Optimal bounds for convergence of expected spectral distributions to the semi-circular law

Let \(\mathbf {X}=(X_{jk})_{j,k=1}^n\) denote a Hermitian random matrix with entries \(X_{jk}\), which are independent for \(1\le j\le k\le n\). We consider the rate of convergence of the empirical spectral distribution function of the matrix \(\mathbf {X}\) to the semi-circular law assuming that \(\mathbf{E}X_{jk}=0\), \(\mathbf{E}X_{jk}^2=1\) and that

$$\begin{aligned} \sup _{n\ge 1}\sup _{1\le j,k\le n}\mathbf{E}|X_{jk}|^4=:\mu _4<\infty , \end{aligned}$$

and

$$\begin{aligned} \sup _{1\le j,k\le n}|X_{jk}|\le D_0n^{\frac{1}{4}}. \end{aligned}$$

By means of a recursion argument it is shown that the Kolmogorov distance between the expected spectral distribution of the Wigner matrix \(\mathbf {W}=\frac{1}{\sqrt{n}}\mathbf {X}\) and the semicircular law is of order \(O(n^{-1})\).

     

The fixed points of the multivariate smoothing transform

Let \((\mathbf {T}_1, \mathbf {T}_2, \ldots )\) be a sequence of random \(d\times d\) matrices with nonnegative entries, and let Q be a random vector with nonnegative entries. Consider random vectors \(X\) with nonnegative entries, satisfying

$$\begin{aligned} X\mathop {=}\limits ^{{\mathcal {L}}}\sum _{i \ge 1} \mathbf {T}_i X_i + Q, \end{aligned}$$

(*)

where \(\mathop {=}\limits ^{{\mathcal {L}}}\) denotes equality of the corresponding laws, \((X_i)_{i \ge 1}\) are i.i.d. copies of \(X\) and independent of \((Q, \mathbf {T}_1, \mathbf {T}_2, \ldots )\). For \(d=1\), this equation, known as fixed point equation of the smoothing transform, has been intensively studied. Under assumptions similar to the one-dimensional case, we obtain a complete characterization of all solutions \(X\) to (*) in the non-critical case, and existence results in the critical case.

     

Local uniqueness and periodicity for the prescribed scalar curvature problem of fractional operator in $${\mathbb {R}}^{N}$$

Consider the following prescribed scalar curvature problem involving the fractional Laplacian with critical exponent:

$$\begin{aligned} \left\{ \begin{array}{ll}(-\Delta )^{\sigma }u=K(y)u^{\frac{N+2\sigma }{N-2\sigma }} \text { in }~ {\mathbb {R}}^{N},\\ ~u>0, \quad y\in {\mathbb {R}}^{N}.\end{array}\right. \end{aligned}$$

(0.1)

For \(N\ge 4\) and \(\sigma \in (\frac{1}{2}, 1),\) we prove a local uniqueness result for bubbling solutions of (0.1). Such a result implies that some bubbling solutions preserve the symmetry from the scalar curvature K(y).

     

Symmetry breaking of an elliptic equation in expanding annuli

We obtain non-radial bifurcation from radial solutions of a semilinear elliptic equation in expanding annuli of \(\mathbb {R}^N\). To obtain the main results, we use a blow-up argument via the Morse index of the regular entire solutions of the equation

$$\begin{aligned} -\Delta u=\lambda u^p \quad \text {in}\quad \mathbb {R}^N. \end{aligned}$$

(0.1)

The main results of this paper can be seen as applications of the properties of regular entire solutions of (0.1).

     

A family of polylog-trigonometric integrals

In this paper, we study the sequences

$$\begin{aligned} I_n=\int _0^1\mathrm {Li}_n(\sin \pi x)\mathrm {d}x\quad \text{ and }\quad J_n=\int _0^1\mathrm {Li}_n(\cos \pi x)\mathrm {d}x, \end{aligned}$$

where \(\mathrm {Li}_n\) is the nth polylogarithm function. Among others, we determine their generating functions, asymptotic behaviour and their connection to the well-known log-sine integrals

$$\begin{aligned} S_n=(-1)^n\int _0^1\log ^n(\sin \pi x)\mathrm {d}x. \end{aligned}$$

With the help of the explicit forms of \(I_n\) and \(J_n\), we deduce closed-form evaluations for a number of polylog-trigonometric definite integrals.

