On uniqueness problems related to the Fokker–Planck–Kolmogorov equation for measures

We survey recent results related to uniqueness problems for parabolic equations for measures. We consider equations of the form ∂ _t μ = L ^* μ for bounded Borel measures on ℝ ^d  × (0, T), where L is a second order elliptic operator, for example, Lu = D_xu + ( b,?_xu ) Lu = {\Delta_x}u + \left( {b,{\nabla_x}u} \right) , and the equation is understood as the identity

On the Lyapunov Exponent of a Multidimensional Stochastic Flow

Let X _t be a reversible and positive recurrent diffusion in ℝ^d described by

Properties of Riesz fractional derivatives for Korteweg–de Vries solitons

Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (?Δ) ^α/2 for ${\alpha \in \mathbb{R}}Riesz fractional derivatives are defined as fractional powers of the Laplacian, D ^α  = (−Δ) ^α/2 for a ? \mathbbR{\alpha \in \mathbb{R}}. For the soliton solution of the Korteweg–de Vries equation, u ₀(X) with X = x − 4t, these derivatives, u _α (X) = D ^α u ₀(X), and their Hilbert transforms, v _α (X) = −HD ^α u ₀(X), can be expressed in terms of the full range Hurwitz Zeta functions ζ₊(s, a) and ζ₋(s, a), respectively. New properties are established for u _α (X) and v _α (X). It is proved that the functions w _α (X) = u _α (X) + iv _α (X) with α > −1 are solutions of the differential equation

