Lower order eigenvalues of a system of equations of the drifting Laplacian on the metric measure spaces

Let \(\Omega \) be a bounded domain with smooth boundary in an n-dimensional metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and let \(\mathbf {u}=(u^1, \ldots , u^n)\) be a vector-valued function from \(\Omega \) to \(\mathbb {R}^n\). In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of the drifting Laplacian: \(\mathbb {L}_{\phi } \mathbf {u} + \alpha [ \nabla (\mathrm {div}\mathbf { u}) -\nabla \phi \mathrm {div} \mathbf {u}]= - \widetilde{\sigma } \mathbf {u}\), in \( \Omega \), and \(u|_{\partial \Omega }=0,\) where \(\mathbb {L}_{\phi } = \Delta - \nabla \phi \cdot \nabla \) is the drifting Laplacian and \(\alpha \) is a nonnegative constant. We establish some universal inequalities for lower order eigenvalues of this problem on the metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and the Gaussian shrinking soliton \((\mathbb {R}^n, \langle ,\rangle _{\mathrm {can}}, e^{-\frac{|x|^2}{4}}dv, \frac{1}{2})\). Moreover, we give an estimate for the upper bound of the second eigenvalue of this problem in terms of its first eigenvalue on the gradient product Ricci soliton \((\Sigma \times \mathbb {R}, \langle ,\rangle , e^{-\frac{\kappa t^2}{2}}dv, \kappa )\), where \( \Sigma \) is an Einstein manifold with constant Ricci curvature \(\kappa \).     

On Reachable Families of the Loewner Differential Equation in Several Complex Variables

Graham, Hamada, Kohr and Kohr studied the normalized time \(T\) reachable families \(\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},\Omega )\) of the Loewner differential equation, which are generated by the Carathéodory mappings with values in a subfamily \(\Omega \) of the Carathéodory family \({\mathcal {N}}_A\) for the Euclidean unit ball \({\mathbb {B}}^n\), where \(A\) is a linear operator with \(k_+(A)<2m(A)\) (\(k_+(A)\) is the Lyapunov index of \(A\) and \(m(A)=\min \{\mathfrak {R}\left\langle Az,z\right\rangle \big |z\in {\mathbb {C}}^n,\Vert z\Vert =1\}\)). They obtained some compactness and density results, as generalizations of related results due to Roth, and conjectured that if \(\Omega \) is compact and convex, then \(\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},\Omega )\) is compact and \(\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},ex\,\Omega )\) is dense in \(\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},\Omega )\), where \(ex\,\Omega \) denotes the corresponding set of extreme points and \(T\in [0,\infty ]\). We confirm this, by embedding the Carathéodory mappings in a suitable Bochner space.     

The identification problem for complex-valued Ornstein–Uhlenbeck operators in $$L^p(\mathbb {R}^d,\mathbb {C}^N)$$

In this paper we study perturbed Ornstein–Uhlenbeck operators

$$\begin{aligned} \left[ \mathcal {L}_{\infty } v\right] (x)=A\triangle v(x) + \left\langle Sx,\nabla v(x)\right\rangle -B v(x),\,x\in \mathbb {R}^d,\,d\geqslant 2, \end{aligned}$$

for simultaneously diagonalizable matrices \(A,B\in \mathbb {C}^{N,N}\). The unbounded drift term is defined by a skew-symmetric matrix \(S\in \mathbb {R}^{d,d}\). Differential operators of this form appear when investigating rotating waves in time-dependent reaction diffusion systems. We prove under certain conditions that the maximal domain \(\mathcal {D}(A_p)\) of the generator \(A_p\) belonging to the Ornstein–Uhlenbeck semigroup coincides with the domain of \(\mathcal {L}_{\infty }\) in \(L^p(\mathbb {R}^d,\mathbb {C}^N)\) given by

$$\begin{aligned} \mathcal {D}^p_{\mathrm {loc}}(\mathcal {L}_0)=\left\{ v\in W^{2,p}_{\mathrm {loc}}\cap L^p\mid A\triangle v + \left\langle S\cdot ,\nabla v\right\rangle \in L^p\right\} ,\,1<p<\infty . \end{aligned}$$

