Coupling and harmonic functions in the case of continuous time Markov processes

Consider two transient Markov processes (X^v_t)_tεR·, (X^μ_t)_tεR· with the same transition semigroup and initial distributions v and μ. The probability spaces supporting the processes each are also assumed to support an exponentially distributed random variable independent of the process.

We show that there exist (randomized) stopping times S for (X^v_t), T for (X^μ_t) with common final distribution, L(X^v_S|S < ∞) = L(X^μ_T|T < ∞), and the property that for t < S, resp. t < T, the processes move in disjoint portions of the state space. For such a coupling (S, T) it is shown

where denotes the bounded harmonic functions of the Markov transition semigroup. Extensions, consequences and applications of this result are discussed.     

Asymptotic L-based comparison of Tikhonov regularization and Landweber iteration

The distance between the Tikhonov and Landweber regularized solutions of a linear inverse problem is partly controlled by the L^∞-norm of the difference in their corresponding singular value filters. For large Landweber iteration number, the regularization parameter of the closest Tikhonov filter to a given Landweber filter is determined. This asymptotically computed parameter compares well with the numerically computed value even for moderate sized iteration number. A consequence of the analysis is to determine the range of singular values to which the difference in regularized solutions is most sensitive.     

The thermal equilibrium state of semiconductor devices

The thermal equilibrium state of two oppositely charged gases confined to a bounded domain , m = 1,2 or m = 3, is entirely described by the gases' particle densities p, n minimizing the total energy (p, n). it is shown that for given P, N > 0 the energy functional admits a unique minimizer in {(p, n) ε L²(Ω) x L ²(Ω) : p, n ≥ 0, _Ωp = P, _Ωn = N} and that p, n ε C(Ω) ∩ L^∞(Ω).

The analysis is applied to the hydrodynamic semiconductor device equations. These equations in general possess more than one thermal equilibrium solution, but only the unique solution of the corresponding variational problem minimizes the total energy. It is equivalent to prescribe boundary data for electrostatic potential and particle densities satisfying the usual compatibility relations and to prescribe V^e and P, N for the variational problem.     

A dense planar point set from iterated line intersections

Given S₁, a starting set of points in the plane, not all on a line, we define a sequence of planar point sets {S_i}_i=1^∞ as follows. With S_i already determined, let L_i be the set of all the lines determined by pairs of points from S_i, and let S_i+1 be the set of all the intersection points of lines in L_i. We show that with the exception of some very particular starting configurations, the limiting point set _i=1^∞S_i is everywhere dense in the plane.     

L2-approximation of real-valued functions with interpolatory constraints

We construct the polynomial p_m,n^* of degree m which interpolates a given real-valued function f L²[a, b] at pre-assigned n distinct nodes and is the best approximant to f in the L₂-sense over all polynomials of degree m with the same interpolatory character. It is shown that the L₂-error p_m,n^* − f → 0 as m → ∞ if f C[a, b].     

H least-squares method for the velocity–pressure–stress formulation of Stokes equations

This paper studies the discrete H⁻¹-norm least-squares method for the incompressible Stokes equations based on the velocity–pressure–stress formulation by the least-squares functional defined as the sum of L²-norms and H⁻¹-norm of the residual equations. Some computational experiments by multigrid method and preconditioning conjugate gradient method (PCGM) on this method are shown by taking efficient and β in the discrete solution operator T_h=h²I+βB_h corresponding to the minus one norm. We also propose a new method and compare it with PCGM and multigrid method through the analysis of numerical experiments depending on the choice of β.     

