Extremal eigenvalues and eigenvectors of deformed Wigner matrices

We consider random matrices of the form \(H = W + \lambda V, \lambda \in {\mathbb {R}}^+\), where \(W\) is a real symmetric or complex Hermitian Wigner matrix of size \(N\) and \(V\) is a real bounded diagonal random matrix of size \(N\) with i.i.d. entries that are independent of \(W\). We assume subexponential decay of the distribution of the matrix entries of \(W\) and we choose \(\lambda \sim 1\), so that the eigenvalues of \(W\) and \(\lambda V\) are typically of the same order. Further, we assume that the density of the entries of \(V\) is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is \(\lambda _+\in {\mathbb {R}}^+\) such that the largest eigenvalues of \(H\) are in the limit of large \(N\) determined by the order statistics of \(V\) for \(\lambda >\lambda _+\). In particular, the largest eigenvalue of \(H\) has a Weibull distribution in the limit \(N\rightarrow \infty \) if \(\lambda >\lambda _+\). Moreover, for \(N\) sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for \(\lambda >\lambda _+\), while they are completely delocalized for \(\lambda <\lambda _+\). Similar results hold for the lowest eigenvalues.     

Generating Hyperbolic Singularities in Semitoric Systems Via Hopf Bifurcations

Let \((M,\Omega )\) be a connected symplectic 4-manifold and let \(F=(J,H) :M\rightarrow \mathbb {R}^2\) be a completely integrable system on M with only non-degenerate singularities. Assume that F does not have singularities with hyperbolic blocks and that \(p_1,\ldots ,p_n\) are the focus–focus singularities of F. For each subset \(S=\{i_1,\ldots ,i_j\}\), we will show how to modify F locally around any \(p_i, i \in S\), in order to create a new integrable system \(\widetilde{F}=(J, \widetilde{H}) :M \rightarrow \mathbb {R}^2\) such that its classical spectrum \(\widetilde{F}(M)\) contains j smooth curves of singular values corresponding to non-degenerate transversally hyperbolic singularities of \(\widetilde{F}\). Moreover the focus–focus singularities of \(\widetilde{F}\) are precisely \(p_i\), \(i \in \{1,\ldots ,n\} \setminus S\). The proof is based on Eliasson’s linearization theorem for non-degenerate singularities, and properties of the Hamiltonian Hopf bifurcation.     

Asymptotic Expansions of the Contact Angle in Nonlocal Capillarity Problems

We consider a family of nonlocal capillarity models, where surface tension is modeled by exploiting the family of fractional interaction kernels \(|z|^{-n-s}\), with \(s\in (0,1)\) and n the dimension of the ambient space. The fractional Young’s law (contact angle condition) predicted by these models coincides, in the limit as \(s\rightarrow 1^-\), with the classical Young’s law determined by the Gauss free energy. Here we refine this asymptotics by showing that, for s close to 1, the fractional contact angle is always smaller than its classical counterpart when the relative adhesion coefficient \(\sigma \) is negative, and larger if \(\sigma \) is positive. In addition, we address the asymptotics of the fractional Young’s law in the limit case \(s\rightarrow 0^+\) of interaction kernels with heavy tails. Interestingly, near \(s=0\), the dependence of the contact angle from the relative adhesion coefficient becomes linear.     

Multiple positive solutions of the stationary Keller–Segel system

We consider the stationary Keller–Segel equation

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta v+v=\lambda e^v, \quad v>0 \quad &{} \text {in }\Omega ,\\ \partial _\nu v=0 &{}\text {on } \partial \Omega , \end{array}\right. } \end{aligned}$$

where \(\Omega \) is a ball. In the regime \(\lambda \rightarrow 0\), we study the radial bifurcations and we construct radial solutions by a gluing variational method. For any given \(n\in \mathbb {N}_0\), we build a solution having multiple layers at \(r_1,\ldots ,r_n\) by which we mean that the solutions concentrate on the spheres of radii \(r_i\) as \(\lambda \rightarrow 0\) (for all \(i=1,\ldots ,n\)). A remarkable fact is that, in opposition to previous known results, the layers of the solutions do not accumulate to the boundary of \(\Omega \) as \(\lambda \rightarrow 0\). Instead they satisfy an optimal partition problem in the limit.