One key assumption is a new \(L^p\)-dissipativity condition

$$\begin{aligned} |z|^2\mathrm {Re}\,\left\langle w,Aw\right\rangle + (p-2)\mathrm {Re}\,\left\langle w,z\right\rangle \mathrm {Re}\,\left\langle z,Aw\right\rangle \geqslant \gamma _A |z|^2|w|^2\;\forall \,z,w\in \mathbb {C}^N \end{aligned}$$

for some \(\gamma _A>0\). The proof utilizes the following ingredients. First we show the closedness of \(\mathcal {L}_{\infty }\) in \(L^p\) and derive \(L^p\)-resolvent estimates for \(\mathcal {L}_{\infty }\). Then we prove that the Schwartz space is a core of \(A_p\) and apply an \(L^p\)-solvability result of the resolvent equation for \(A_p\). In addition, we derive \(W^{1,p}\)-resolvent estimates. Our results may be considered as extensions of earlier works by Metafune, Pallara and Vespri to the vector-valued complex case.

     

Lengths of words in transformation semigroups generated by digraphs

Given a simple digraph D on n vertices (with \(n\ge 2\)), there is a natural construction of a semigroup of transformations \(\langle D\rangle \). For any edge (a, b) of D, let \(a\rightarrow b\) be the idempotent of rank \(n-1\) mapping a to b and fixing all vertices other than a; then, define \(\langle D\rangle \) to be the semigroup generated by \(a \rightarrow b\) for all \((a,b) \in E(D)\). For \(\alpha \in \langle D\rangle \), let \(\ell (D,\alpha )\) be the minimal length of a word in E(D) expressing \(\alpha \). It is well known that the semigroup \(\mathrm {Sing}_n\) of all transformations of rank at most \(n-1\) is generated by its idempotents of rank \(n-1\). When \(D=K_n\) is the complete undirected graph, Howie and Iwahori, independently, obtained a formula to calculate \(\ell (K_n,\alpha )\), for any \(\alpha \in \langle K_n\rangle = \mathrm {Sing}_n\); however, no analogous non-trivial results are known when \(D \ne K_n\). In this paper, we characterise all simple digraphs D such that either \(\ell (D,\alpha )\) is equal to Howie–Iwahori’s formula for all \(\alpha \in \langle D\rangle \), or \(\ell (D,\alpha ) = n - \mathrm {fix}(\alpha )\) for all \(\alpha \in \langle D\rangle \), or \(\ell (D,\alpha ) = n - \mathrm {rk}(\alpha )\) for all \(\alpha \in \langle D\rangle \). We also obtain bounds for \(\ell (D,\alpha )\) when D is an acyclic digraph or a strong tournament (the latter case corresponds to a smallest generating set of idempotents of rank \(n-1\) of \(\mathrm {Sing}_n\)). We finish the paper with a list of conjectures and open problems.     

Mapping prefer-opposite to prefer-one de Bruijn sequences

We present a mapping of the binary prefer-opposite de Bruijn sequence of order n onto the binary prefer-one de Bruijn sequence of order \(n-1\). The mapping is based on the differentiation operator \(D(\langle {b_1,\ldots ,b_l}\rangle ) = \langle b_2-b_1, b_3-b_2,\ldots , b_{l}-b_{l-1} \rangle \) where bit subtraction is modulo two. We show that if we take the prefer-opposite sequence \(\langle {b_1,b_2,\ldots ,b_{2^n}}\rangle \), apply D to get the sequence \(\langle {\hat{b}_1, \ldots , \hat{b}_{2^n-1}}\rangle \) and drop all the bits \(\hat{b}_i\) such that \(\langle {\hat{b}_i,\ldots ,\hat{b}_{i+n-1}}\rangle \) is a substring of \(\langle {\hat{b}_1,\ldots ,\hat{b}_{i+n-2}}\rangle \), we get the prefer-one de Bruijn sequence of order \(n-1\).     

