Two-step estimation of ergodic Lévy driven SDE

We consider high frequency samples from ergodic Lévy driven stochastic differential equation with drift coefficient \(a(x,\alpha )\) and scale coefficient \(c(x,\gamma )\) involving unknown parameters \(\alpha \) and \(\gamma \). We suppose that the Lévy measure \(\nu _{0}\), has all order moments but is not fully specified. We will prove the joint asymptotic normality of some estimators of \(\alpha \), \(\gamma \) and a class of functional parameter \(\int \varphi (z)\nu _0(dz)\), which are constructed in a two-step manner: first, we use the Gaussian quasi-likelihood for estimation of \((\alpha ,\gamma )\); and then, for estimating \(\int \varphi (z)\nu _0(dz)\) we make use of the method of moments based on the Euler-type residual with the the previously obtained quasi-likelihood estimator.     

Complete Continuity of Eigen-Pairs of Weighted Dirichlet Eigenvalue Problem

In this paper, we will study the dependence of eigen-pairs \((\lambda _k(\rho ), \varphi _k(x,\rho ))\) of weighted Dirichlet eigenvalue problem on weights \(\rho \). It will be shown that \(\lambda _k(\rho )\) and \(\varphi _k(x,\rho )\) are completely continuous (CC) in \(\rho \). Precisely, when \(\rho _n\) is weakly convergent to \(\rho \) in some Lebesgue space, \(\lambda _k(\rho _n)\) is convergent to \(\lambda _k(\rho )\). As for the convergence of eigenfunctions, since eigenvalues may have multiple eigenfunctions, it will be shown that the distance from \(\varphi _k(x,\rho _n)\) to the eigen space \(V_k(\rho )\) of \(\lambda _k(\rho )\) is tending to zero. As applications, the CC dependence of solutions of linear inhomogeneous equations and the CC dependence of the heat kernels on coefficients will be given.     

The First Passage Time Problem Over a Moving Boundary for Asymptotically Stable Lévy Processes

We study the asymptotic tail behaviour of the first passage time over a moving boundary for asymptotically \(\alpha \)-stable Lévy processes with \(\alpha <1\). Our main result states that if the left tail of the Lévy measure is regularly varying with index \(- \alpha \), and the moving boundary is equal to \(1 - t^{\gamma }\) for some \(\gamma <1/\alpha \), then the probability that the process stays below the moving boundary has the same asymptotic polynomial order as in the case of a constant boundary. The same is true for the increasing boundary \(1 + t^{\gamma }\) with \(\gamma <1/\alpha \) under the assumption of a regularly varying right tail with index \(-\alpha \).     

Singular Values Distribution of Squares of Elliptic Random Matrices and Type B Narayana Polynomials

We consider Gaussian elliptic random matrices X of a size \(N \times N\) with parameter \(\rho \), i.e., matrices whose pairs of entries \((X_{ij}, X_{ji})\) are mutually independent Gaussian vectors with \(\mathbb {E}\,X_{ij} = 0\), \(\mathbb {E}\,X^2_{ij} = 1\) and \(\mathbb {E}\,X_{ij} X_{ji} = \rho \). We are interested in the asymptotic distribution of eigenvalues of the matrix \(W =\frac{1}{N^2} X^2 X^{*2}\). We show that this distribution is determined by its moments, and we provide a recurrence relation for these moments. We prove that the (symmetrized) asymptotic distribution is determined by its free cumulants, which are Narayana polynomials of type B:

$$\begin{aligned} c_{2n} = \sum _{k=0}^n {\left( {\begin{array}{c}n\\ k\end{array}}\right) }^2 \rho ^{2k}. \end{aligned}$$

     

Reducibility of operator semigroups and values of vector states

Let \(\mathcal S\) be a multiplicative semigroup of bounded linear operators on a complex Hilbert space \(\mathcal H\), and let \(\Omega \) be the range of a vector state on \(\mathcal S\) so that \(\Omega = \{ \langle S \xi , \xi \rangle \,{:}\,S \in \mathcal S\}\) for some fixed unit vector \(\xi \in \mathcal H\). We study the structure of sets \(\Omega \) of cardinality two coming from irreducible semigroups \(\mathcal S\). This leads us to sufficient conditions for reducibility and, in some cases, for the existence of common fixed points for \(\mathcal S\). This is made possible by a thorough investigation of the structure of maximal families \(\mathcal F\) of unit vectors in \(\mathcal H\) with the property that there exists a fixed constant \(\rho \in \mathbb C\) for which \(\langle x, y \rangle = \rho \) for all distinct pairs x and y in \(\mathcal F\).     