     

On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where

$$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$

is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and

$$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$

then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.

     

An Erdös–Révész Type Law of the Iterated Logarithm for Order Statistics of a Stationary Gaussian Process

Let \(\{X(t):t\in \mathbb R_+\}\) be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, \(\mathbb E X(t) = 0, \mathbb E X^2(t) = 1\) and correlation function satisfying (i) \(r(t) = 1 - C|t|^{\alpha } + o(|t|^{\alpha })\) as \(t\rightarrow 0\) for some \(0\le \alpha \le 2\) and \(C>0\); (ii) \(\sup _{t\ge s}|r(t)|<1\) for each \(s>0\) and (iii) \(r(t) = O(t^{-\lambda })\) as \(t\rightarrow \infty \) for some \(\lambda >0\). For any \(n\ge 1\), consider n mutually independent copies of X and denote by \(\{X_{r:n}(t):t\ge 0\}\) the rth smallest order statistics process, \(1\le r\le n\). We provide a tractable criterion for assessing whether, for any positive, non-decreasing function \(f, \mathbb P(\mathscr {E}_f)=\mathbb P(X_{r:n}(t) > f(t)\, \text { i.o.})\) equals 0 or 1. Using this criterion we find, for a family of functions \(f_p(t)\) such that \(z_p(t)=\mathbb P(\sup _{s\in [0,1]}X_{r:n}(s)>f_p(t))=O((t\log ^{1-p} t)^{-1})\), that \(\mathbb P(\mathscr {E}_{f_p})= 1_{\{p\ge 0\}}\). Consequently, with \(\xi _p (t) = \sup \{s:0\le s\le t, X_{r:n}(s)\ge f_p(s)\}\), for \(p\ge 0\) we have \(\lim _{t\rightarrow \infty }\xi _p(t)=\infty \) and \(\limsup _{t\rightarrow \infty }(\xi _p(t)-t)=0\) a.s. Complementarily, we prove an Erdös–Révész type law of the iterated logarithm lower bound on \(\xi _p(t)\), namely, that \(\liminf _{t\rightarrow \infty }(\xi _p(t)-t)/h_p(t) = -1\) a.s. for \(p>1\) and \(\liminf _{t\rightarrow \infty }\log (\xi _p(t)/t)/(h_p(t)/t) = -1\) a.s. for \(p\in (0,1]\), where \(h_p(t)=(1/z_p(t))p\log \log t\).     

A note on Wilson’s functional equation

Let S be a semigroup, and \(\mathbb {F}\) a field of characteristic \(\ne 2\). If the pair \(f,g:S \rightarrow \mathbb {F}\) is a solution of Wilson’s \(\mu \)-functional equation such that \(f \ne 0\), then g satisfies d’Alembert’s \(\mu \)-functional equation.     

Upper and lower fast Khintchine spectra in continued fractions

For an irrational number \(x\in [0,1)\), let \(x=[a_1(x), a_2(x),\ldots ]\) be its continued fraction expansion. Let \(\psi : \mathbb {N} \rightarrow \mathbb {N}\) be a function with \(\psi (n)/n\rightarrow \infty \) as \(n\rightarrow \infty \). The (upper, lower) fast Khintchine spectrum for \(\psi \) is defined as the Hausdorff dimension of the set of numbers \(x\in (0,1)\) for which the (upper, lower) limit of \(\frac{1}{\psi (n)}\sum _{j=1}^n\log a_j(x)\) is equal to 1. The fast Khintchine spectrum was determined by Fan, Liao, Wang, and Wu. We calculate the upper and lower fast Khintchine spectra. These three spectra can be different.     