-\fracddX(Pa(X)\fracdwdX)+Qa(X)w = lra(X)w,       X ? \mathbbR,-\frac{\rm d}{{\rm d}X}\left(P_{\alpha}(X)\frac{{\rm d}w}{{\rm d}X}\right)+Q_{\alpha}(X)w = \lambda\rho_{\alpha}(X)w,\qquad X \in \mathbb{R},      4. Uniform dimension results for Gaussian random fields      DongSheng Wu  YiMin Xiao 《中国科学A辑(英文版)》2009,52(7):1478-1496 Let X = {X(t), t ∈ ℝ N } be a Gaussian random field with values in ℝ d defined by ((1)) . The properties of space and time anisotropy of X and their connections to uniform Hausdorff dimension results are discussed. It is shown that in general the uniform Hausdorff dimension result does not hold for the image sets of a space-anisotropic Gaussian random field X. When X is an (N, d)-Gaussian random field as in (1), where X 1,...,X d are independent copies of a real valued, centered Gaussian random field X 0 which is anisotropic in the time variable. We establish uniform Hausdorff dimension results for the image sets of X. These results extend the corresponding results on one-dimensional Brownian motion, fractional Brownian motion and the Brownian sheet.       5. Nonparametric estimation of trend for stochastic differential equations driven by fractional Brownian motion      M. N. Mishra  B. L. S. Prakasa Rao 《Statistical Inference for Stochastic Processes》2011,14(2):101-109 Consider a stochastic process {X t , 0 ≤ t ≤ T} governed by a stochastic differential equation given by dXt = S(Xt)   dt + e  dWtH,    X0=x0,    0 ￡ t ￡ T dX_t= S(X_t) \;dt + \epsilon \; dW_t^H,\quad X_0=x_0,\quad 0 \leq t \leq T      6. Viability for nonlinear multi-valued reaction–diffusion systems      Daniela Roşu 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(4):479-496 In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system { lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      7. On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data      Ryosuke Hyakuna  Masayoshi Tsutsumi 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(3):309-327 We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      8. Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions      Sihua Liang  Yueqiang Song 《Lithuanian Mathematical Journal》2012,52(1):62-76 In this paper, we consider the following nonlinear fractional three-point boundary-value problem: *20c D0 + a u(t) + f( t,u(t) ) = 0,    0 < t < 1, u(0) = u￠(0) = 0,    u￠(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array}      9. Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps      Feng Dai  Heping Wang 《Constructive Approximation》2010,31(1):1-36 Let Π n d denote the space of all spherical polynomials of degree at most n on the unit sphere $\mathbb{S}^{d}Let Π n d denote the space of all spherical polynomials of degree at most n on the unit sphere \mathbbSd\mathbb{S}^{d} of ℝ d+1, and let d(x,y) denote the geodesic distance arccos x⋅y between x,y ? \mathbbSdx,y\in\mathbb{S}^{d} . Given a spherical cap B(e,a)={x ? \mathbbSd:d(x,e) ￡ a}    (e ? \mathbbSd, a ? (0,p) is bounded awayfrom p),B(e,\alpha)=\big\{x\in\mathbb{S}^{d}:d(x,e)\leq\alpha\big\}\quad \bigl(e\in\mathbb{S}^{d},\ \alpha\in(0,\pi)\ \mbox{is bounded awayfrom}\ \pi\bigr),      10. On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type      Zhiting Xu 《Monatshefte für Mathematik》2007,57(5):157-171 Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x￠(t))￠+q1(t)| y(t-s1)|a sgn y(t-s1) +q2(t)| y(t-s2)|b sgn y(t-s2)=0,    t 3 t0,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ1 and σ2 are nonnegative constants, α > 0, β > 0, and a, p, q 1, q2 ? C([t0, ￥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.      11. Canonical State/Signal Shift Realizations of Passive Continuous Time Behaviors      Damir Z. Arov  Mikael Kurula  Olof J. Staffans 《Complex Analysis and Operator Theory》2011,5(2):331-402 This work is devoted to the construction of canonical passive and conservative state/signal shift realizations of arbitrary passive continuous time behaviors. By definition, a passive future continuous time behavior is a maximal nonnegative right-shift invariant subspace of the Kreĭn space L2([0,￥);W){L^2([0,\infty);\mathcal W)}, where W{\mathcal W} is a Kreĭn space, and the inner product in L2([0,￥);W){L^2([0,\infty);\mathcal W)} is the one inherited from W{\mathcal W}. A state/signal system S = (V;X,W){\Sigma=(V;\mathcal X,\mathcal W)}, with a Hilbert state space X{\mathcal X} and a Kreĭn signal space W{\mathcal W}, is a dynamical system whose classical trajectories (x, w) on [0, ∞) satisfy x ? C1([0,￥);X){x\in C^1([0,\infty);\mathcal X)}, w ? C([0,￥);W){w \in C([0,\infty);\mathcal W)}, and ([(x)\dot](t),x(t),w(t)) ? V,    t ? [0,￥), (\dot x(t),x(t),w(t))\in V,\quad t \in [0,\infty),      12. The Size of Exponential Sums on Intervals of the Real Line      Tamás Erdélyi  Kaveh Khodjasteh  Lorenza Viola 《Constructive Approximation》2012,35(1):123-136 We prove that there is a constant c>0 depending only on M≥1 and μ≥0 such that òyy+a |g(t)|  dt 3 exp(-c/(ad)),     ad ? (0,p],\int_y^{y+a}{ \bigl|g(t)\bigr| \, dt} \geq \exp \bigl(-c/(a\delta)\bigr), \quad a\delta \in (0,\pi],      13. Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations      Qingliu Yao 《Acta Appl Math》2010,110(2):871-883 This paper studies the existence of a positive solution to the second-order periodic boundary value problem u￠￠(t)+l(t)u(t)=f(t,u(t)),    0 < t < 2p,  u(0)=u(2p), u￠(0)=u￠(2p),u^{\prime \prime }(t)+\lambda (t)u(t)=f\bigl(t,u(t)\bigr),\quad 0相似文献    14. On sets of linear differential systems to which perturbed linear systems cannot be reduced      N. A. Izobov  S. A. Mazanik 《Differential Equations》2011,47(11):1563-1568 The paper [2] defines the noncoinciding irreducibility sets N 2(a, σ) and N 3(a, σ), σ ∈ (0, 2a], of all n-dimensional linear differential systems with piecewise continuous coefficient matrices A(t) such that ‖A(t)‖ ≤ a < + ∞ for t ∈ [0,+∞) and there exists a linear differential system that is not Lyapunov reducible to the original system and has coefficient matrix B(t) satisfying [for the case of N 2(a, σ)] the condition || B(t) - A(t) || \leqslant const ×e - st ,t \geqslant 0,\left\| {B(t) - A(t)} \right\| \leqslant const \times e^{ - \sigma t} ,t \geqslant 0,      15. Highly time-oscillating solutions for very fast diffusion equations      Juan Luis V��zquez  Michael Winkler 《Journal of Evolution Equations》2011,11(3):725-742 This work is concerned with the fast diffusion equation ut = ?·(um-1 ?u)        (*) u_t = \nabla \cdot \big(u^{m-1} \nabla u\big) \qquad (\star)      16. Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients      Jitsuro Sugie  Saori Hata 《Monatshefte für Mathematik》2012,190(3):255-280 The following system considered in this paper: x￠ = - e(t)x + f(t)fp*(y),        y￠ = - (p-1)g(t)fp(x) - (p-1)h(t)y,x' = -\,e(t)x + f(t)\phi_{p^*}(y), \qquad y'= -\,(p-1)g(t)\phi_p(x) - (p-1)h(t)y,      17. Packing-type Measures of the Sample Paths of Fractional Brownian Motion      Zhen-long Chen  San-yang Liu  Ci-wen Xu 《应用数学学报(英文版)》2005,21(2):335-352 Abstract   Let Λ = {λ k } be an infinite increasing sequence of positive integers with λ k →∞. Let X = {X(t), t ∈? R N } be a multi-parameter fractional Brownian motion of index α(0 < α < 1) in R d . Subject to certain hypotheses, we prove that if N < αd, then there exist positive finite constants K 1 and K 2 such that, with unit probability, if and only if there exists γ > 0 such that where ϕ(s) = s N/α (log log 1/s) N/(2α), ϕ-p Λ(E) is the Packing-type measure of E,X([0, 1]) N is the image and GrX([0, 1] N ) = {(t,X(t)); ∈? [0, 1] N } is the graph of X, respectively. We also establish liminf type laws of the iterated logarithm for the sojourn measure of X. Supported by the National Natural Science Foundation of China (No.10471148), Sci-tech Innovation Item for Excellent Young and Middle-Aged University Teachers and Major Item of Educational Department of Hubei (No.2003A005)      18. Drift estimation for a periodic mean reversion process      Herold Dehling  Brice Franke  Thomas Kott 《Statistical Inference for Stochastic Processes》2010,13(3):175-192 In this paper we propose a periodic, mean-reverting Ornstein–Uhlenbeck process of the form dXt=(L(t)-a Xt) dt + s dBt,     t 3 0, dX_t=(L(t)-\alpha\, X_t)\, dt + \sigma\, dB_t, \quad t\geq 0,      19. Results on Local Times of a Class of Multiparameter Gaussian Processes      Zong-mao Cheng Xiu-yun Wang Zheng-yan Lin 《应用数学学报(英文版)》2006,22(1):81-90 In this paper, we introduce a class of Gaussian processes Y={Y（t）：t∈R^N},the so called hifractional Brownian motion with the indcxes H=（H1,…,HN）and α. We consider the （N, d, H, α） Gaussian random field x（t） = （x1 （t）,..., xd（t））,where X1 （t）,…, Xd（t） are independent copies of Y（t）, At first we show the existence and join continuity of the local times of X = {X（t）, t ∈ R＋^N}, then we consider the HSlder conditions for the local times.      20. Nonuniform Sampling and Recovery of?Multidimensional Bandlimited Functions by?Gaussian Radial-Basis Functions      B. A. Bailey  T. Schlumprecht  N. Sivakumar 《Journal of Fourier Analysis and Applications》2011,17(3):519-533 Let S⊂ℝ d be a bounded subset with positive Lebesgue measure. The Paley-Wiener space associated to S, PW S , is defined to be the set of all square-integrable functions on ℝ d whose Fourier transforms vanish outside S. A sequence (x j :j∈ℕ) in ℝ d is said to be a Riesz-basis sequence for L 2(S) (equivalently, a complete interpolating sequence for PW S ) if the sequence (e-iáxj,·?:j ? \mathbb N)(e^{-i\langle x_{j},\cdot \rangle }:j\in \mathbb {N}) of exponential functions forms a Riesz basis for L 2(S). Let (x j :j∈ℕ) be a Riesz-basis sequence for L 2(S). Given λ>0 and f∈PW S , there is a unique sequence (a j ) in ℓ 2 such that the function Il(f)(x):=?j ? \mathbb Naje-l||x-xj||22,    x ? \mathbb Rd,I_\lambda(f)(x):=\sum_{j\in \mathbb {N}}a_je^{-\lambda \|x-x_j\|_2^2},\quad x\in \mathbb {R}^d,      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Uniform dimension results for Gaussian random fields