Stable limits of empirical processes of moving averages with infinite variance

The paper obtains a functional limit theorem for the empirical process of a stationary moving average process X_t with i.i.d. innovations belonging to the domain of attraction of a symmetric -stable law, 1<<2, with weights b_j decaying as j^−β, 1<β<2/. We show that the empirical process (normalized by N^1/β) weakly converges, as the sample size N increases, to the process c_x⁺L⁺+c_x⁻L⁻, where L⁺,L⁻ are independent totally skewed β-stable random variables, and c_x⁺,c_x⁻ are some deterministic functions. We also show that, for any bounded function H, the weak limit of suitably normalized partial sums of H(X_s) is an β-stable Lévy process with independent increments. This limiting behavior is quite different from the behavior of the corresponding empirical processes in the parameter regions 1/<β<1 and 2/<β studied in Koul and Surgailis (Stochastic Process. Appl. 91 (2001) 309) and Hsing (Ann. Probab. 27 (1999) 1579), respectively.     

MEROMORPHIC FUNCTIONS SHARING TWO FINITE SETS

Let S1 ＝ {∞} and S2 ＝ {w: Ps(w)＝ 0}, Ps(w) being a uniqueness polynomial under some restricted conditions. Then, for any given nonconstant meromorphic function f, there exist at most finitely many nonconstant meromorphic functions g such that f-1(Si) ＝ g-1(Si)(i ＝ 1,2), where f-1(Si) and g-1(Si) denote the pull-backs of Si considered as a divisor, namely, the inverse images of Si counted with multiplicities, by f and g respectively.     

More on θ-compact fuzzy topological spaces

Recently, El-Naschie has shown that the notion of fuzzy topology may be relevant to quantum particle physics in connection with string theory and ε^∞ theory. In 2005, Caldas and Jafari have introduced θ-compact fuzzy topological spaces. The purpose of this paper is to investigate further properties of θ-compact fuzzy topological spaces. Moreover, the notion of θ-closed fuzzy topological spaces is introduced and properties of it are obtained.     

Karp complexity and classes with the independence property

A class K of structures is controlled if for all cardinals λ, the relation of L_∞,λ-equivalence partitions K into a set of equivalence classes (as opposed to a proper class). We prove that no pseudo-elementary class with the independence property is controlled. By contrast, there is a pseudo-elementary class with the strict order property that is controlled (see Arch. Math. Logic 40 (2001) 69–88).     

H∞ estimation for fuzzy membership function optimization

Given a fuzzy logic system, how can we determine the membership functions that will result in the best performance? If we constrain the membership functions to a specific shape (e.g., triangles or trapezoids) then each membership function can be parameterized by a few variables and the membership optimization problem can be reduced to a parameter optimization problem. The parameter optimization problem can then be formulated as a nonlinear filtering problem. In this paper we solve the nonlinear filtering problem using H_∞ state estimation theory. However, the membership functions that result from this approach are not (in general) sum normal. That is, the membership function values do not add up to one at each point in the domain. We therefore modify the H_∞ filter with the addition of state constraints so that the resulting membership functions are sum normal. Sum normality may be desirable not only for its intuitive appeal but also for computational reasons in the real time implementation of fuzzy logic systems. The methods proposed in this paper are illustrated on a fuzzy automotive cruise controller and compared to Kalman filtering based optimization.     

A C finite element method for an inverse problem

A C⁰ finite element method is presented for an inverse problem in which the coefficient in the differential operator is to be determined from the measurement of the solution of a boundary value problem. The unknown in the inverse problem is approximated by a minimizer of a cost function that includes both the output error and equation error. Error estimates in a weighted H⁻¹ norm and L² are given. Numerical examples are presented to show features of the method.     

Lower bounds on entropy via the Conley index with application to time series

A proof of a localized version of the proven entropy conjecture for C^∞ smooth maps is given. This allows for computational methods for bounding topological entropy through properties of the Conley index. Chaos can then be determined by a non-global index calculation robust under possibly large (and noisy) perturbations. In addition, a proof of a Wazewski's Principle for time series analysis is given which allows for lifting of entropy to the observed dynamical system under certain conditions.     