Convergence of empirical spectral distributions of large dimensional quaternion sample covariance matrices

In this paper, we establish the limit of empirical spectral distributions of quaternion sample covariance matrices. Motivated by Bai and Silverstein (Spectral analysis of large dimensional random matrices, Springer, New York, 2010) and Mar?enko and Pastur (Matematicheskii Sb, 114:507–536, 1967), we can extend the results of the real or complex sample covariance matrix to the quaternion case. Suppose \(\mathbf X_n = ({x_{jk}^{(n)}})_{p\times n}\) is a quaternion random matrix. For each \(n\), the entries \(\{x_{ij}^{(n)}\}\) are independent random quaternion variables with a common mean \(\mu \) and variance \(\sigma ^2>0\). It is shown that the empirical spectral distribution of the quaternion sample covariance matrix \(\mathbf S_n=n^{-1}\mathbf X_n\mathbf X_n^*\) converges to the Mar?enko–Pastur law as \(p\rightarrow \infty \), \(n\rightarrow \infty \) and \(p/n\rightarrow y\in (0,+\infty )\).     

Representation of real Riesz maps on a strong <Emphasis Type="Italic">f</Emphasis>-ring by prime elements of a frame

In classical topology, it is proved that for a topological space X, every bounded Riesz map \(\varphi :C (X) \rightarrow {\mathbb {R}}\) is of the from \({\hat{x}}\) for a point \(x\in X\). In this paper, our main purpose is to prove a version of this result by lattice-valued maps. A ring representation of the from \(A\rightarrow {\mathbb {R}}\) is constructed. This representation is denoted by \(\widetilde{p_c}\) that is an onto f-ring homomorphism for every \(p\in \Sigma L\), where its index c, denotes a cozero lattice-valued map. Also, it is shown that for every Riesz map \(\phi :A\rightarrow {\mathbb {R}} \) and \(c\in F(A, L)\) with specific properties, there exists \(p\in \Sigma L\) such that \(\phi =\phi (1)\widetilde{p_c}\).     

$$u\tau $$-convergence in locally solid vector lattices

Let \((x_\alpha )\) be a net in a locally solid vector lattice \((X,\tau )\); we say that \((x_\alpha )\) is unbounded \(\tau \)-convergent to a vector \(x\in X\) if \(|x_\alpha -x |\wedge w \xrightarrow {\tau } 0\) for all \(w\in X_+\). In this paper, we study general properties of unbounded \(\tau \)-convergence (shortly \(u\tau \)-convergence). \(u\tau \)-convergence generalizes unbounded norm convergence and unbounded absolute weak convergence in normed lattices that have been investigated recently. We introduce \(u\tau \)-topology and briefly study metrizability and completeness of this topology.     

Estimates for large solutions of a quasilinear elliptic equation

Let \(\Omega \) be a bounded smooth domain of \(R^{n}\). We study the asymptotic behaviour of the solutions to the equation \(\triangle u-|Du|^{q}=f(u)\) in \(\Omega , 1<q<2,\) which satisfy the boundary condition \(u(x)\rightarrow \infty \) as \(x\rightarrow \partial \Omega \). These solutions are called large or blowup solutions. Near the boundary we give lower and upper bounds for the ratio \(\psi (u)/\delta \), where \(\psi (u) = \int _{u}^{\infty }1/\sqrt{2F}dt\), \(F'=f\), \(\delta =dist(x,\partial \Omega )\) or for the ratio \(u/\delta ^{(2-q)/(1-q)}\). When in particular the ratio \(f/F^{q/2}\)is regular at infinity, we find again known results (Bandle and Giarrusso, in Adv Diff Equ 1, 133–150, 1996; Giarrusso, in Comptes Rendus de l’Acad Sci 331, 777–782 2000).     