     

Multiple complex-valued solutions for nonlinear magnetic Schrödinger equations

We study, in the semiclassical limit, the singularly perturbed nonlinear Schrödinger equations

$$\begin{aligned} L^{\hbar }_{A,V} u = f(|u|^2)u \quad \hbox {in}\quad \mathbb {R}^N \end{aligned}$$

(0.1)

where \(N \ge 3\), \(L^{\hbar }_{A,V}\) is the Schrödinger operator with a magnetic field having source in a \(C^1\) vector potential A and a scalar continuous (electric) potential V defined by

$$\begin{aligned} L^{\hbar }_{A,V}= -\hbar ^2 \Delta -\frac{2\hbar }{i} A \cdot \nabla + |A|^2- \frac{\hbar }{i}\mathrm{div}A + V(x). \end{aligned}$$

(0.2)

Here, f is a nonlinear term which satisfies the so-called Berestycki-Lions conditions. We assume that there exists a bounded domain \(\Omega \subset \mathbb {R}^N\) such that

$$\begin{aligned} m_0 \equiv \inf _{x \in \Omega } V(x) < \inf _{x \in \partial \Omega } V(x) \end{aligned}$$

and we set \(K = \{ x \in \Omega \ | \ V(x) = m_0\}\). For \(\hbar >0\) small we prove the existence of at least \({\mathrm{cupl}}(K) + 1\) geometrically distinct, complex-valued solutions to (0.1) whose moduli concentrate around K as \(\hbar \rightarrow 0\).

     

Maximal monotone operator theory and its applications to thin film equation in epitaxial growth on vicinal surface

In this work we consider

$$\begin{aligned} w_t=\left[ \left( w_{hh}+c_0\right) ^{-3}\right] _{hh},\qquad w(0)=w^0, \end{aligned}$$

(1)

which is derived from a thin film equation for epitaxial growth on vicinal surface. We formulate the problem as the gradient flow of a suitably-defined convex functional in a non-reflexive space. Then by restricting it to a Hilbert space and proving the uniqueness of its sub-differential, we can apply the classical maximal monotone operator theory. The mathematical difficulty is due to the fact that \(w_{hh}\) can appear as a positive Radon measure. We prove the existence of a global strong solution with hidden singularity. In particular, (1) holds almost everywhere when \(w_{hh}\) is replaced by its absolutely continuous part.

     

Congruences for 7 and 49-regular partitions modulo powers of 7

Let \(b_{k}(n)\) denote the number of k-regular partitions of n. In this paper, we prove Ramanujan-type congruences modulo powers of 7 for \(b_{7}(n)\) and \(b_{49}(n)\). For example, for all \(j\ge 1\) and \(n\ge 0\), we prove that

$$\begin{aligned} b_{7}\Bigg (7^{2j-1}n+\frac{3\cdot 7^{2j-1}-1}{4}\Bigg )\equiv 0\pmod {7^{j}} \end{aligned}$$

and

$$\begin{aligned} b_{49}\Big (7^{j}n+7^{j}-2\Big )\equiv 0\pmod {7^{j}}. \end{aligned}$$

     

Multiplicity results for elliptic problems with super-critical concave and convex nonlinearties

We shall prove a multiplicity result for semilinear elliptic problems with a super-critical nonlinearity of the form,

$$\begin{aligned} \left\{ \begin{array}{ll} -\,\Delta u =|u|^{p-2} u+\mu |u|^{q-2}u, &{}\quad x \in \Omega \\ u=0, &{}\quad x \in \partial \Omega \end{array} \right. \end{aligned}$$

(1)

where \(\Omega \subset \mathbb {R}^n\) is a bounded domain with \(C^2\)-boundary and \(1<q< 2<p.\) As a consequence of our results we shall show that, for each \(p>2\), there exists \(\mu ^*>0\) such that for each \(\mu \in (0, \mu ^*)\) problem (1) has a sequence of solutions with a negative energy. This result is already known for the subcritical values of p. In this paper, we shall extend it to the supercritical values of p as well. Our methodology is based on a new variational principle established by one of the authors that allows one to deal with problems beyond the usual locally compactness structure.

     

On a problem of Bourgain concerning the $$L_p$$ norms of exponential sums

For \(n \ge 1\) let

$$\begin{aligned} {\mathcal {A}}_n := \bigg \{ P: P(z) = \sum \limits _{j=1}^n{z^{k_j}}: 0 \le k_1 < k_2 < \cdots < k_n, k_j \in {\mathbb {Z}} \bigg \}, \end{aligned}$$

that is, \({\mathcal {A}}_n\) is the collection of all sums of \(n\) distinct monomials. These polynomials are also called Newman polynomials. Let

$$\begin{aligned} M_{p}(Q) := \left( \int _{0}^{1}{\left| Q(e^{i2\pi t}) \right| ^p\,dt} \right) ^{1/p}, \qquad p > 0. \end{aligned}$$

