Large Deviations for Two-Time-Scale Diffusions, with Delays

We consider the problem of large deviations for a two-time-scale reflected diffusion process, possibly with delays in the dynamical terms. The Dupuis-Ellis weak convergence approach is used. It is perhaps the most intuitive and simplest for the problems of concern. The results have applications to the problem of approximating optimal controls for two-time-scale systems via use of the averaged equation.     

Limit Property for Regular and Weak Generalized Convolution

We denote by ? \((\mathcal{P_{+}})\) the set of all probability measures defined on the Borel subsets of the real line (the positive half-line [0,∞)). K. Urbanik defined the generalized convolution as a commutative and associative ?₊-valued binary operation ? on ? ₊ ² which is continuous in each variable separately. This convolution is distributive with respect to convex combinations and scale changes T _a (a>0) with δ ₀ as the unit element. The key axiom of a generalized convolution is the following: there exist norming constants c _n and a measure ν other than δ ₀ such that \(T_{c_{n}}\delta_{1}^{\bullet n}\to\nu\).In Sect. 2 we discuss basic properties of the generalized convolution on ? which hold for the convolutions without the key axiom. This rather technical discussion is important for the weak generalized convolution where the key axiom is not a natural assumption. In Sect. 4 we show that if the weak generalized convolution defined by a weakly stable measure μ has this property, then μ is a factor of strictly stable distribution.     

Fluctuation Identities Applied to the Hitting Time of a Half-line in the Plane

We study asymptotic estimates that contain the hitting time and the hitting place of a half-line by a two-dimensional random walk. Fluctuation identities are used without resorting to pair annihilation but by interchanging summation and integration. The same method applies to the hitting of the half-space by a one-dimensional random walk. The author was supported by a grant (no. 18740053) of Japan Society for the Promotion of Science.     

Star-Configurations in ℙ2 Having Generic Hilbert Function and the Weak Lefschetz Property

We prove that a star-configuration 𝕏 in ?² is defined by general forms of degrees ≤2 if and only if 𝕏 has generic Hilbert function. We also show that if 𝕏 and 𝕐 are star-configurations in ?² defined by general forms of degrees ≤2 and σ(𝕏) ≠ σ(𝕐), then the ring R/(I _𝕏 + I _𝕐) has the Weak Lefschetz property. These two results generalize results of Ahn and Shin [3 Ahn , J. , Shin , Y. S. ( 2012 ). The minimal free resolution of a star-configuration in ? ⁿ and the weak lefschetz property . J. Korean Math. Soc. 49 ( 2 ): 405 – 417 . [Google Scholar]]. Furthermore, we find the Lefschetz element of the graded Artinian ring R/(I _𝕏 + I _?) precisely when 𝕏 and ? are two star-configurations in ?² defined by general forms F ₁,…, F _s , and G ₁,…, G _s , L, respectively, with deg F _i  = deg G _i  = 2 for every i ≥ 1, and deg L = 1 with s ≥ 3.     

On the Winding of Stationary Diffusions

In this paper, the winding of stationary diffusions are studied. It is shown that, differently from Brownian motions,stationary diffusions wind up to time t angles in order of t, while Brownian motions wind up angles in order of logt,and the winding rates tend to , where c, an-i c2 are constants and is a Cauchy random variable.Moreover, it is proved that a stationary diffusion is reversible iff c2=0 for the winding around every point.     

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}.     

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.     

A hybrid simulation-optimization algorithm for the Hamiltonian cycle problem

In this paper, we propose a new hybrid algorithm for the Hamiltonian cycle problem by synthesizing the Cross Entropy method and Markov decision processes. In particular, this new algorithm assigns a random length to each arc and alters the Hamiltonian cycle problem to the travelling salesman problem. Thus, there is now a probability corresponding to each arc that denotes the probability of the event “this arc is located on the shortest tour.” Those probabilities are then updated as in cross entropy method and used to set a suitable linear programming model. If the solution of the latter yields any tour, the graph is Hamiltonian. Numerical results reveal that when the size of graph is small, say less than 50 nodes, there is a high chance the algorithm will be terminated in its cross entropy component by simply generating a Hamiltonian cycle, randomly. However, for larger graphs, in most of the tests the algorithm terminated in its optimization component (by solving the proposed linear program).     

Continuous dependence on the boundary conditions for the Wentzell Laplacian

The solution u of the well-posed problem

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.     

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.     

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}).     

Property(ω) and the Single-valued Extension Property

By the new spectrum originated from the single-valued extension property, we give the necessary and sufficient conditions for a bounded linear operator defined on a Banach space for which property(ω) holds. Meanwhile, the relationship between hypercyclic property(or supercyclic property)and property(ω) is discussed.     

Critical Landscape Topology for Optimization on the Symplectic Group

Optimization problems over compact Lie groups have been studied extensively due to their broad applications in linear programming and optimal control. This paper analyzes an optimization problem over a noncompact symplectic Lie group Sp(2N,ℝ), i.e., minimizing the Frobenius distance from a target symplectic transformation, which can be used to assess the fidelity function over dynamical transformations in classical mechanics and quantum optics. The topology of the set of critical points is proven to have a unique local minimum and a number of saddlepoint submanifolds, exhibiting the absence of local suboptima that may hinder the search for ultimate optimal solutions. Compared with those of previously studied problems on compact Lie groups, such as the orthogonal and unitary groups, the topology is more complicated due to the significant nonlinearity brought by the incompatibility of the Frobenius norm with the pseudo-Riemannian structure on the symplectic group.     

A comparison of problem decomposition techniques for the FAP

This paper proposes a problem decomposition approach to solve hard Frequency Assignment Problem instances with standard meta-heuristics. The proposed technique aims to divide the initial problem into a number of easier subproblems, solve them and then recompose the partial solutions into one of the original problem. We consider the COST-259 MI-FAP instances and other Cardiff University test problems in order to simulate larger and more realistic networks. For both benchmarks the standard implementations of meta-heuristics do not generally produce a satisfactory performance within reasonable times of execution. However, the decomposed assignment approach can improve their results, both in terms of solution quality and runtime.     

Moderate and Large Deviation Estimate for the Markov-Binomial Distribution

Considering the Markov-binomial distribution, we obtain the moderate deviation and large deviation estimates for the number of successes S _n and the number of experiments Y(r) until the rth success.     

On the Higher-Order Method for the Solution of a Nonlinear Scalar Equation

In this paper, a higher-order method for the solution of a nonlinear scalar equation is presented. It is proved that the new method is locally convergent with an order of (m+2), where m is the highest order derivative used in the iterative formula. Some numerical examples are used to demonstrate the new method.