Green’s function for elliptic systems: existence and Delmotte–Deuschel bounds

This paper is divided into two parts: In the main deterministic part, we prove that for an open domain \(D \subset \mathbb {R}^d\) with \(d \ge 2\), for every (measurable) uniformly elliptic tensor field a and for almost every point \(y \in D\), there exists a unique Green’s function centred in y associated to the vectorial operator \(-\nabla \cdot a\nabla \) in D. This result implies the existence of the fundamental solution for elliptic systems when \(d>2\), i.e. the Green function for \(-\nabla \cdot a\nabla \) in \(\mathbb {R}^d\). In the second part, we introduce a shift-invariant ensemble \(\langle \cdot \rangle \) over the set of uniformly elliptic tensor fields, and infer for the fundamental solution G some pointwise bounds for \(\langle |G(\cdot ; x,y)|\rangle \), \(\langle |\nabla _x G(\cdot ; x,y)|\rangle \) and \(\langle |\nabla _x\nabla _y G(\cdot ; x,y)|\rangle \). These estimates scale optimally in space and provide a generalisation to systems of the bounds obtained by Delmotte and Deuschel for the scalar case.     

Local Maximum Modulus Property for Polyanalytic Functions

Let \(\Omega \subset {\mathbb {C}}\) be an open subset and let \({\mathcal {F}}\) be a space of functions defined on \(\Omega \). \({\mathcal {F}}\) is said to have the local maximum modulus property if: for every \(f\in {\mathcal {F}},p_0\in \Omega ,\) and for every sufficiently small domain \(D\subset \Omega ,\) with \(p_0\in D,\) it holds true that \(\max _{z\in \overline{D}}\left| f(z)\right| = \max _{z\in \Sigma \cup \partial D}\left| f(z)\right| ,\) where \(\Sigma \subset \Omega \) denotes the set of points at which \(\left| f\right| \) attains strict local maximum. This property fails for \({\mathcal {F}}=C^{\infty }.\) We verify it however for the set of complex-valued functions whose real and imaginary parts are real analytic. We show by example that the property cannot be improved upon whenever \({\mathcal {F}}\) is the set of n-analytic functions on \(\Omega \), \(n\ge 2,\) in the sense that locality cannot be removed as a condition and independently \(\Sigma \) cannot be removed from the conclusion.     

Multilicity results for some nonlinear ellitic roblems with asymtotically $${{\varvec{}}}$$-linear terms

Taking any \(p > 1\), we consider the asymptotically p-linear problem

$$\begin{aligned} \left\{ \begin{array}{ll} - {{\mathrm{div}}}(a(x,u,\nabla u)) + A_t(x,u,\nabla u)\ = \ \lambda ^\infty |u|^{p-2}u + g^\infty (x,u) &{}\quad \hbox {in}\;\Omega ,\\ u\ = \ 0 &{}\quad \hbox {on}\;\partial \Omega , \end{array} \right. \end{aligned}$$

where \(\Omega \) is a bounded domain in \(\mathbb R^N\), \(N\ge 2\), \(A(x,t,\xi )\) is a real function on \(\Omega \times \mathbb R\times \mathbb R^N\) which grows with power p with respect to \(\xi \) and has partial derivatives \(A_t(x,t,\xi ) = \frac{\partial A}{\partial t}(x,t,\xi )\), \(a(x,t,\xi ) = \nabla _\xi A(x,t,\xi )\). If \(A(x,t,\xi ) \rightarrow A^\infty (x,t)\) and \(\frac{g^\infty (x,t)}{|t|^{p-1}} \rightarrow 0\) as \(|t| \rightarrow +\infty \), suitable assumptions, variational methods and either the cohomological index theory or its related pseudo-index one, allow us to prove the existence of multiple nontrivial bounded solutions in the non-resonant case, i.e. if \(\lambda ^\infty \) is not an eigenvalue of the operator associated to \(\nabla _\xi A^\infty (x,\xi )\). In particular, while in [14] the model problem \(A(x,t,\xi ) = \mathcal{A}(x,t) |\xi |^p\) with \(p > N\) is studied, here our goal is twofold: extending such results not only to a more general family of functions \(A(x,t,\xi )\), but also to the more difficult case \(1 < p \le N\).