On the structure of modules of vector-valued modular forms

If \(\rho \) denotes a finite-dimensional complex representation of \(\mathbf {SL}_{2}(\mathbf {Z})\), then it is known that the module \(M(\rho )\) of vector-valued modular forms for \(\rho \) is free and of finite rank over the ring M of scalar modular forms of level one. This paper initiates a general study of the structure of \(M(\rho )\). Among our results are absolute upper and lower bounds, depending only on the dimension of \(\rho \), on the weights of generators for \(M(\rho )\), as well as upper bounds on the multiplicities of weights of generators of \(M(\rho )\). We provide evidence, both computational and theoretical, that a stronger three-term multiplicity bound might hold. An important step in establishing the multiplicity bounds is to show that there exists a free basis for \(M(\rho )\) in which the matrix of the modular derivative operator does not contain any copies of the Eisenstein series \(E_6\) of weight six.     

Functional equations for the Stieltjes constants

The Stieltjes constants \(\gamma _k(a)\) appear as the coefficients in the regular part of the Laurent expansion of the Hurwitz zeta function \(\zeta (s,a)\) about \(s=1\). We present the evaluation of \(\gamma _1(a)\) and \(\gamma _2(a)\) at rational arguments, this being of interest to theoretical and computational analytic number theory and elsewhere. We give multiplication formulas for \(\gamma _0(a)\), \(\gamma _1(a)\), and \(\gamma _2(a)\), and point out that these formulas are cases of an addition formula previously presented. We present certain integral evaluations generalizing Gauss’ formula for the digamma function at rational argument. In addition, we give the asymptotic form of \(\gamma _k(a)\) as \(a \rightarrow 0\) as well as a novel technique for evaluating integrals with integrands with \(\ln (-\ln x)\) and rational factors.     

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.     

Couplings of Brownian Motions of Deterministic Distance in Model Spaces of Constant Curvature

We consider the model space \(\mathbb {M}^{n}_{K}\) of constant curvature K and dimension \(n\ge 1\) (Euclidean space for \(K=0\), sphere for \(K>0\) and hyperbolic space for \(K<0\)), and we show that given a function \(\rho :[0,\infty )\rightarrow [0, \infty )\) with \(\rho (0)=\mathrm {dist}(x,y)\) there exists a coadapted coupling (X(t), Y(t)) of Brownian motions on \(\mathbb {M}^{n}_{K}\) starting at (x, y) such that \(\rho (t)=\mathrm {dist}(X(t),Y(t))\) for every \(t\ge 0\) if and only if \(\rho \) is continuous and satisfies for almost every \(t\ge 0\) the differential inequality

$$\begin{aligned} -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) \le \rho '(t)\le -(n-1)\sqrt{K}\tan \left( \tfrac{\sqrt{K}\rho (t)}{2}\right) +\tfrac{2(n-1)\sqrt{K}}{\sin (\sqrt{K}\rho (t))}. \end{aligned}$$

In other words, we characterize all coadapted couplings of Brownian motions on the model space \(\mathbb {M}^{n}_{K}\) for which the distance between the processes is deterministic. In addition, the construction of the coupling is explicit for every choice of \(\rho \) satisfying the above hypotheses.

     

Packing Chromatic Number of Base-3 Sierpiński Graphs

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that there exists a k-vertex coloring of G in which any two vertices receiving color i are at distance at least \(i+1\). Let \(S^n\) be the base-3 Sierpiński graph of dimension n. It is proved that \(\chi _{\rho }(S^1) = 3\), \(\chi _{\rho }(S^2) = 5\), \(\chi _{\rho }(S^3) = \chi _{\rho }(S^4) = 7\), and that \(8\le \chi _\rho (S^n) \le 9\) holds for any \(n\ge 5\).     

Improved Upper Bounds on the Domination Number of Graphs With Minimum Degree at Least Five

An algorithmic upper bound on the domination number \(\gamma \) of graphs in terms of the order n and the minimum degree \(\delta \) is proved. It is demonstrated that the bound improves best previous bounds for any \(5\le \delta \le 50\). In particular, for \(\delta =5\), Xing et al. (Graphs Comb. 22:127–143, 2006) proved that \(\gamma \le 5n/14 < 0.3572 n\). This bound is improved to 0.3440 n. For \(\delta =6\), Clark et al. (Congr. Numer. 132:99–123, 1998) established \(\gamma <0.3377 n\), while Biró et al. (Bull. Inst. Comb. Appl. 64:73–83, 2012) recently improved it to \(\gamma <0.3340 n\). Here the bound is further improved to \(\gamma < 0.3159n\). For \(\delta =7\), the best earlier bound 0.3088n is improved to \(\gamma < 0.2927n\).     