Let X = {X(t), t ∈ ℝ ^N } be a Gaussian random field with values in ℝ ^d defined by

Nonparametric estimation of trend for stochastic differential equations driven by fractional Brownian motion

Consider a stochastic process {X _t , 0 ≤ t ≤ T} governed by a stochastic differential equation given by

Viability for nonlinear multi-valued reaction–diffusion systems

In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system

{ lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      7. On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data      Ryosuke Hyakuna  Masayoshi Tsutsumi 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(3):309-327 We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      8. Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions      Sihua Liang  Yueqiang Song 《Lithuanian Mathematical Journal》2012,52(1):62-76 In this paper, we consider the following nonlinear fractional three-point boundary-value problem: *20c D0 + a u(t) + f( t,u(t) ) = 0,    0 < t < 1, u(0) = u￠(0) = 0,    u￠(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array}      9. Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps      Feng Dai  Heping Wang 《Constructive Approximation》2010,31(1):1-36 Let Π n d denote the space of all spherical polynomials of degree at most n on the unit sphere $\mathbb{S}^{d}Let Π n d denote the space of all spherical polynomials of degree at most n on the unit sphere \mathbbSd\mathbb{S}^{d} of ℝ d+1, and let d(x,y) denote the geodesic distance arccos x⋅y between x,y ? \mathbbSdx,y\in\mathbb{S}^{d} . Given a spherical cap B(e,a)={x ? \mathbbSd:d(x,e) ￡ a}    (e ? \mathbbSd, a ? (0,p) is bounded awayfrom p),B(e,\alpha)=\big\{x\in\mathbb{S}^{d}:d(x,e)\leq\alpha\big\}\quad \bigl(e\in\mathbb{S}^{d},\ \alpha\in(0,\pi)\ \mbox{is bounded awayfrom}\ \pi\bigr),      10. On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type      Zhiting Xu 《Monatshefte für Mathematik》2007,57(5):157-171 Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x￠(t))￠+q1(t)| y(t-s1)|a sgn y(t-s1) +q2(t)| y(t-s2)|b sgn y(t-s2)=0,    t 3 t0,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ1 and σ2 are nonnegative constants, α > 0, β > 0, and a, p, q 1, q2 ? C([t0, ￥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.      11. Canonical State/Signal Shift Realizations of Passive Continuous Time Behaviors      Damir Z. Arov  Mikael Kurula  Olof J. Staffans 《Complex Analysis and Operator Theory》2011,5(2):331-402 This work is devoted to the construction of canonical passive and conservative state/signal shift realizations of arbitrary passive continuous time behaviors. By definition, a passive future continuous time behavior is a maximal nonnegative right-shift invariant subspace of the Kreĭn space L2([0,￥);W){L^2([0,\infty);\mathcal W)}, where W{\mathcal W} is a Kreĭn space, and the inner product in L2([0,￥);W){L^2([0,\infty);\mathcal W)} is the one inherited from W{\mathcal W}. A state/signal system S = (V;X,W){\Sigma=(V;\mathcal X,\mathcal W)}, with a Hilbert state space X{\mathcal X} and a Kreĭn signal space W{\mathcal W}, is a dynamical system whose classical trajectories (x, w) on [0, ∞) satisfy x ? C1([0,￥);X){x\in C^1([0,\infty);\mathcal X)}, w ? C([0,￥);W){w \in C([0,\infty);\mathcal W)}, and ([(x)\dot](t),x(t),w(t)) ? V,    t ? [0,￥), (\dot x(t),x(t),w(t))\in V,\quad t \in [0,\infty),      12. The Size of Exponential Sums on Intervals of the Real Line      Tamás Erdélyi  Kaveh Khodjasteh  Lorenza Viola 《Constructive Approximation》2012,35(1):123-136 We prove that there is a constant c>0 depending only on M≥1 and μ≥0 such that òyy+a |g(t)|  dt 3 exp(-c/(ad)),     ad ? (0,p],\int_y^{y+a}{ \bigl|g(t)\bigr| \, dt} \geq \exp \bigl(-c/(a\delta)\bigr), \quad a\delta \in (0,\pi],      13. Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations      Qingliu Yao 《Acta Appl Math》2010,110(2):871-883 This paper studies the existence of a positive solution to the second-order periodic boundary value problem u￠￠(t)+l(t)u(t)=f(t,u(t)),    0 < t < 2p,  u(0)=u(2p), u￠(0)=u￠(2p),u^{\prime \prime }(t)+\lambda (t)u(t)=f\bigl(t,u(t)\bigr),\quad 0相似文献    14. On sets of linear differential systems to which perturbed linear systems cannot be reduced      N. A. Izobov  S. A. Mazanik 《Differential Equations》2011,47(11):1563-1568 The paper [2] defines the noncoinciding irreducibility sets N 2(a, σ) and N 3(a, σ), σ ∈ (0, 2a], of all n-dimensional linear differential systems with piecewise continuous coefficient matrices A(t) such that ‖A(t)‖ ≤ a < + ∞ for t ∈ [0,+∞) and there exists a linear differential system that is not Lyapunov reducible to the original system and has coefficient matrix B(t) satisfying [for the case of N 2(a, σ)] the condition || B(t) - A(t) || \leqslant const ×e - st ,t \geqslant 0,\left\| {B(t) - A(t)} \right\| \leqslant const \times e^{ - \sigma t} ,t \geqslant 0,      15. Highly time-oscillating solutions for very fast diffusion equations      Juan Luis V��zquez  Michael Winkler 《Journal of Evolution Equations》2011,11(3):725-742 This work is concerned with the fast diffusion equation ut = ?·(um-1 ?u)        (*) u_t = \nabla \cdot \big(u^{m-1} \nabla u\big) \qquad (\star)      16. Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients      Jitsuro Sugie  Saori Hata 《Monatshefte für Mathematik》2012,190(3):255-280 The following system considered in this paper: x￠ = - e(t)x + f(t)fp*(y),        y￠ = - (p-1)g(t)fp(x) - (p-1)h(t)y,x' = -\,e(t)x + f(t)\phi_{p^*}(y), \qquad y'= -\,(p-1)g(t)\phi_p(x) - (p-1)h(t)y,      17. Packing-type Measures of the Sample Paths of Fractional Brownian Motion      Zhen-long Chen  San-yang Liu  Ci-wen Xu 《应用数学学报(英文版)》2005,21(2):335-352 Abstract   Let Λ = {λ k } be an infinite increasing sequence of positive integers with λ k →∞. Let X = {X(t), t ∈? R N } be a multi-parameter fractional Brownian motion of index α(0 < α < 1) in R d . Subject to certain hypotheses, we prove that if N < αd, then there exist positive finite constants K 1 and K 2 such that, with unit probability, if and only if there exists γ > 0 such that where ϕ(s) = s N/α (log log 1/s) N/(2α), ϕ-p Λ(E) is the Packing-type measure of E,X([0, 1]) N is the image and GrX([0, 1] N ) = {(t,X(t)); ∈? [0, 1] N } is the graph of X, respectively. We also establish liminf type laws of the iterated logarithm for the sojourn measure of X. Supported by the National Natural Science Foundation of China (No.10471148), Sci-tech Innovation Item for Excellent Young and Middle-Aged University Teachers and Major Item of Educational Department of Hubei (No.2003A005)      18. Drift estimation for a periodic mean reversion process      Herold Dehling  Brice Franke  Thomas Kott 《Statistical Inference for Stochastic Processes》2010,13(3):175-192 In this paper we propose a periodic, mean-reverting Ornstein–Uhlenbeck process of the form dXt=(L(t)-a Xt) dt + s dBt,     t 3 0, dX_t=(L(t)-\alpha\, X_t)\, dt + \sigma\, dB_t, \quad t\geq 0,      19. Results on Local Times of a Class of Multiparameter Gaussian Processes      Zong-mao Cheng Xiu-yun Wang Zheng-yan Lin 《应用数学学报(英文版)》2006,22(1):81-90 In this paper, we introduce a class of Gaussian processes Y={Y（t）：t∈R^N},the so called hifractional Brownian motion with the indcxes H=（H1,…,HN）and α. We consider the （N, d, H, α） Gaussian random field x（t） = （x1 （t）,..., xd（t））,where X1 （t）,…, Xd（t） are independent copies of Y（t）, At first we show the existence and join continuity of the local times of X = {X（t）, t ∈ R＋^N}, then we consider the HSlder conditions for the local times.      20. Nonuniform Sampling and Recovery of?Multidimensional Bandlimited Functions by?Gaussian Radial-Basis Functions      B. A. Bailey  T. Schlumprecht  N. Sivakumar 《Journal of Fourier Analysis and Applications》2011,17(3):519-533 Let S⊂ℝ d be a bounded subset with positive Lebesgue measure. The Paley-Wiener space associated to S, PW S , is defined to be the set of all square-integrable functions on ℝ d whose Fourier transforms vanish outside S. A sequence (x j :j∈ℕ) in ℝ d is said to be a Riesz-basis sequence for L 2(S) (equivalently, a complete interpolating sequence for PW S ) if the sequence (e-iáxj,·?:j ? \mathbb N)(e^{-i\langle x_{j},\cdot \rangle }:j\in \mathbb {N}) of exponential functions forms a Riesz basis for L 2(S). Let (x j :j∈ℕ) be a Riesz-basis sequence for L 2(S). Given λ>0 and f∈PW S , there is a unique sequence (a j ) in ℓ 2 such that the function Il(f)(x):=?j ? \mathbb Naje-l||x-xj||22,    x ? \mathbb Rd,I_\lambda(f)(x):=\sum_{j\in \mathbb {N}}a_je^{-\lambda \|x-x_j\|_2^2},\quad x\in \mathbb {R}^d,      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data