     

On the principal components of sample covariance matrices

We introduce a class of \(M \times M\) sample covariance matrices \({\mathcal {Q}}\) which subsumes and generalizes several previous models. The associated population covariance matrix \(\Sigma = \mathbb {E}{\mathcal {Q}}\) is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of \(\Sigma - I_M\) may depend on \(M\) in an arbitrary fashion. We investigate the principal components, i.e. the top eigenvalues and eigenvectors, of \({\mathcal {Q}}\). We derive precise large deviation estimates on the generalized components \(\langle {\mathbf{{w}}} , {\varvec{\xi }_i}\rangle \) of the outlier and non-outlier eigenvectors \(\varvec{\xi }_i\). Our results also hold near the so-called BBP transition, where outliers are created or annihilated, and for degenerate or near-degenerate outliers. We believe the obtained rates of convergence to be optimal. In addition, we derive the asymptotic distribution of the generalized components of the non-outlier eigenvectors. A novel observation arising from our results is that, unlike the eigenvalues, the eigenvectors of the principal components contain information about the subcritical spikes of \(\Sigma \). The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix \({\mathcal {Q}}\), satisfying \(\mathbb {E}{\mathcal {Q}} = I_M\), as input: the isotropic local Marchenko–Pastur law established in Bloemendal et al. (Electron J Probab 19:1–53, 2014), level repulsion, and quantum unique ergodicity of the eigenvectors. The latter is a special case of a new universality result for the joint eigenvalue–eigenvector distribution.     

Existence of self-Cheeger sets on Riemannian manifolds

Let \(({\mathcal M},g)\) be a smooth compact Riemannian manifold of dimension \(N\ge 2\). We prove the existence of a family \((\Omega _\varepsilon )_{\varepsilon \in (0,\varepsilon _0)}\) of self-Cheeger sets in \(({\mathcal M},g)\). The domains \(\Omega _\varepsilon \subset {\mathcal M}\) are perturbations of geodesic balls of radius \(\varepsilon \) centered at \(p \in {\mathcal M}\), and in particular, if \(p_0\) is a non-degenerate critical point of the scalar curvature of g, then the family \((\partial \Omega _\varepsilon )_{\varepsilon \in (0,\varepsilon _0)}\) constitutes a smooth foliation of a neighborhood of \(p_0\).     

On the Rationality of the Spectrum

Let \(\Omega \subset {\mathbb R}\) be a compact set with measure 1. If there exists a subset \(\Lambda \subset {\mathbb R}\) such that the set of exponential functions \(E_{\Lambda }:=\{e_\lambda (x) = e^{2\pi i \lambda x}|_\Omega :\lambda \in \Lambda \}\) is an orthonormal basis for \(L^2(\Omega )\), then \(\Lambda \) is called a spectrum for the set \(\Omega \). A set \(\Omega \) is said to tile \({\mathbb R}\) if there exists a set \(\mathcal T\) such that \(\Omega + \mathcal T = {\mathbb R}\), the set \(\mathcal T\) is called a tiling set. A conjecture of Fuglede suggests that spectra and tiling sets are related. Lagarias and Wang (Invent Math 124(1–3):341–365, 1996) proved that tiling sets are always periodic and are rational. That any spectrum is also a periodic set was proved in Bose and Madan (J Funct Anal 260(1):308–325, 2011) and Iosevich and Kolountzakis (Anal PDE 6:819–827, 2013). In this paper, we give some partial results to support the rationality of the spectrum.     

