On the Wave Equation Associated to the Hermite and the Twisted Laplacian

The dispersive properties of the wave equation u _tt +Au=0 are considered, where A is either the Hermite operator −Δ+|x|² or the twisted Laplacian −(∇ _x −iy)²/2−(∇ _y +ix)²/2. In both cases we prove optimal L ¹−L ^∞ dispersive estimates. More generally, we give some partial results concerning the flows exp (itL ^ν ) associated to fractional powers of the twisted Laplacian for 0<ν<1.     

Vector Interpretation of the Matrix Orthogonality on the Real Line

In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials satisfy three-term recurrence relations with matrix coefficients that do not obey to any type of symmetry. In this sense the vectorial reinterpretation allows us to study a non-symmetric case of the matrix orthogonality. We also prove that our systems of polynomials are indeed orthonormal with respect to a complex measure of orthogonality. Approximation problems of Hermite-Padé type are also discussed. Finally, a Markov’s type theorem is presented.     

The Poincaré series of divisorial valuations in the plane defines the topology of the set of divisors

With a plane curve singularity one associates a multi-index filtration on the ring of germs of functions of two variables defined by the orders of a function on irreducible components of the curve. The Poincaré series of this filtration turns out to coincide with the Alexander polynomial of the curve germ. For a finite set of divisorial valuations on the ring corresponding to some components of the exceptional divisor of a modification of the plane, in a previous paper there was obtained a formula for the Poincaré series of the corresponding multi-index filtration similar to the one associated with plane germs. Here we show that the Poincaré series of a set of divisorial valuations on the ring of germs of functions of two variables defines “the topology of the set of the divisors” in the sense that it defines the minimal resolution of this set up to combinatorial equivalence. For the plane curve singularity case, we also give a somewhat simpler proof of the statement by Yamamoto which shows that the Alexander polynomial is equivalent to the embedded topology.     

Remarks on the Gagliardo-Nirenberg Type Inequality in the Besov and the Triebel-Lizorkin Spaces in the Limiting Case

We consider the generalized Gagliardo-Nirenberg inequality in $\Bbb{R}^{n}$ including homogeneous Besov space $\dot{B}^{s}_{r,\rho}(\Bbb{R}^{n})$ with the critical order s=n/r, which describes the continuous embedding such as $L^{p}(\Bbb{R}^{n})\cap\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})\subset L^{q}(\Bbb{R}^{n})$ for all q with p ≦ q<∞, where 1≦ p ≦ r<∞ and 1<ρ ≦∞. Indeed, the following inequality holds: $$\|u\|_{L^{q}(\Bbb{R}^{n})}\leqq C\,q^{1-1/\rho}\|u\|_{L^{p}(\Bbb{R}^{n})}^{p/q}\|u\|_{\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})}^{1-p/q},$$ where C is a constant depending only on r. In this inequality, we have the exact order 1?1/ρ of divergence to the power q tending to the infinity. Furthermore, as a corollary of this inequality, we obtain the Gagliardo-Nirenberg inequality with the homogeneous Triebel-Lizorkin space $\dot{F}^{n/r}_{r,\rho}(\Bbb{R}^{n})$ , which implies the usual Sobolev imbedding with the critical Sobolev space $\dot{H}^{n/r}_{r}(\Bbb{R}^{n})$ . Moreover, as another corollary, we shall prove the Trudinger-Moser type inequality in $\dot{B}^{n/r}_{r,\rho}(\Bbb{R}^{n})$ .     

A presentation for the singular part of the full transformation semigroup

The (full) transformation semigroup T_n\mathcal{T}_{n} is the semigroup of all functions from the finite set {1,…,n} to itself, under the operation of composition. The symmetric group S_n í T_n{\mathcal{S}_{n}\subseteq \mathcal{T}_{n}} is the group of all permutations on {1,…,n} and is the group of units of T_n\mathcal{T}_{n}. The complement T_n\S_n\mathcal{T}_{n}\setminus \mathcal{S}_{n} is a subsemigroup (indeed an ideal) of T_n\mathcal{T}_{n}. In this article we give a presentation, in terms of generators and relations, for T_n\S_n\mathcal{T}_{n}\setminus \mathcal{S}_{n}, the so-called singular part of T_n\mathcal{T}_{n}.     