We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations

l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      8. Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions      Sihua Liang  Yueqiang Song 《Lithuanian Mathematical Journal》2012,52(1):62-76 In this paper, we consider the following nonlinear fractional three-point boundary-value problem: *20c D0 + a u(t) + f( t,u(t) ) = 0,    0 < t < 1, u(0) = u￠(0) = 0,    u￠(1) = ò0h u(s)\textds, \begin{array}{*{20}{c}} {D_{0 + }^\alpha u(t) + f\left( {t,u(t)} \right) = 0,\,\,\,\,0 < t < 1,} \\ {u(0) = u'(0) = 0,\,\,\,\,u'(1) = \int\limits_0^\eta {u(s){\text{d}}s,} } \\ \end{array}      9. Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps      Feng Dai  Heping Wang 《Constructive Approximation》2010,31(1):1-36 Let Π n d denote the space of all spherical polynomials of degree at most n on the unit sphere $\mathbb{S}^{d}Let Π n d denote the space of all spherical polynomials of degree at most n on the unit sphere \mathbbSd\mathbb{S}^{d} of ℝ d+1, and let d(x,y) denote the geodesic distance arccos x⋅y between x,y ? \mathbbSdx,y\in\mathbb{S}^{d} . Given a spherical cap B(e,a)={x ? \mathbbSd:d(x,e) ￡ a}    (e ? \mathbbSd, a ? (0,p) is bounded awayfrom p),B(e,\alpha)=\big\{x\in\mathbb{S}^{d}:d(x,e)\leq\alpha\big\}\quad \bigl(e\in\mathbb{S}^{d},\ \alpha\in(0,\pi)\ \mbox{is bounded awayfrom}\ \pi\bigr),      10. On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type      Zhiting Xu 《Monatshefte für Mathematik》2007,57(5):157-171 Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x￠(t))￠+q1(t)| y(t-s1)|a sgn y(t-s1) +q2(t)| y(t-s2)|b sgn y(t-s2)=0,    t 3 t0,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ1 and σ2 are nonnegative constants, α > 0, β > 0, and a, p, q 1, q2 ? C([t0, ￥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.      11. Canonical State/Signal Shift Realizations of Passive Continuous Time Behaviors      Damir Z. Arov  Mikael Kurula  Olof J. Staffans 《Complex Analysis and Operator Theory》2011,5(2):331-402 This work is devoted to the construction of canonical passive and conservative state/signal shift realizations of arbitrary passive continuous time behaviors. By definition, a passive future continuous time behavior is a maximal nonnegative right-shift invariant subspace of the Kreĭn space L2([0,￥);W){L^2([0,\infty);\mathcal W)}, where W{\mathcal W} is a Kreĭn space, and the inner product in L2([0,￥);W){L^2([0,\infty);\mathcal W)} is the one inherited from W{\mathcal W}. A state/signal system S = (V;X,W){\Sigma=(V;\mathcal X,\mathcal W)}, with a Hilbert state space X{\mathcal X} and a Kreĭn signal space W{\mathcal W}, is a dynamical system whose classical trajectories (x, w) on [0, ∞) satisfy x ? C1([0,￥);X){x\in C^1([0,\infty);\mathcal X)}, w ? C([0,￥);W){w \in C([0,\infty);\mathcal W)}, and ([(x)\dot](t),x(t),w(t)) ? V,    t ? [0,￥), (\dot x(t),x(t),w(t))\in V,\quad t \in [0,\infty),      12. The Size of Exponential Sums on Intervals of the Real Line      Tamás Erdélyi  Kaveh Khodjasteh  Lorenza Viola 《Constructive Approximation》2012,35(1):123-136 We prove that there is a constant c>0 depending only on M≥1 and μ≥0 such that òyy+a |g(t)|  dt 3 exp(-c/(ad)),     ad ? (0,p],\int_y^{y+a}{ \bigl|g(t)\bigr| \, dt} \geq \exp \bigl(-c/(a\delta)\bigr), \quad a\delta \in (0,\pi],      13. Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations      Qingliu Yao 《Acta Appl Math》2010,110(2):871-883 This paper studies the existence of a positive solution to the second-order periodic boundary value problem u￠￠(t)+l(t)u(t)=f(t,u(t)),    0 < t < 2p,  u(0)=u(2p), u￠(0)=u￠(2p),u^{\prime \prime }(t)+\lambda (t)u(t)=f\bigl(t,u(t)\bigr),\quad 0相似文献    14. On sets of linear differential systems to which perturbed linear systems cannot be reduced      N. A. Izobov  S. A. Mazanik 《Differential Equations》2011,47(11):1563-1568 The paper [2] defines the noncoinciding irreducibility sets N 2(a, σ) and N 3(a, σ), σ ∈ (0, 2a], of all n-dimensional linear differential systems with piecewise continuous coefficient matrices A(t) such that ‖A(t)‖ ≤ a < + ∞ for t ∈ [0,+∞) and there exists a linear differential system that is not Lyapunov reducible to the original system and has coefficient matrix B(t) satisfying [for the case of N 2(a, σ)] the condition || B(t) - A(t) || \leqslant const ×e - st ,t \geqslant 0,\left\| {B(t) - A(t)} \right\| \leqslant const \times e^{ - \sigma t} ,t \geqslant 0,      15. Highly time-oscillating solutions for very fast diffusion equations      Juan Luis V��zquez  Michael Winkler 《Journal of Evolution Equations》2011,11(3):725-742 This work is concerned with the fast diffusion equation ut = ?·(um-1 ?u)        (*) u_t = \nabla \cdot \big(u^{m-1} \nabla u\big) \qquad (\star)      16. Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients      Jitsuro Sugie  Saori Hata 《Monatshefte für Mathematik》2012,190(3):255-280 The following system considered in this paper: x￠ = - e(t)x + f(t)fp*(y),        y￠ = - (p-1)g(t)fp(x) - (p-1)h(t)y,x' = -\,e(t)x + f(t)\phi_{p^*}(y), \qquad y'= -\,(p-1)g(t)\phi_p(x) - (p-1)h(t)y,      17. Packing-type Measures of the Sample Paths of Fractional Brownian Motion      Zhen-long Chen  San-yang Liu  Ci-wen Xu 《应用数学学报(英文版)》2005,21(2):335-352 Abstract   Let Λ = {λ k } be an infinite increasing sequence of positive integers with λ k →∞. Let X = {X(t), t ∈? R N } be a multi-parameter fractional Brownian motion of index α(0 < α < 1) in R d . Subject to certain hypotheses, we prove that if N < αd, then there exist positive finite constants K 1 and K 2 such that, with unit probability, if and only if there exists γ > 0 such that where ϕ(s) = s N/α (log log 1/s) N/(2α), ϕ-p Λ(E) is the Packing-type measure of E,X([0, 1]) N is the image and GrX([0, 1] N ) = {(t,X(t)); ∈? [0, 1] N } is the graph of X, respectively. We also establish liminf type laws of the iterated logarithm for the sojourn measure of X. Supported by the National Natural Science Foundation of China (No.10471148), Sci-tech Innovation Item for Excellent Young and Middle-Aged University Teachers and Major Item of Educational Department of Hubei (No.2003A005)      18. Drift estimation for a periodic mean reversion process      Herold Dehling  Brice Franke  Thomas Kott 《Statistical Inference for Stochastic Processes》2010,13(3):175-192 In this paper we propose a periodic, mean-reverting Ornstein–Uhlenbeck process of the form dXt=(L(t)-a Xt) dt + s dBt,     t 3 0, dX_t=(L(t)-\alpha\, X_t)\, dt + \sigma\, dB_t, \quad t\geq 0,      19. Results on Local Times of a Class of Multiparameter Gaussian Processes      Zong-mao Cheng Xiu-yun Wang Zheng-yan Lin 《应用数学学报(英文版)》2006,22(1):81-90 In this paper, we introduce a class of Gaussian processes Y={Y（t）：t∈R^N},the so called hifractional Brownian motion with the indcxes H=（H1,…,HN）and α. We consider the （N, d, H, α） Gaussian random field x（t） = （x1 （t）,..., xd（t））,where X1 （t）,…, Xd（t） are independent copies of Y（t）, At first we show the existence and join continuity of the local times of X = {X（t）, t ∈ R＋^N}, then we consider the HSlder conditions for the local times.      20. Nonuniform Sampling and Recovery of?Multidimensional Bandlimited Functions by?Gaussian Radial-Basis Functions      B. A. Bailey  T. Schlumprecht  N. Sivakumar 《Journal of Fourier Analysis and Applications》2011,17(3):519-533 Let S⊂ℝ d be a bounded subset with positive Lebesgue measure. The Paley-Wiener space associated to S, PW S , is defined to be the set of all square-integrable functions on ℝ d whose Fourier transforms vanish outside S. A sequence (x j :j∈ℕ) in ℝ d is said to be a Riesz-basis sequence for L 2(S) (equivalently, a complete interpolating sequence for PW S ) if the sequence (e-iáxj,·?:j ? \mathbb N)(e^{-i\langle x_{j},\cdot \rangle }:j\in \mathbb {N}) of exponential functions forms a Riesz basis for L 2(S). Let (x j :j∈ℕ) be a Riesz-basis sequence for L 2(S). Given λ>0 and f∈PW S , there is a unique sequence (a j ) in ℓ 2 such that the function Il(f)(x):=?j ? \mathbb Naje-l||x-xj||22,    x ? \mathbb Rd,I_\lambda(f)(x):=\sum_{j\in \mathbb {N}}a_je^{-\lambda \|x-x_j\|_2^2},\quad x\in \mathbb {R}^d,      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Existence and uniqueness of positive solutions to nonlinear fractional differential equation with integral boundary conditions