Mean curvature bounds and eigenvalues of Robin Laplacians

We consider the Laplacian with attractive Robin boundary conditions,

$$\begin{aligned} Q^\Omega _\alpha u=-\Delta u, \quad \dfrac{\partial u}{\partial n}=\alpha u \text { on } \partial \Omega , \end{aligned}$$

in a class of bounded smooth domains \(\Omega \in \mathbb {R}^\nu \); here \(n\) is the outward unit normal and \(\alpha >0\) is a constant. We show that for each \(j\in \mathbb {N}\) and \(\alpha \rightarrow +\infty \), the \(j\)th eigenvalue \(E_j(Q^\Omega _\alpha )\) has the asymptotics

$$\begin{aligned} E_j(Q^\Omega _\alpha )=-\alpha ^2 -(\nu -1)H_\mathrm {max}(\Omega )\,\alpha +{\mathcal O}(\alpha ^{2/3}), \end{aligned}$$

where \(H_\mathrm {max}(\Omega )\) is the maximum mean curvature at \(\partial \Omega \). The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of \(H_\mathrm {max}\). In particular, we show that the ball is the strict minimizer of \(H_\mathrm {max}\) among the smooth star-shaped domains of a given volume, which leads to the following result: if \(B\) is a ball and \(\Omega \) is any other star-shaped smooth domain of the same volume, then for any fixed \(j\in \mathbb {N}\) we have \(E_j(Q^B_\alpha )>E_j(Q^\Omega _\alpha )\) for large \(\alpha \). An open question concerning a larger class of domains is formulated.

     

A note on generalized skew derivations on Lie ideals

Let \(\mathcal {R}\) be a prime ring, \(\mathcal {Z(R)}\) its center, \(\mathcal {C}\) its extended centroid, \(\mathcal {L}\) a Lie ideal of \(\mathcal {R}, \mathcal {F}\) a generalized skew derivation associated with a skew derivation d and automorphism \(\alpha \). Assume that there exist \(t\ge 1\) and \(m,n\ge 0\) fixed integers such that \( vu = u^m\mathcal {F}(uv)^tu^n\) for all \(u,v \in \mathcal {L}\). Then it is shown that either \(\mathcal {L}\) is central or \(\mathrm{char}(\mathcal {R})=2, \mathcal {R}\subseteq \mathcal {M}_2(\mathcal {C})\), the ring of \(2\times 2\) matrices over \(\mathcal {C}, \mathcal {L}\) is commutative and \(u^2\in \mathcal {Z(R)}\), for all \(u\in \mathcal {L}\). In particular, if \(\mathcal {L}=[\mathcal {R,R}]\), then \(\mathcal {R}\) is commutative.     

A potential well for an integral functional with singular Lagrangian

This paper concerns a functional of the form

$$\begin{aligned} \Phi (u)=\int _\Omega L(x,u(x),\nabla u(x))\, dx \end{aligned}$$

on the Sobolev space \(H_0^1(\Omega )\) where \(\Omega \) is a bounded open subset of \({\mathbb {R}}^N\) with \(N\ge 3\) and \(0\in \Omega \). The hypotheses on L ensure that \(u\equiv 0\) is a critical point of \(\Phi \), but allow the Lagrangian to be singular at \(x=0\). It is shown that, under these assumptions, the usual conditions associated with Jacobi (positive definiteness of the second variation of \(\Phi \) at \(u\equiv 0\)), Legendre (ellipticity at \(u\equiv 0\)) and Weierstrass [strict convexity of \(L(x,s,\xi )\) with respect to \(\xi \)] from the calculus of variations are not sufficient ensure that \(u\equiv 0\) is a local minimum of \(\Phi \). Using recent criteria for the existence of a potential well of a \(C^1\)-functional on a real Hilbert space, conditions implying that \(u\equiv 0\) lies in a potential well of \(\Phi \) are established. They are shown to be sharp in some cases.