Moving-average representation of autoregressive approximations

We study the properties of an MA(∞)-representation of an autoregressive approximation for a stationary, real-valued process. In doing so we give an extension of Wiener's theorem in the deterministic approximation setup. When dealing with data, we can use this new key result to obtain insight into the structure of MA(∞)-representations of fitted autoregressive models where the order increases with the sample size. In particular, we give a uniform bound for estimating the moving-average coefficients via autoregressive approximation being uniform over all integers.     

On an optimal stopping problem of Gusein-Zade

We study the problem of selecting one of the r best of n rankable individuals arriving in random order, in which selection must be made with a stopping rule based only on the relative ranks of the successive arrivals. For each r up to r=25, we give the limiting (as n→∞) optimal risk (probability of not selecting one of the r best) and the limiting optimal proportion of individuals to let go by before being willing to stop. (The complete limiting form of the optimal stopping rule is presented for each r up to r=10, and for r=15, 20 and 25.) We show that, for large n and r, the optical risk is approximately (1−t^*)^r, where t^*≈0.2834 is obtained as the roof of a function which is the solution to a certain differential equation. The optimal stopping rule τ_r,n lets approximately t^*n arrivals go by and then stops ‘almost immediately’, in the sense that τ_r,n/n→t^* in probability as n→∞, r→∞     

3-Homogeneous latin trades

Let T be a partial latin square and L be a latin square with TL. We say that T is a latin trade if there exists a partial latin square T^′ with T^′∩T= such that (LT)T^′ is a latin square. A k-homogeneous latin trade is one which intersects each row, each column and each entry either 0 or k times. In this paper, we construct 3-homogeneous latin trades from hexagonal packings of the plane with circles. We show that 3-homogeneous latin trades of size 3 m exist for each m3. This paper discusses existence results for latin trades and provides a glueing construction which is subsequently used to construct all latin trades of finite order greater than three.     

Conservative schemes for the symmetric Regularized Long Wave equations

In this paper, we study the Symmetric Regularized Long Wave (SRLW) equations by finite difference method. We design some numerical schemes which preserve the original conservative properties for the equations. The first scheme is two-level and nonlinear-implicit. Existence of its difference solutions are proved by Brouwer fixed point theorem. It is proved by the discrete energy method that the scheme is uniquely solvable, unconditionally stable and second-order convergent for U in L_∞ norm, and for N in L₂ norm on the basis of the priori estimates. The second scheme is three-level and linear-implicit. Its stability and second-order convergence are proved. Both of the two schemes are conservative so can be used for long time computation. However, they are coupled in computing so need more CPU time. Thus we propose another three-level linear scheme which is not only conservative but also uncoupled in computation, and give the numerical analysis on it. Numerical experiments demonstrate that the schemes are accurate and efficient.     

On an explicit formula for the distribution of the supremum

Let M_∞ be the supremum of a random walk drifting to -∞ which is generated by the partial sums of a sequence of independent identically distributed random variables with a common distribution F. We prove that the moment generating function E exp(sM_∞) is a rational function if and only if the function ∫₀^∞ exp(sx)F(dx) is rational.     

Least-squares Galerkin procedures for parabolic integro-differential equations

Two least-squares Galerkin finite element schemes are formulated to solve parabolic integro-differential equations. The advantage of this method is that it is not subject to the LBB condition. The convergence analysis shows that the least-squares mixed element schemes yield the approximate solution with optimal accuracy in H(div;Ω)×H¹(Ω) and (L²(Ω))²×L²(Ω), respectively.     

Proof of a conjecture of Fiedler and Markham

Let A be an n×n nonsingular M-matrix. For the Hadamard product AA⁻¹, M. Fiedler and T.L. Markham conjectured in [Linear Algebra Appl. 10l (1988) 1] that q(AA⁻¹)2/n, where q(AA⁻¹) is the smallest eigenvalue (in modulus) of AA⁻¹. We considered this conjecture in [Linear Algebra Appl. 288 (1999) 259] having observed an incorrect proof in [Linear Algebra Appl. 144 (1991) 171] and obtained that q(AA⁻¹)(2/n)(n−1)/n. The present paper gives a proof for this conjecture.