We define

$$\begin{aligned} S_{n,p} := \sup _{Q \in {\mathcal {A}}_n}{\frac{M_p(Q)}{\sqrt{n}}} \qquad \text{ and } \qquad S_p := \liminf _{n \rightarrow \infty }{S_{n,p}} \le \Sigma _p := \limsup _{n \rightarrow \infty }{S_{n,p}}. \end{aligned}$$

We show that

$$\begin{aligned} \Sigma _p \ge \Gamma (1+p/2)^{1/p}, \qquad p \in (0,2). \end{aligned}$$

The special case \(p=1\) recaptures a recent result of Aistleitner [1], the best known lower bound for \(\Sigma _1\).

     

Absolutely minimising generalised solutions to the equations of vectorial calculus of variations in $$L^\infty $$

Consider the supremal functional

$$\begin{aligned} E_\infty (u,A) := \Vert \mathscr {L}(\cdot ,u,\mathrm {D}u)\Vert _{L^\infty (A)},\quad A\subseteq \Omega , \end{aligned}$$

(1)

applied to \(W^{1,\infty }\) maps \(u:\Omega \subseteq \mathbb {R}\longrightarrow \mathbb {R}^N\), \(N\ge 1\). Under certain assumptions on \(\mathscr {L}\), we prove for any given boundary data the existence of a map which is:

1. (i)
   a vectorial Absolute Minimiser of (1) in the sense of Aronsson,
    
2. (ii)
   a generalised solution to the ODE system associated to (1) as the analogue of the Euler-Lagrange equations,
    
3. (iii)
   a limit of minimisers of the respective \(L^p\) functionals as \(p\rightarrow \infty \) for any \(q\ge 1\) in the strong \(W^{1,q}\) topology and
    
4. (iv)
   partially \(C^2\) on \(\Omega \) off an exceptional compact nowhere dense set.
    

Our method is based on \(L^p\) approximations and stable a priori partial regularity estimates. For item ii) we utilise the recently proposed by the author notion of \(\mathcal {D}\)-solutions in order to characterise the limit as a generalised solution. Our results are motivated from and apply to Data Assimilation in Meteorology.

     

Positive solutions of nonlinear multi-point boundary value problems

This paper deals with the existence of positive solutions of nonlinear differential equation

$$\begin{aligned} u^{\prime \prime }(t)+ a(t) f(u(t) )=0,\quad 0<t <1, \end{aligned}$$

subject to the boundary conditions

$$\begin{aligned} u(0)=\sum _{i=1}^{m-2} a_i u (\xi _i) ,\quad u^{\prime } (1) = \sum _{i=1}^{m-2} b_i u^{\prime } (\xi _i), \end{aligned}$$

where \( \xi _i \in (0,1) \) with \( 0< \xi _1<\xi _2< \cdots<\xi _{m-2} < 1,\) and \(a_i,b_i \) satisfy   \(a_i,b_i\in [0,\infty ),~~ 0< \sum _{i=1}^{m-2} a_i <1,\) and \( \sum _{i=1}^{m-2} b_i <1. \) By using Schauder’s fixed point theorem, we show that it has at least one positive solution if f is nonnegative and continuous. Positive solutions of the above boundary value problem satisfy the Harnack inequality

$$\begin{aligned} \displaystyle \inf _{0 \le t \le 1} u(t) \ge \gamma \Vert u\Vert _\infty . \end{aligned}$$

     

A remark on exceptional sets in Erdös-Rényi limit theorem

Let \((x_i)_{i=1}^{+\infty }\) be the digits sequence in the unique terminating dyadic expansion of \(x\in [0,1)\). The run-length function \(l_n(x)\) is defined by

$$\begin{aligned} l_n(x):=\max \left\{ j:x_{i+1}=x_{i+2}=\cdots =x_{i+j}=1\ \text {for some}\ 0\le i\le n-j\right\} . \end{aligned}$$

Erdös and Rényi proved that

$$\begin{aligned} \lim _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=1, \text {a.e.}\ x\in [0,1). \end{aligned}$$

In this note, we show that for each pair of numbers \(\alpha ,\beta \in [0,+\infty ]\) with \(\alpha \le \beta \), the following exceptional set

$$\begin{aligned} E_{\alpha ,\beta }=\left\{ x\in [0,1):\liminf _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=\alpha ,\ \limsup _{n\rightarrow +\infty }\frac{l_n(x)}{\log _2{n}}=\beta \right\} \end{aligned}$$

has Hausdorff dimension one.