     

A Completion of the Spectrum for the Overlarge Sets of Pure Mendelsohn Triple Systems

A pure Mendelsohn triple system of order v, denoted by PMTS(v), is a pair \((X,\mathcal {B})\) where X is a v-set and \(\mathcal {B}\) is a collection of cyclic triples on X such that every ordered pair of X belongs to exactly one triple of \(\mathcal {B}\) and if \(\langle a,b,c\rangle \in \mathcal {B}\) implies \(\langle c,b,a\rangle \notin \mathcal {B}\). An overlarge set of PMTS(v), denoted by OLPMTS(v), is a collection \(\{(Y{\setminus }\{y_i\},{\mathcal {A}}_i)\}_i\), where Y is a \((v+1)\)-set, \(y_i\in Y\), each \((Y{\setminus }\{y_i\},{\mathcal {A}}_i)\) is a PMTS(v) and these \({\mathcal {A}}_i\)s form a partition of all cyclic triples on Y. It is shown in [3] that there exists an OLPMTS(v) for \(v\equiv 1,3\) (mod 6), \(v>3\), or \(v \equiv 0,4\) (mod 12). In this paper, we shall discuss the existence problem of OLPMTS(v)s for \(v\equiv 6,10\) (mod 12) and get the following conclusion: there exists an OLPMTS(v) if and only if \(v\equiv 0,1\) (mod 3), \(v>3\) and \(v\ne 6\).     

Large Deviations for Sums of Random Vectors Attracted to Operator Semi-Stable Laws

Let \(\{X_i, i\ge 1\}\) be i.i.d. \(\mathbb {R}^d\)-valued random vectors attracted to operator semi-stable laws and write \(S_n=\sum _{i=1}^{n}X_i\). This paper investigates precise large deviations for both the partial sums \(S_n\) and the random sums \(S_{N(t)}\), where N(t) is a counting process independent of the sequence \(\{X_i, i\ge 1\}\). In particular, we show for all unit vectors \(\theta \) the asymptotics

$$\begin{aligned} {\mathbb P}(|\langle S_n,\theta \rangle |>x)\sim n{\mathbb P}(|\langle X,\theta \rangle |>x) \end{aligned}$$

which holds uniformly for x-region \([\gamma _n, \infty )\), where \(\langle \cdot , \cdot \rangle \) is the standard inner product on \(\mathbb {R}^d\) and \(\{\gamma _n\}\) is some monotone sequence of positive numbers. As applications, the precise large deviations for random sums of real-valued random variables with regularly varying tails and \(\mathbb {R}^d\)-valued random vectors with weakly negatively associated occurrences are proposed. The obtained results improve some related classical ones.

     

CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups

Let \(\mathcal S\) be an abelian group of automorphisms of a probability space \((X, {\mathcal A}, \mu )\) with a finite system of generators \((A_1, \ldots , A_d).\) Let \(A^{{\underline{\ell }}}\) denote \(A_1^{\ell _1} \ldots A_d^{\ell _d}\), for \({{\underline{\ell }}}= (\ell _1, \ldots , \ell _d).\) If \((Z_k)\) is a random walk on \({\mathbb {Z}}^d\), one can study the asymptotic distribution of the sums \(\sum _{k=0}^{n-1} \, f \circ A^{\,{Z_k(\omega )}}\) and \(\sum _{{\underline{\ell }}\in {\mathbb {Z}}^d} {\mathbb {P}}(Z_n= {\underline{\ell }}) \, A^{\underline{\ell }}f\), for a function f on X. In particular, given a random walk on commuting matrices in \(SL(\rho , {\mathbb {Z}})\) or in \({\mathcal M}^*(\rho , {\mathbb {Z}})\) acting on the torus \({\mathbb {T}}^\rho \), \(\rho \ge 1\), what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on \({\mathbb {T}}^\rho \) after normalization? In this paper, we prove a central limit theorem when X is a compact abelian connected group G endowed with its Haar measure (e.g., a torus or a connected extension of a torus), \(\mathcal S\) a totally ergodic d-dimensional group of commuting algebraic automorphisms of G and f a regular function on G. The proof is based on the cumulant method and on preliminary results on random walks.