In this paper, we consider the following nonlinear fractional three-point boundary-value problem:

Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps

Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere $\mathbb{S}^{d}Let Π _n ^d denote the space of all spherical polynomials of degree at most n on the unit sphere \mathbbS^d\mathbb{S}^{d} of ℝ ^d+1, and let d(x,y) denote the geodesic distance arccos x⋅y between x,y ? \mathbbS^dx,y\in\mathbb{S}^{d} . Given a spherical cap

B(e,a)={x ? \mathbbSd:d(x,e) ￡ a}    (e ? \mathbbSd, a ? (0,p) is bounded awayfrom p),B(e,\alpha)=\big\{x\in\mathbb{S}^{d}:d(x,e)\leq\alpha\big\}\quad \bigl(e\in\mathbb{S}^{d},\ \alpha\in(0,\pi)\ \mbox{is bounded awayfrom}\ \pi\bigr),      10. On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type      Zhiting Xu 《Monatshefte für Mathematik》2007,57(5):157-171 Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x￠(t))￠+q1(t)| y(t-s1)|a sgn y(t-s1) +q2(t)| y(t-s2)|b sgn y(t-s2)=0,    t 3 t0,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ1 and σ2 are nonnegative constants, α > 0, β > 0, and a, p, q 1, q2 ? C([t0, ￥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.      11. Canonical State/Signal Shift Realizations of Passive Continuous Time Behaviors      Damir Z. Arov  Mikael Kurula  Olof J. Staffans 《Complex Analysis and Operator Theory》2011,5(2):331-402 This work is devoted to the construction of canonical passive and conservative state/signal shift realizations of arbitrary passive continuous time behaviors. By definition, a passive future continuous time behavior is a maximal nonnegative right-shift invariant subspace of the Kreĭn space L2([0,￥);W){L^2([0,\infty);\mathcal W)}, where W{\mathcal W} is a Kreĭn space, and the inner product in L2([0,￥);W){L^2([0,\infty);\mathcal W)} is the one inherited from W{\mathcal W}. A state/signal system S = (V;X,W){\Sigma=(V;\mathcal X,\mathcal W)}, with a Hilbert state space X{\mathcal X} and a Kreĭn signal space W{\mathcal W}, is a dynamical system whose classical trajectories (x, w) on [0, ∞) satisfy x ? C1([0,￥);X){x\in C^1([0,\infty);\mathcal X)}, w ? C([0,￥);W){w \in C([0,\infty);\mathcal W)}, and ([(x)\dot](t),x(t),w(t)) ? V,    t ? [0,￥), (\dot x(t),x(t),w(t))\in V,\quad t \in [0,\infty),      12. The Size of Exponential Sums on Intervals of the Real Line      Tamás Erdélyi  Kaveh Khodjasteh  Lorenza Viola 《Constructive Approximation》2012,35(1):123-136 We prove that there is a constant c>0 depending only on M≥1 and μ≥0 such that òyy+a |g(t)|  dt 3 exp(-c/(ad)),     ad ? (0,p],\int_y^{y+a}{ \bigl|g(t)\bigr| \, dt} \geq \exp \bigl(-c/(a\delta)\bigr), \quad a\delta \in (0,\pi],      13. Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations      Qingliu Yao 《Acta Appl Math》2010,110(2):871-883 This paper studies the existence of a positive solution to the second-order periodic boundary value problem u￠￠(t)+l(t)u(t)=f(t,u(t)),    0 < t < 2p,  u(0)=u(2p), u￠(0)=u￠(2p),u^{\prime \prime }(t)+\lambda (t)u(t)=f\bigl(t,u(t)\bigr),\quad 0相似文献    14. On sets of linear differential systems to which perturbed linear systems cannot be reduced      N. A. Izobov  S. A. Mazanik 《Differential Equations》2011,47(11):1563-1568 The paper [2] defines the noncoinciding irreducibility sets N 2(a, σ) and N 3(a, σ), σ ∈ (0, 2a], of all n-dimensional linear differential systems with piecewise continuous coefficient matrices A(t) such that ‖A(t)‖ ≤ a < + ∞ for t ∈ [0,+∞) and there exists a linear differential system that is not Lyapunov reducible to the original system and has coefficient matrix B(t) satisfying [for the case of N 2(a, σ)] the condition || B(t) - A(t) || \leqslant const ×e - st ,t \geqslant 0,\left\| {B(t) - A(t)} \right\| \leqslant const \times e^{ - \sigma t} ,t \geqslant 0,      15. Highly time-oscillating solutions for very fast diffusion equations      Juan Luis V��zquez  Michael Winkler 《Journal of Evolution Equations》2011,11(3):725-742 This work is concerned with the fast diffusion equation ut = ?·(um-1 ?u)        (*) u_t = \nabla \cdot \big(u^{m-1} \nabla u\big) \qquad (\star)      16. Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients      Jitsuro Sugie  Saori Hata 《Monatshefte für Mathematik》2012,190(3):255-280 The following system considered in this paper: x￠ = - e(t)x + f(t)fp*(y),        y￠ = - (p-1)g(t)fp(x) - (p-1)h(t)y,x' = -\,e(t)x + f(t)\phi_{p^*}(y), \qquad y'= -\,(p-1)g(t)\phi_p(x) - (p-1)h(t)y,      17. Packing-type Measures of the Sample Paths of Fractional Brownian Motion      Zhen-long Chen  San-yang Liu  Ci-wen Xu 《应用数学学报(英文版)》2005,21(2):335-352 Abstract   Let Λ = {λ k } be an infinite increasing sequence of positive integers with λ k →∞. Let X = {X(t), t ∈? R N } be a multi-parameter fractional Brownian motion of index α(0 < α < 1) in R d . Subject to certain hypotheses, we prove that if N < αd, then there exist positive finite constants K 1 and K 2 such that, with unit probability, if and only if there exists γ > 0 such that where ϕ(s) = s N/α (log log 1/s) N/(2α), ϕ-p Λ(E) is the Packing-type measure of E,X([0, 1]) N is the image and GrX([0, 1] N ) = {(t,X(t)); ∈? [0, 1] N } is the graph of X, respectively. We also establish liminf type laws of the iterated logarithm for the sojourn measure of X. Supported by the National Natural Science Foundation of China (No.10471148), Sci-tech Innovation Item for Excellent Young and Middle-Aged University Teachers and Major Item of Educational Department of Hubei (No.2003A005)      18. Drift estimation for a periodic mean reversion process      Herold Dehling  Brice Franke  Thomas Kott 《Statistical Inference for Stochastic Processes》2010,13(3):175-192 In this paper we propose a periodic, mean-reverting Ornstein–Uhlenbeck process of the form dXt=(L(t)-a Xt) dt + s dBt,     t 3 0, dX_t=(L(t)-\alpha\, X_t)\, dt + \sigma\, dB_t, \quad t\geq 0,      19. Results on Local Times of a Class of Multiparameter Gaussian Processes      Zong-mao Cheng Xiu-yun Wang Zheng-yan Lin 《应用数学学报(英文版)》2006,22(1):81-90 In this paper, we introduce a class of Gaussian processes Y={Y（t）：t∈R^N},the so called hifractional Brownian motion with the indcxes H=（H1,…,HN）and α. We consider the （N, d, H, α） Gaussian random field x（t） = （x1 （t）,..., xd（t））,where X1 （t）,…, Xd（t） are independent copies of Y（t）, At first we show the existence and join continuity of the local times of X = {X（t）, t ∈ R＋^N}, then we consider the HSlder conditions for the local times.      20. Nonuniform Sampling and Recovery of?Multidimensional Bandlimited Functions by?Gaussian Radial-Basis Functions      B. A. Bailey  T. Schlumprecht  N. Sivakumar 《Journal of Fourier Analysis and Applications》2011,17(3):519-533 Let S⊂ℝ d be a bounded subset with positive Lebesgue measure. The Paley-Wiener space associated to S, PW S , is defined to be the set of all square-integrable functions on ℝ d whose Fourier transforms vanish outside S. A sequence (x j :j∈ℕ) in ℝ d is said to be a Riesz-basis sequence for L 2(S) (equivalently, a complete interpolating sequence for PW S ) if the sequence (e-iáxj,·?:j ? \mathbb N)(e^{-i\langle x_{j},\cdot \rangle }:j\in \mathbb {N}) of exponential functions forms a Riesz basis for L 2(S). Let (x j :j∈ℕ) be a Riesz-basis sequence for L 2(S). Given λ>0 and f∈PW S , there is a unique sequence (a j ) in ℓ 2 such that the function Il(f)(x):=?j ? \mathbb Naje-l||x-xj||22,    x ? \mathbb Rd,I_\lambda(f)(x):=\sum_{j\in \mathbb {N}}a_je^{-\lambda \|x-x_j\|_2^2},\quad x\in \mathbb {R}^d,      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