Diagonalization of the Finite Hilbert Transform on Two Adjacent Intervals

We continue the study of stability of solving the interior problem of tomography. The starting point is the Gelfand–Graev formula, which converts the tomographic data into the finite Hilbert transform (FHT) of an unknown function f along a collection of lines. Pick one such line, call it the x-axis, and assume that the function to be reconstructed depends on a one-dimensional argument by restricting f to the x-axis. Let \(I_1\) be the interval where f is supported, and \(I_2\) be the interval where the Hilbert transform of f can be computed using the Gelfand–Graev formula. The equation to be solved is \(\left. {\mathcal {H}}_1 f=g\right| _{I_2}\), where \({\mathcal {H}}_1\) is the FHT that integrates over \(I_1\) and gives the result on \(I_2\), i.e. \({\mathcal {H}}_1: L^2(I_1)\rightarrow L^2(I_2)\). In the case of complete data, \(I_1\subset I_2\), and the classical FHT inversion formula reconstructs f in a stable fashion. In the case of interior problem (i.e., when the tomographic data are truncated), \(I_1\) is no longer a subset of \(I_2\), and the inversion problems becomes severely unstable. By using a differential operator L that commutes with \({\mathcal {H}}_1\), one can obtain the singular value decomposition of \({\mathcal {H}}_1\). Then the rate of decay of singular values of \({\mathcal {H}}_1\) is the measure of instability of finding f. Depending on the available tomographic data, different relative positions of the intervals \(I_{1,2}\) are possible. The cases when \(I_1\) and \(I_2\) are at a positive distance from each other or when they overlap have been investigated already. It was shown that in both cases the spectrum of the operator \({\mathcal {H}}_1^*{\mathcal {H}}_1\) is discrete, and the asymptotics of its eigenvalues \(\sigma _n\) as \(n\rightarrow \infty \) has been obtained. In this paper we consider the case when the intervals \(I_1=(a_1,0)\) and \(I_2=(0,a_2)\) are adjacent. Here \(a_1 < 0 < a_2\). Using recent developments in the Titchmarsh–Weyl theory, we show that the operator L corresponding to two touching intervals has only continuous spectrum and obtain two isometric transformations \(U_1\), \(U_2\), such that \(U_2{\mathcal {H}}_1 U_1^*\) is the multiplication operator with the function \(\sigma (\lambda )\), \(\lambda \ge (a_1^2+a_2^2)/8\). Here \(\lambda \) is the spectral parameter. Then we show that \(\sigma (\lambda )\rightarrow 0\) as \(\lambda \rightarrow \infty \) exponentially fast. This implies that the problem of finding f is severely ill-posed. We also obtain the leading asymptotic behavior of the kernels involved in the integral operators \(U_1\), \(U_2\) as \(\lambda \rightarrow \infty \). When the intervals are symmetric, i.e. \(-a_1=a_2\), the operators \(U_1\), \(U_2\) are obtained explicitly in terms of hypergeometric functions.     

Eigenvalue Counting Function for Robin Laplacians on Conical Domains

We study the discrete spectrum of the Robin Laplacian \(Q^{\Omega }_\alpha \) in \(L^2(\Omega )\), \(u\mapsto -\Delta u, \quad D_n u=\alpha u \text { on }\partial \Omega \), where \(D_n\) is the outer unit normal derivative and \(\Omega \subset {\mathbb {R}}^{3}\) is a conical domain with a regular cross-section \(\Theta \subset {\mathbb {S}}^2\), n is the outer unit normal, and \(\alpha >0\) is a fixed constant. It is known from previous papers that the bottom of the essential spectrum of \(Q^{\Omega }_\alpha \) is \(-\alpha ^2\) and that the finiteness of the discrete spectrum depends on the geometry of the cross-section. We show that the accumulation of the discrete spectrum of \(Q^\Omega _\alpha \) is determined by the discrete spectrum of an effective Hamiltonian defined on the boundary and far from the origin. By studying this model operator, we prove that the number of eigenvalues of \(Q^{\Omega }_\alpha \) in \((-\infty ,-\alpha ^2-\lambda )\), with \(\lambda >0\), behaves for \(\lambda \rightarrow 0\) as

$$\begin{aligned} \dfrac{\alpha ^2}{8\pi \lambda } \int _{\partial \Theta } \kappa _+(s)^2\mathrm {d}s +o\left( \frac{1}{\lambda }\right) , \end{aligned}$$

where \(\kappa _+\) is the positive part of the geodesic curvature of the cross-section boundary.

     

On the distribution of the van der Corput sequence in arbitrary base

A central limit theorem with explicit error bound, and a large deviation result are proved for a sequence of weakly dependent random variables of a special form. As a corollary, under certain conditions on the function \(f:[0,1] \rightarrow \mathbb {R}\) a central limit theorem and a large deviation result are obtained for the sum \(\sum _{n=0}^{N-1} f(x_n)\), where \(x_n\) is the base b van der Corput sequence for an arbitrary integer \(b \ge 2\). Similar results are also proved for the \(L^p\) discrepancy of the same sequence for \(1 \le p < \infty \). The main methods used in the proofs are the Berry–Esseen theorem and Fourier analysis.     