A Characterization Theorem for the $$L^{2}$$-Discrepancy of Integer Points in Dilated Polygons

Let C be a convex d-dimensional body. If \(\rho \) is a large positive number, then the dilated body \(\rho C\) contains \(\rho ^{d}\left| C\right| +\mathcal {O}\left( \rho ^{d-1}\right) \) integer points, where \(\left| C\right| \) denotes the volume of C. The above error estimate \(\mathcal {O}\left( \rho ^{d-1}\right) \) can be improved in several cases. We are interested in the \(L^{2}\)-discrepancy \(D_{C}(\rho )\) of a copy of \(\rho C\) thrown at random in \(\mathbb {R}^{d}\). More precisely, we consider

Open image in new window

where \(\mathbb {T}^{d}=\) \(\mathbb {R}^{d}/\mathbb {Z}^{d}\) is the d-dimensional flat torus and \(SO\left( d\right) \) is the special orthogonal group of real orthogonal matrices of determinant 1. An argument of Kendall shows that \(D_{C}(\rho )\le c\ \rho ^{(d-1)/2}\). If C also satisfies the reverse inequality \(\ D_{C}(\rho )\ge c_{1} \ \rho ^{(d-1)/2}\), we say that C is \(L^{2}\) -regular. Parnovski and Sobolev proved that, if \(d>1\), a d-dimensional unit ball is \(L^{2} \)-regular if and only if \(d\not \equiv 1\ ({\text {mod}}4)\). In this paper we characterize the \(L^{2}\)-regular convex polygons. More precisely, we prove that a convex polygon is not \(L^{2}\)-regular if and only if it can be inscribed in a circle and it is symmetric about the centre.

     

Flatness implies smoothness for solutions of the porous medium equation

One of the major problems in the theory of the porous medium equation \(\partial _t\rho =\Delta _x\rho ^m,\,m > 1\), is the regularity of the solutions \(\rho (t,x)\ge 0\) and the free boundaries \(\Gamma =\partial \{(t,x): \rho >0\}\). Here we assume flatness of the solution and derive \(C^\infty \) regularity of the interface after a small time, as well as \(C^\infty \) regularity of the solution in the positivity set and up to the free boundary for some time interval. The proof starts from Caffarelli’s blueprint of an improvement of flatness by rescaling, and combines it with the Carleson measure approach applied to the degenerate subelliptic equation satisfied by the pressure of the porous medium equation in transformed coordinates. The improvement of flatness finally hinges on Gaussian estimates for the subelliptic problem. We use these facts to prove the following eventual regularity result: solutions defined in the whole space with compactly supported initial data are smooth after a finite time \(T_r\) that depends on \(\rho _0\). More precisely, we prove that for \(t \ge T_r\) the pressure \(\rho ^{m-1}\) is \(C^\infty \) in the positivity set and up to the free boundary, which is a \(C^\infty \) hypersurface. Moreover, \(T_r\) can be estimated in terms of only the initial mass and the initial support radius. This regularity result eliminates the assumption of non-degeneracy on the initial data that has been carried on for decades in the literature. Let us recall that regularization for small times is false, and that as \(t\rightarrow \infty \) the solution increasingly resembles a Barenblatt function and the support looks like a ball.     

Normalized bound states for the nonlinear Schrödinger equation in bounded domains

Given \(\rho >0\), we study the elliptic problem

$$\begin{aligned} \text {find } (U,\lambda )\in H^1_0(\Omega )\times {\mathbb {R}}\text { such that } {\left\{ \begin{array}{ll} -\Delta U+\lambda U=|U|^{p-1}U\\ \int _{\Omega } U^2\, dx=\rho , \end{array}\right. } \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^N\) is a bounded domain and \(p>1\) is Sobolev-subcritical, searching for conditions (about \(\rho \), N and p) for the existence of solutions. By the Gagliardo-Nirenberg inequality it follows that, when p is \(L^2\)-subcritical, i.e. \(1<p<1+4/N\), the problem admits solutions for every \(\rho >0\). In the \(L^2\)-critical and supercritical case, i.e. when \(1+4/N \le p < 2^*-1\), we show that, for any \(k\in {\mathbb {N}}\), the problem admits solutions having Morse index bounded above by k only if \(\rho \) is sufficiently small. Next we provide existence results for certain ranges of \(\rho \), which can be estimated in terms of the Dirichlet eigenvalues of \(-\Delta \) in \(H^1_0(\Omega )\), extending to changing sign solutions and to general domains some results obtained in Noris et al. in Anal. PDE 7:1807–1838, 2014 for positive solutions in the ball.

