ring of continuous functions on $X$ to define a module on the measure algebra \(M_c(X)\) of a commutative hypergroup $X$. Derivatives $D$ are continuous linear operators on \(M_c(X)\) and indeed continuous module homomorphisms of \(M_c(X)\) (over the ring of continuous functions \(C(X)\)) satisfying the following conditions:
\begin{itemize}
\item
$D(\mu*\nu)=D\mu*\nu+\mu*D\nu$ for each $\mu,\nu$ in \(M_c(X)\),
\item
$D(\mu+\nu)=D\mu+D\nu$ and
\item
$D(\phi\mu)=\phi D\mu$.
\end{itemize}
The authors also define higher derivatives and establish the connections between the higher derivatives. Examples are provided to illustrate those results.
Reviewer: Norbert Youmbi (Loretto)Poisson representation and Furstenberg entropy of hypergroupshttps://zbmath.org/1530.430042024-04-15T15:10:58.286558Z"Forghani, Behrang"https://zbmath.org/authors/?q=ai:forghani.behrang"Mallahi-Karai, Keivan"https://zbmath.org/authors/?q=ai:karai.keivan-mallahiTopological hypergroups generalize in many ways locally compact groups. But unlike groups the underlying space \(X\) of a hypergroup does not require an algebraic structure making it difficult to generalize results from groups to hypergroups. In order to do so most authors make use of the algebraic structure inherited from the convolution of measures on \(M_c(X)\) to extend results from groups to hypergroups.
In this article the authors consider the theory of Poisson boundary, tail boundary and the associated entropy theory of groups and extend it to the class of discrete hypergroups. At first the authors set the foundational work needed for developing such a theory for hypergroups, essentially defining the basic concepts of random walk for topological hypergroups which will ease the introduction of Poisson boundary, tail boundary and the associated entropy theory. Basic results in connection with such extensions are established. Essentially the authors solve the identification problem for the Poisson boundary of finite range random walks on permutation hypergroups associated with affine groups of homogenous trees. Examples of random walks on these hypergroups that have a countable infinite Poisson boundary are also provided.
Reviewer: Norbert Youmbi (Loretto)\(L^p\)-boundedness of pseudo-differential operators on rank one Riemannian symmetric spaces of noncompact typehttps://zbmath.org/1530.430052024-04-15T15:10:58.286558Z"Pusti, Sanjoy"https://zbmath.org/authors/?q=ai:pusti.sanjoy"Rana, Tapendu"https://zbmath.org/authors/?q=ai:rana.tapenduSummary: We study the boundedness properties of pseudo-differential operators associated with a symbol on rank one Riemannian symmetric spaces of noncompact type, where the symbol satisfies Hörmander-type conditions near infinity. We use a generalized transference principle by Coifman-Weiss to establish a connection between the \(L^p\) operator norm of the local part of the pseudo-differential operators on symmetric space with the Euclidean pseudo-differential operators. We also define a class of pseudo-differential operators with symbols having no regularity conditions in the space variable and explore their \(L^p\)-boundedness properties.Projections in Lipschitz-free spaces induced by group actionshttps://zbmath.org/1530.460092024-04-15T15:10:58.286558Z"Cúth, Marek"https://zbmath.org/authors/?q=ai:cuth.marek"Doucha, Michal"https://zbmath.org/authors/?q=ai:doucha.michalGiven a metric space \(\mathcal M\) and a group \(G\) acting by isometries on \(\mathcal M\), one can consider the space \(\mathcal M/G\) of the closures of the orbits endowed with the Hausdorff metric. The authors show that the space of Lipschitz functions \(\operatorname{Lip}_0(\mathcal M/G)\) is isometric to the subspace of \(\operatorname{Lip}_0(\mathcal M)\) made up of the \(G\)-invariant functions, and that the Lipschitz-free space \(\mathcal F(\mathcal M/G)\) is isometric to a quotient of \(\mathcal F(\mathcal M)\). The main results of the paper provide conditions on \(\mathcal M\) and \(G\) ensuring that \(\operatorname{Lip}_0(\mathcal M/G)\) is complemented in \(\operatorname{Lip}_0(\mathcal M)\) or that \(\mathcal F(\mathcal M/G)\) is complemented in \(\mathcal F(\mathcal M)\). For instance,
\begin{itemize}
\item if \(G\) is compact, then \(\mathcal F(\mathcal M/G)\) is complemented in \(\mathcal F(\mathcal M)\);
\item if \(G\) is abelian or locally compact and amenable, and the orbits are bounded, then \(\operatorname{Lip}_0(\mathcal M/G)\) is complemented in \(\operatorname{Lip}_0(\mathcal M)\).
\end{itemize}
The case in which \(\mathcal F(\mathcal M)\) is a dual space is also analysed.
The results are interesting for the study of properties preserved by complemented subspaces, like the bounded approximation property. The authors address these applications in the last section, posing also several open questions.
It is remarkable that the results are proven from an abstract study of the projection associated with the action of an amenable group on a Banach space that might be useful for other settings.