     

Sharp Moser–Trudinger Inequalities on Hyperbolic Spaces with Exact Growth Condition

Let \(\Phi _{n}(x)=e^x-\sum _{j=0}^{n-2}\frac{x^j}{j!}\) and \(\alpha _{n} =n\omega _{n-1}^{\frac{1}{n-1}}\) be the sharp constant in Moser’s inequality (where \(\omega _{n-1}\) is the area of the surface of the unit \(n\)-ball in \(\mathbb {R}^n\)), and \(dV\) be the volume element on the \(n\)-dimensional hyperbolic space \((\mathbb {H}^n, g)\) (\(n\ge {2}\)). In this paper, we establish the following sharp Moser–Trudinger type inequalities with the exact growth condition on \(\mathbb {H}^n\):

For any \(u\in {W^{1,n}(\mathbb {H}^n)}\) satisfying \(\Vert \nabla _{g}u\Vert _{n}\le {1}\), there exists a constant \(C(n)>0\) such that

$$\begin{aligned} \int _{\mathbb {H}^n}\frac{\Phi _{n}(\alpha _{n}|u|^{\frac{n}{n-1}})}{(1+|u|)^{\frac{n}{n-1}}}dV \le {C(n)\Vert u\Vert _{L^n}^{n}}. \end{aligned}$$

The power \(\frac{n}{n-1}\) and the constant \(\alpha _{n}\) are optimal in the following senses:

1. (i)
   If the power \(\frac{n}{n-1}\) in the denominator is replaced by any \(p<\frac{n}{n-1}\), then there exists a sequence of functions \(\{u_{k}\}\) such that \(\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}\), but

   $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha _{n}(|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV \rightarrow {\infty }. \end{aligned}$$
    
2. (ii)
   If \(\alpha >\alpha _{n}\), then there exists a sequence of function \(\{u_{k}\}\) such that \(\Vert \nabla _{g}u_{k}\Vert _{n}\le {1}\), but

   $$\begin{aligned} \frac{1}{\Vert u_{k}\Vert _{L^n}^{n}}\int _{\mathbb {H}^n} \frac{\Phi _{n}(\alpha (|u_{k}|)^{\frac{n}{n-1}})}{(1+|u_{k}|)^{p}}dV\rightarrow {\infty }, \end{aligned}$$

   for any \(p\ge {0}\).
    

This result sharpens the earlier work of the authors Lu and Tang (Adv Nonlinear Stud 13(4):1035–1052, 2013) on best constants for the Moser–Trudinger inequalities on hyperbolic spaces.

     

On 3-regular partitions with odd parts distinct

We consider \(\text {pod}_3(n)\), the number of 3-regular partitions with odd parts distinct, whose generating function is

$$\begin{aligned} \sum _{n\ge 0}\text {pod}_3(n)q^n=\frac{(-q;q^2)_\infty (q^6;q^6)_\infty }{(q^2;q^2)_\infty (-q^3;q^3)_\infty }=\frac{\psi (-q^3)}{\psi (-q)}, \end{aligned}$$

where

$$\begin{aligned} \psi (q)=\sum _{n\ge 0}q^{(n^2+n)/2}=\sum _{-\infty }^\infty q^{2n^2+n}. \end{aligned}$$

For each \(\alpha >0\), we obtain the generating function for

$$\begin{aligned} \sum _{n\ge 0}\text {pod}_3\left( 3^{\alpha }n+\delta _\alpha \right) q^n, \end{aligned}$$

where \(4\delta _\alpha \equiv {-1}\pmod {3^{\alpha }}\) if \(\alpha \) is even, \(4\delta _\alpha \equiv {-1}\pmod {3^{\alpha +1}}\) if \(\alpha \) is odd.

We show that the sequence {\(\text {pod}_3(n)\)} satisfies the internal congruences

$$\begin{aligned} \text {pod}_3(9n+2)\equiv \text {pod}_3(n)\pmod 9, \end{aligned}$$

(0.1)

$$\begin{aligned} \text {pod}_3(27n+20)\equiv \text {pod}_3(3n+2)\pmod {27} \end{aligned}$$

(0.2)

and

$$\begin{aligned} \text {pod}_3(243n+182)\equiv \text {pod}_3(27n+20)\pmod {81}. \end{aligned}$$

(0.3)

     

On the numerical approximation of $$\infty $$-harmonic mappings

A map \(u : \Omega \subseteq \mathbb {R}^n \longrightarrow \mathbb {R}^N\), is said to be \(\infty \)-harmonic if it satisfies

Open image in new window

(1)

The system (1) is the model of vector-valued Calculus of Variations in \(L^\infty \) and arises as the “Euler-Lagrange” equation in relation to the supremal functional

$$\begin{aligned} E_\infty (u,\Omega )\, :=\, \Vert \text {D}u \Vert _{L^\infty (\Omega )}. \end{aligned}$$

(2)

In this work we provide numerical approximations of solutions to the Dirichlet problem when \(n=2\) and in the vector valued case of \(N=2,3\) for certain carefully selected boundary data on the unit square. Our experiments demonstrate interesting and unexpected phenomena occurring in the vector valued case and provide insights on the structure of general solutions and the natural separation to phases they present.