     

The Hardy–Schrödinger operator with interior singularity: the remaining cases

We consider the remaining unsettled cases in the problem of existence of energy minimizing solutions for the Dirichlet value problem \(L_\gamma u-\lambda u=\frac{u^{2^*(s)-1}}{|x|^s}\) on a smooth bounded domain \(\Omega \) in \({\mathbb {R}}^n\) (\(n\ge 3\)) having the singularity 0 in its interior. Here \(\gamma <\frac{(n-2)^2}{4}\), \(0\le s <2\), \(2^*(s):=\frac{2(n-s)}{n-2}\) and \(0\le \lambda <\lambda _1(L_\gamma )\), the latter being the first eigenvalue of the Hardy–Schrödinger operator \(L_\gamma :=-\Delta -\frac{\gamma }{|x|^2}\). There is a threshold \(\lambda ^*(\gamma , \Omega ) \ge 0\) beyond which the minimal energy is achieved, but below which, it is not. It is well known that \(\lambda ^*(\Omega )=0\) in higher dimensions, for example if \(0\le \gamma \le \frac{(n-2)^2}{4}-1\). Our main objective in this paper is to show that this threshold is strictly positive in “lower dimensions” such as when \( \frac{(n-2)^2}{4}-1<\gamma <\frac{(n-2)^2}{4}\), to identify the critical dimensions (i.e., when the situation changes), and to characterize it in terms of \(\Omega \) and \(\gamma \). If either \(s>0\) or if \(\gamma > 0\), i.e., in the truly singular case, we show that in low dimensions, a solution is guaranteed by the positivity of the “Hardy-singular internal mass” of \(\Omega \), a notion that we introduce herein. On the other hand, and just like the case when \(\gamma =s=0\) studied by Brezis and Nirenberg (Commun Pure Appl Math 36:437–477, 1983) and completed by Druet (Ann Inst H Poincaré Anal Non Linéaire 19(2):125–142, 2002), \(n=3\) is the critical dimension, and the classical positive mass theorem is sufficient for the merely singular case, that is when \(s=0\), \(\gamma \le 0\).     

Approximation by modified Szász-Durrmeyer operators

The main goal of this paper is to introduce Durrmeyer modifications for the generalized Szász–Mirakyan operators defined in (Aral et al., in Results Math 65:441–452, 2014). The construction of the new operators is based on a function \(\rho \) which is continuously differentiable \(\infty \) times on \( \left[ 0,\infty \right) ,\) such that \(\rho \left( 0\right) =0\) and \( \inf _{x\in \left[ 0,\infty \right) }\rho ^{\prime }\left( x\right) \ge 1.\) Involving the weighted modulus of continuity constructed using the function \( \rho \), approximation properties of the operators are explored: uniform convergence over unbounded intervals is established and a quantitative Voronovskaya theorem is given. Moreover, we obtain direct approximation properties of the operators in terms of the moduli of smoothness. Our results show that the new operators are sensitive to the rate of convergence to f,  depending on the selection of \(\rho .\) For the particular case \(\rho \left( x\right) =x\), the previous results for classical Szász-Durrmeyer operators are captured.     

On the Diophantine equation $$X^{2N}+2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5$$

We prove that for each prime p, positive integer \(\alpha \), and non-negative integers \(\beta \) and \(\gamma \), the Diophantine equation \(X^{2N} + 2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5\) has no solution with N, X, \(Z\in \mathbb {Z}^+\), \(N > 1\), and \(\gcd (X,Z) = 1\).     

Scalar conservation laws with monotone pure-jump Markov initial conditions

In 2010 Menon and Srinivasan published a conjecture for the statistical structure of solutions \(\rho \) to scalar conservation laws with certain Markov initial conditions, proposing a kinetic equation that should suffice to describe \(\rho (x,t)\) as a stochastic process in x with t fixed. In this article we verify an analogue of the conjecture for initial conditions which are bounded, monotone, and piecewise constant. Our argument uses a particle system representation of \(\rho (x,t)\) over \(0 \le x \le L\) for \(L > 0\), with a suitable random boundary condition at \(x = L\).     

Inscribed and circumscribed polygons that characterize inner product spaces

Let X be a real normed space with unit sphere S. We prove that X is an inner product space if and only if there exists a real number \(\rho =\sqrt{(1+\cos \frac{2k\pi }{2m+1})/2}, (k=1,2,\ldots ,m; m=1,2,\ldots )\), such that every chord of S that supports \(\rho S\) touches \(\rho S\) at its middle point. If this condition holds, then every point \(u\in S\) is a vertex of a regular polygon that is inscribed in S and circumscribed about \(\rho S\).