Reviewer: Luis C. García Lirola (Zaragoza)Hölder continuity of the traces of Sobolev functions to hypersurfaces in Carnot groups and the \(\mathcal{P} \)-differentiability of Sobolev mappingshttps://zbmath.org/1530.460322024-04-15T15:10:58.286558Z"Basalaev, S. G."https://zbmath.org/authors/?q=ai:basalaev.sergey-g"Vodopyanov, S. K."https://zbmath.org/authors/?q=ai:vodopyanov.serguei-kSummary: We study the behavior of Sobolev functions and mappings on the Carnot groups with the left invariant sub-Riemannian metric. We obtain some sufficient conditions for a Sobolev function to be locally Hölder continuous (in the Carnot-Carathéodory metric) on almost every hypersurface of a given foliation. As an application of these results we show that a quasimonotone contact mapping of class \(W^{1,\nu}\) of Carnot groups is continuous, \( \mathcal{P} \)-differentiable almost everywhere, and has the \(\mathcal{N} \)-Luzin property.Linear preservers on idempotents of Fourier algebrashttps://zbmath.org/1530.470472024-04-15T15:10:58.286558Z"Lin, Ying-Fen"https://zbmath.org/authors/?q=ai:lin.ying-fen"Oi, Shiho"https://zbmath.org/authors/?q=ai:oi.shihoSummary: In this article, we give a representation of bounded complex linear operators that preserve idempotent elements on the Fourier algebra of a locally compact group. When such an operator is, moreover, positive or contractive, we show that the operator is induced by either a continuous group homomorphism or a continuous group antihomomorphism. If the groups are totally disconnected, bounded homomorphisms on the Fourier algebra can be realized by the idempotent preserving operators.Random walk on finite extensions of latticeshttps://zbmath.org/1530.600442024-04-15T15:10:58.286558Z"Oussa, Vignon"https://zbmath.org/authors/?q=ai:oussa.vignon-s|oussa.vignonIt is known that the probability that a simple random walk on \(\mathbb{Z}^d\) returns to its initial position in \(2n\) steps is equal to
\[
\frac{1}{d^{2n}}\int_{[0,\,1]^d}\big(\cos(2\pi\lambda_1)+\ldots+\cos(2\pi\lambda_d)\big)^{2n}\mathrm{d}\lambda.
\]
An analogue of this formula is given for a simple random walk on the semidirect product \(G:=A\rtimes H\), where \(A\) is a full rank lattice in \(\mathbb{R}^d\) and \(H\) is a finite subgroup of \(\mathrm{GL}(A)\). The group \(G\) is equipped with the operation
\[
(x,z)(x^\prime, z^\prime)=(x+zx^\prime, zz^\prime)
\]
for \((x,z)\), \((x^\prime, z^\prime)\in A\times H\).
Reviewer: Alexander Iksanov (Kyïv)The efficient computation of Fourier transforms on semisimple algebrashttps://zbmath.org/1530.651892024-04-15T15:10:58.286558Z"Maslen, David"https://zbmath.org/authors/?q=ai:maslen.david-keith"Rockmore, Daniel N."https://zbmath.org/authors/?q=ai:rockmore.daniel-n"Wolff, Sarah"https://zbmath.org/authors/?q=ai:wolff.sarahIn this article, the problem of the efficient computation of a Fourier transform on a finite-dimensional complex semisimple algebra is discussed. The authors present a general approach to the construction of efficient algorithms for computing a Fourier transform on a semisimple algebra and give a general result (Theorem 4.5) to find efficient Fourier transforms on a finite dimensional semisimple algebra with special subalgebra structure. Particular results include highly efficient algorithms for the Brauer, Temperley-Lieb and Birman-Murakami-Wenzl algebras. To obtain these results the authors use a connection between Bratteli diagrams, the derived path algebra and the construction of Gelfand-Tsetlin bases.
Reviewer: S. F. Lukomskii (Saratov)On the existence of bound states of a system of two fermions on the two-dimensional cubic latticehttps://zbmath.org/1530.810802024-04-15T15:10:58.286558Z"Abdukhakimov, S. Kh."https://zbmath.org/authors/?q=ai:abdukhakimov.s-kh"Lakaev, S. N."https://zbmath.org/authors/?q=ai:lakaev.saidakhmat-n|lakaev.saidakhmat-norjigitovichSummary: We construct a two-particle discrete Schrödinger-type operator \(\widehat{H}_{\mu}(k)=\widehat{H}_0(k)+\mu\widehat{V}\), \(k\in\mathbb{T}^2\) associated to a system of two fermions on the two-dimensional cubic lattice \(\mathbb{Z}^2\) interacting via short-range potential, where the non-perturbed part \(\widehat{H}_0(k)\), \(k\in\mathbb{T}^2\) is a convolution type operator with dispersion relation \(\mathcal{E}_k(\cdot)\) defined on the torus \(\mathbb{T}^2\) and having a degenerate minimum at \(0\in\mathbb{T}^2 \). The existence of eigenvalues below the essential spectrum of the operator \(\widehat{H}_{\mu}(k)\) is proved in the following two cases: in the case \(k=0\) for all \(\mu>0\) and in the case of \(k\neq 0\) for large \(\mu>0\).