     

Uniform Bounds on the Relative Error in the Approximation of Upper Quantiles for Sums of Arbitrary Independent Random Variables

Fix any \(n\ge 1\). Let \(\tilde{X}_1,\ldots ,\tilde{X}_n\) be independent random variables. For each \(1\le j \le n\), \(\tilde{X}_j\) is transformed in a canonical manner into a random variable \(X_j\). The \(X_j\) inherit independence from the \(\tilde{X}_j\). Let \(s_y\) and \(s_y^*\) denote the upper \(\frac{1}{y}{\underline{\text{ th }}}\) quantile of \(S_n=\sum _{j=1}^nX_j\) and \(S^*_n=\sup _{1\le k\le n}S_k\), respectively. We construct a computable quantity \(\underline{Q}_y\) based on the marginal distributions of \(X_1,\ldots ,X_n\) to produce upper and lower bounds for \(s_y\) and \(s_y^*\). We prove that for \(y\ge 8\)

$$\begin{aligned} 6^{-1} \gamma _{3y/16}\underline{Q}_{3y/16}\le s^*_{y}\le \underline{Q}_y \end{aligned}$$

where

$$\begin{aligned} \gamma _y=\frac{1}{2w_y+1} \end{aligned}$$

and \(w_y\) is the unique solution of

$$\begin{aligned} \Big (\frac{w_y}{e\ln (\frac{y}{y-2})}\Big )^{w_y}=2y-4 \end{aligned}$$

for \(w_y>\ln (\frac{y}{y-2})\), and for \(y\ge 37\)

$$\begin{aligned} \frac{1}{9}\gamma _{u(y)}\underline{Q}_{u(y)}<s_y \le \underline{Q}_y \end{aligned}$$

where

$$\begin{aligned} u(y)=\frac{3y}{32} \left( 1+\sqrt{1-\frac{64}{3y}}\right) . \end{aligned}$$

The distribution of \(S_n\) is approximately centered around zero in that \(P(S_n\ge 0) \ge \frac{1}{18}\) and \(P(S_n\le 0)\ge \frac{1}{65}\). The results extend to \(n=\infty \) if and only if for some (hence all) \(a>0\)

$$\begin{aligned} \sum _{j=1}^{\infty }E\{(\tilde{X}_j-m_j)^2\wedge a^2\}<\infty . \end{aligned}$$

(1)

     

Difference operators for partitions under the Littlewood decomposition

Let n be a positive integer. Let \(\delta _3(n)\) denote the difference between the number of (positive) divisors of n congruent to 1 modulo 3 and the number of those congruent to 2 modulo 3. In 2004, Farkas proved that the arithmetic convolution sum

$$\begin{aligned} D_3(n):=\sum _{j=1}^{n-1}\delta _3(j)\delta _3(n-j) \end{aligned}$$

satisfies the relation

$$\begin{aligned} 3D_3(n)+\delta _3(n)={\sum _{\mathop {_{d \mid n}}\limits _{3 \not \mid d}}}d. \end{aligned}$$

In this paper, we use a result about binary quadratic forms to prove a general arithmetic convolution identity which contains Farkas’ formula and two other similar known formulas as special cases. From our identity, we deduce a number of analogous new convolution formulas.

     

Eldan’s Stochastic Localization and Tubular Neighborhoods of Complex-Analytic Sets

Let \(f: \mathbb {C}^n \rightarrow \mathbb {C}^k\) be a holomorphic function and set \(Z = f^{-1}(0)\). Assume that Z is non-empty. We prove that for any \(r > 0\),

$$\begin{aligned} \gamma _n(Z + r) \ge \gamma _n(E + r), \end{aligned}$$

where \(Z + r\) is the Euclidean r-neighborhood of Z; \(\gamma _n\) is the standard Gaussian measure in \(\mathbb {C}^n\), and \(E \subseteq \mathbb {C}^n\) is an \((n-k)\)-dimensional, affine, complex subspace whose distance from the origin is the same as the distance of Z from the origin.