Note on the Pair-crossing Number and the Odd-crossing Number

The crossing number of a graph G is the minimum possible number of edge-crossings in a drawing of G, the pair-crossing number is the minimum possible number of crossing pairs of edges in a drawing of G, and the odd-crossing number is the minimum number of pairs of edges that cross an odd number of times. Clearly, . We construct graphs with . This improves the bound of Pelsmajer, Schaefer and Štefankovič. Our construction also answers an old question of Tutte. Slightly improving the bound of Valtr, we also show that if the pair-crossing number of G is k, then its crossing number is at most O(k ²/log ² k). G. Tóth’s research was supported by the Hungarian Research Fund grant OTKA-K-60427 and the Research Foundation of the City University of New York.     

Initialization of the Shooting Method via the Hamilton-Jacobi-Bellman Approach

The aim of this paper is to investigate from the numerical point of view the coupling of the Hamilton-Jacobi-Bellman (HJB) equation and the Pontryagin minimum principle (PMP) to solve some control problems. A rough approximation of the value function computed by the HJB method is used to obtain an initial guess for the PMP method. The advantage of our approach over other initialization techniques (such as continuation or direct methods) is to provide an initial guess close to the global minimum. Numerical tests involving multiple minima, discontinuous control, singular arcs and state constraints are considered.     

Continuous dependence on the boundary conditions for the Wentzell Laplacian

The solution u of the well-posed problem

Maximum Principle of Optimal Control of the Primitive Equations of the Ocean with Two Point Boundary State Constraint

We consider a class of stochastic impulse control problems of general stochastic processes i.e. not necessarily Markovian. Under fairly general conditions we establish existence of an optimal impulse control. We also prove existence of combined optimal stochastic and impulse control of a fairly general class of diffusions with random coefficients. Unlike, in the Markovian framework, we cannot apply quasi-variational inequalities techniques. We rather derive the main results using techniques involving reflected BSDEs and the Snell envelope.     

How to get the timing right. A computational model of the effects of the timing of contacts on team cohesion in demographically diverse teams

Lau and Murnighan’s faultline theory explains negative effects of demographic diversity on team performance as consequence of strong demographic faultlines. If demographic differences between group members are correlated across various dimensions, the team is likely to show a “subgroup split” that inhibits communication and effective collaboration between team members. Our paper proposes a rigorous formal and computational reconstruction of the theory. Our model integrates four elementary mechanisms of social interaction, homophily, heterophobia, social influence and rejection into a computational representation of the dynamics of both opinions and social relations in the team. Computational experiments demonstrate that the central claims of faultline theory are consistent with the model. We show furthermore that the model highlights a new structural condition that may give managers a handle to temper the negative effects of strong demographic faultlines. We call this condition the timing of contacts. Computational analyses reveal that negative effects of strong faultlines critically depend on who is when brought in contact with whom in the process of social interactions in the team. More specifically, we demonstrate that faultlines have hardly negative effects when teams are initially split into demographically homogeneous subteams that are merged only when a local consensus has developed.     

Pointwise Estimates for Laplace Equation. Applications to the Free Boundary of the Obstacle Problem with Dini Coefficients

In this paper we are interested in pointwise regularity of solutions to elliptic equations. In a first result, we prove that if the modulus of mean oscillation of Δu at the origin is Dini (in L ^p average), then the origin is a Lebesgue point of continuity (still in L ^p average) for the second derivatives D ² u. We extend this pointwise regularity result to the obstacle problem for the Laplace equation with Dini right hand side at the origin. Under these assumptions, we prove that the solution to the obstacle problem has a Taylor expansion up to the order 2 (in the L ^p average). Moreover we get a quantitative estimate of the error in this Taylor expansion for regular points of the free boundary. In the case where the right hand side is moreover double Dini at the origin, we also get a quantitative estimate of the error for singular points of the free boundary. Our method of proof is based on some decay estimates obtained by contradiction, using blow-up arguments and Liouville Theorems. In the case of singular points, our method uses moreover a refined monotonicity formula.      

Perturbation resilience for the facility location problem

We analyze a simple local search heuristic for the facility location problem using the notion of perturbation resilience: an instance is $\gamma$-perturbation resilient if all costs can be perturbed by a factor of $\gamma$ without changing the optimal solution.We prove that local search for FLP succeeds in finding the optimal solution for $\gamma$-perturbation resilient instances for $\gamma \ge 3$, and we show that this is tight.     