Qualitative properties of generalized principal eigenvalues for superquadratic viscous Hamilton–Jacobi equations

This paper is concerned with the ergodic problem for superquadratic viscous Hamilton–Jacobi equations with exponent \(m>2\). We prove that the generalized principal eigenvalue of the equation converges to a constant as \(m\rightarrow \infty \), and that the limit coincides with the generalized principal eigenvalue of an ergodic problem with gradient constraint. We also investigate some qualitative properties of the generalized principal eigenvalue with respect to a perturbation of the potential function. It turns out that different situations take place according to \(m=2\), \(2<m<\infty \), and the limiting case \(m=\infty \).     

Surfaces in $$\mathbb {R}^7$$ obtained from harmonic maps in $$S^6$$S6.

We will investigate the local geometry of the surfaces in the 7-dimensional Euclidean space associated to harmonic maps from a Riemann surface \(\varSigma \) into \(S^6\). By applying methods based on the use of harmonic sequences, we will characterize the conformal harmonic immersions \(\varphi :\varSigma \rightarrow S^6\) whose associated immersions \(F:\varSigma \rightarrow \mathbb {R}^7\) belong to certain remarkable classes of surfaces, namely: minimal surfaces in hyperspheres; surfaces with parallel mean curvature vector field; pseudo-umbilical surfaces; isotropic surfaces.     

On the quasi-arithmetic Gauss-type iteration

For a sequence of continuous, monotone functions \(f_1,\ldots ,f_n :I \rightarrow \mathbb {R}\) (I is an interval) we define the mapping \(M :I^n \rightarrow I^n\) as a Cartesian product of quasi-arithmetic means generated by \(f_j\)-s. It is known that, for every initial vector, the iteration sequence of this mapping tends to the diagonal of \(I^n\). We will prove that whenever all \(f_j\)-s are \(\mathcal {C}^2\) with nowhere vanishing first derivative, then this convergence is quadratic. Furthermore, the limit \(\frac{\text {Var}\, M^{k+1}(v)}{(\text {Var}\, M^{k}(v))^2}\) will be calculated in a nondegenerated case.     

Monotonicity of solutions for some nonlocal elliptic problems in half-spaces

In this paper we consider classical solutions u of the semilinear fractional problem \((-\Delta )^s u = f(u)\) in \({\mathbb {R}}^N_+\) with \(u=0\) in \({\mathbb {R}}^N {\setminus } {\mathbb {R}}^N_+\), where \((-\Delta )^s\), \(0<s<1\), stands for the fractional laplacian, \(N\ge 2\), \({\mathbb {R}}^N_+=\{x=(x',x_N)\in {\mathbb {R}}^N{:}\ x_N>0\}\) is the half-space and \(f\in C^1\) is a given function. With no additional restriction on the function f, we show that bounded, nonnegative, nontrivial classical solutions are indeed positive in \({\mathbb {R}}^N_+\) and verify

$$\begin{aligned} \frac{\partial u}{\partial x_N}>0 \quad \hbox {in } {\mathbb {R}}^N_+. \end{aligned}$$

This is in contrast with previously known results for the local case \(s=1\), where nonnegative solutions which are not positive do exist and the monotonicity property above is not known to hold in general even for positive solutions when \(f(0)<0\).

     

Some characterizations of almost limited operators

In this paper we give several characterizations of almost limited operators. Mainly, it is proved that an operator \(T:X\rightarrow E\) from a Banach space X into a \(\sigma \)-Dedekind complete Banach lattice E is almost limited if and only if \(\left\| T^{*}\left( f_{n}\right) \right\| \rightarrow 0\) for every positive weak\(^{*}\) null sequence \(\left( f_{n}\right) \) of \(E^{*}\). Moreover, we present some interesting connections between almost limited, almost Dunford–Pettis and limited operators.