On the Oscillation of Second Order Neutral Differential Equations of Emden-Fowler Type

Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation of Emden-Fowler type (a(t)x￠(t))￠+q₁(t)| y(t-s₁)|^a sgn y(t-s₁) +q₂(t)| y(t-s₂)|^b sgn y(t-s₂)=0,    t 3 t₀,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0, where x(t) = y(t) + p(t)y(t − τ), τ, σ₁ and σ₂ are nonnegative constants, α > 0, β > 0, and a, p, q ₁, q₂ ? C([t₀, ￥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R}) . The results of this paper extend and improve some known results. In particular, two interesting examples that point out the importance of our theorems are also included.     

Canonical State/Signal Shift Realizations of Passive Continuous Time Behaviors

This work is devoted to the construction of canonical passive and conservative state/signal shift realizations of arbitrary passive continuous time behaviors. By definition, a passive future continuous time behavior is a maximal nonnegative right-shift invariant subspace of the Kreĭn space L²([0,￥);W){L^2([0,\infty);\mathcal W)}, where W{\mathcal W} is a Kreĭn space, and the inner product in L²([0,￥);W){L^2([0,\infty);\mathcal W)} is the one inherited from W{\mathcal W}. A state/signal system S = (V;X,W){\Sigma=(V;\mathcal X,\mathcal W)}, with a Hilbert state space X{\mathcal X} and a Kreĭn signal space W{\mathcal W}, is a dynamical system whose classical trajectories (x, w) on [0, ∞) satisfy x ? C¹([0,￥);X){x\in C^1([0,\infty);\mathcal X)}, w ? C([0,￥);W){w \in C([0,\infty);\mathcal W)}, and

The Size of Exponential Sums on Intervals of the Real Line

We prove that there is a constant c>0 depending only on M≥1 and μ≥0 such that

Periodic Positive Solution to a Class of Singular Second-Order Ordinary Differential Equations

This paper studies the existence of a positive solution to the second-order periodic boundary value problem

On sets of linear differential systems to which perturbed linear systems cannot be reduced

The paper [2] defines the noncoinciding irreducibility sets N ₂(a, σ) and N ₃(a, σ), σ ∈ (0, 2a], of all n-dimensional linear differential systems with piecewise continuous coefficient matrices A(t) such that ‖A(t)‖ ≤ a < + ∞ for t ∈ [0,+∞) and there exists a linear differential system that is not Lyapunov reducible to the original system and has coefficient matrix B(t) satisfying [for the case of N ₂(a, σ)] the condition

Highly time-oscillating solutions for very fast diffusion equations

This work is concerned with the fast diffusion equation

Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients

The following system considered in this paper:

Packing-type Measures of the Sample Paths of Fractional Brownian Motion

Abstract   Let Λ = {λ _k } be an infinite increasing sequence of positive integers with λ _k →∞. Let X = {X(t), t ∈? R ^N } be a multi-parameter fractional Brownian motion of index α(0 < α < 1) in R ^d . Subject to certain hypotheses, we prove that if N < αd, then there exist positive finite constants K ₁ and K ₂ such that, with unit probability,

Drift estimation for a periodic mean reversion process

In this paper we propose a periodic, mean-reverting Ornstein–Uhlenbeck process of the form

Results on Local Times of a Class of Multiparameter Gaussian Processes

In this paper, we introduce a class of Gaussian processes Y={Y（t）：t∈R^N},the so called hifractional Brownian motion with the indcxes H=（H1,…,HN）and α. We consider the （N, d, H, α） Gaussian random field x（t） = （x1 （t）,..., xd（t））,where X1 （t）,…, Xd（t） are independent copies of Y（t）, At first we show the existence and join continuity of the local times of X = {X（t）, t ∈ R＋^N}, then we consider the HSlder conditions for the local times.     

Nonuniform Sampling and Recovery of?Multidimensional Bandlimited Functions by?Gaussian Radial-Basis Functions

Let S⊂ℝ ^d be a bounded subset with positive Lebesgue measure. The Paley-Wiener space associated to S, PW _S , is defined to be the set of all square-integrable functions on ℝ ^d whose Fourier transforms vanish outside S. A sequence (x _j :j∈ℕ) in ℝ ^d is said to be a Riesz-basis sequence for L ₂(S) (equivalently, a complete interpolating sequence for PW _S ) if the sequence (e^-iáx_j,·?:j ? \mathbb N)(e^{-i\langle x_{j},\cdot \rangle }:j\in \mathbb {N}) of exponential functions forms a Riesz basis for L ₂(S). Let (x _j :j∈ℕ) be a Riesz-basis sequence for L ₂(S). Given λ>0 and f∈PW _S , there is a unique sequence (a _j ) in ℓ ₂ such that the function