Weak Continuity of the Flow Map for the Benjamin-Ono Equation on the Line

In this paper we show that the flow map of the Benjamin-Ono equation on the line is weakly continuous in L ²(?), using “local smoothing” estimates. L ²(?) is believed to be a borderline space for the local well-posedness theory of this equation. In the periodic case, Molinet (Math. Ann. 337, 353–383, 2007) has recently proved that the flow map of the Benjamin-Ono equation is not weakly continuous in $L^{2}(\mathbb{T})In this paper we show that the flow map of the Benjamin-Ono equation on the line is weakly continuous in L ²(ℝ), using “local smoothing” estimates. L ²(ℝ) is believed to be a borderline space for the local well-posedness theory of this equation. In the periodic case, Molinet (Math. Ann. 337, 353–383, 2007) has recently proved that the flow map of the Benjamin-Ono equation is not weakly continuous in L²(\mathbbT)L^{2}(\mathbb{T}). Our results are in line with previous work on the cubic nonlinear Schr?dinger equation, where Goubet and Molinet (Nonlinear Anal. 71, 317–320, 2009) showed weak continuity in L ²(ℝ) and Molinet (Am. J. Math. 130, 635–683, 2008) showed lack of weak continuity in L²(\mathbbT)L^{2}(\mathbb{T}).     

The Method of Virtual Components in the Multivariate Setting

We describe the so-called method of virtual components for tight wavelet framelets to increase their approximation order and vanishing moments in the multivariate setting. Two examples of the virtual components for tight wavelet frames based on bivariate box splines on three or four direction mesh are given. As a byproduct, a new construction of tight wavelet frames based on box splines under the quincunx dilation matrix is presented.     

Quantization of Pseudo-differential Operators on the Torus

Pseudo-differential and Fourier series operators on the torus \mathbbTⁿ=(\BbbR/2p\BbbZ)ⁿ{{\mathbb{T}}^{n}}=(\Bbb{R}/2\pi\Bbb{Z})^{n} are analyzed by using global representations by Fourier series instead of local representations in coordinate charts. Toroidal symbols are investigated and the correspondence between toroidal and Euclidean symbols of pseudo-differential operators is established. Periodization of operators and hyperbolic partial differential equations is discussed. Fourier series operators, which are analogues of Fourier integral operators on the torus, are introduced, and formulae for their compositions with pseudo-differential operators are derived. It is shown that pseudo-differential and Fourier series operators are bounded on L ² under certain conditions on their phases and amplitudes.     

Maxima and Minima of the Hosoya Index and the Merrifield-Simmons Index

The Hosoya index and the Merrifield-Simmons index are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In recent years, quite a lot of work has been done on the extremal problem for these two indices, i.e., the problem of determining the graphs within certain prescribed classes that maximize or minimize the index value. This survey collects and classifies these results, and also provides some useful auxiliary results, tools and techniques that are frequently used in the study of this type of problem.     

Regularity of the Schrödinger equation for the harmonic oscillator

We consider the Schrödinger equation for the harmonic oscillator i ? _t u=Hu, where H=?Δ+|x|², with initial data in the Hermite–Sobolev space H ^?s/2 L ²(? ⁿ ). We obtain smoothing and maximal estimates and apply these to perturbations of the equation and almost everywhere convergence problems.     

Pointwise Estimates for the Bergman Kernel of the Weighted Fock Space

We prove upper pointwise estimates for the Bergman kernel of the weighted Fock space of entire functions in L ²(e ^−2φ ) where φ is a subharmonic function with Δφ a doubling measure. We derive estimates for the canonical solution operator to the inhomogeneous Cauchy-Riemann equation and we characterize the compactness of this operator in terms of Δφ.     

Revisit of the Theorem of Image Trajectories in the Earth-Moon Space

On the occasion of the 50th anniversary of the theorem of the image trajectories in the Earth-Moon space, the author revisits the theorem and clarifies the relation between the class of image trajectories and the class of symmetric free-return trajectories, which were employed in the Apollo program. In a nutshell, the symmetric free-return trajectories are those image trajectories that intersect the Earth-Moon axis orthogonally at some point above the far side of the Moon. Optimization implications are pointed out.