Mostrando recursos 1 - 20 de 64

  1. Coalescence on Supercritical Bellman-Harris Branching Processes

    Athreya, Krishna B.; Hong, Jyy-I
    We consider a continuous-time single-type age-dependent Bellman-Harris branching process $\{Z(t): t \geq 0\}$ with offspring distribution $\{p_j\}_{j \geq 0}$ and lifetime distribution $G$. Let $k \geq 2$ be a positive integer. If $Z(t) \geq k$, we pick $k$ individuals from those who are alive at time $t$ by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let $D_k(t)$ be the coalescence time (the death time of the most recent common ancestor) and let $X_k(t)$ be the generation number of the most recent common ancestor of these $k$...

  2. Coalescence on Supercritical Bellman-Harris Branching Processes

    Athreya, Krishna B.; Hong, Jyy-I
    We consider a continuous-time single-type age-dependent Bellman-Harris branching process $\{Z(t): t \geq 0\}$ with offspring distribution $\{p_j\}_{j \geq 0}$ and lifetime distribution $G$. Let $k \geq 2$ be a positive integer. If $Z(t) \geq k$, we pick $k$ individuals from those who are alive at time $t$ by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let $D_k(t)$ be the coalescence time (the death time of the most recent common ancestor) and let $X_k(t)$ be the generation number of the most recent common ancestor of these $k$...

  3. Quantitative Recurrence Properties for Systems with Non-uniform Structure

    Zhao, Cao; Chen, Ercai
    Let $X$ be a subshift with non-uniform structure, and $\sigma \colon X \to X$ be a shift map. Further, define \[ R(\psi) := \{x \in X: d(\sigma^{n}x,x) \lt \psi(n) \textrm{ for infinitely many } n\} \] and \[ R(f) := \left\{ x \in X: d(\sigma^{n}x,x) \lt e^{-S_{n} f(x)} \textrm{ for infinitely many } n \right\}, \] where $\psi \colon \mathbb{N} \to \mathbb{R}^{+}$ is a nonincreasing and positive function and $f \colon X \to \mathbb{R}^{+}$ is a continuous positive function. In this paper, we give quantitative estimates of the above sets, that is, $\dim_{H} R(\psi)$ can be expressed by $\psi$ and...

  4. Quantitative Recurrence Properties for Systems with Non-uniform Structure

    Zhao, Cao; Chen, Ercai
    Let $X$ be a subshift with non-uniform structure, and $\sigma \colon X \to X$ be a shift map. Further, define \[ R(\psi) := \{x \in X: d(\sigma^{n}x,x) \lt \psi(n) \textrm{ for infinitely many } n\} \] and \[ R(f) := \left\{ x \in X: d(\sigma^{n}x,x) \lt e^{-S_{n} f(x)} \textrm{ for infinitely many } n \right\}, \] where $\psi \colon \mathbb{N} \to \mathbb{R}^{+}$ is a nonincreasing and positive function and $f \colon X \to \mathbb{R}^{+}$ is a continuous positive function. In this paper, we give quantitative estimates of the above sets, that is, $\dim_{H} R(\psi)$ can be expressed by $\psi$ and...

  5. General Decay for a Viscoelastic Wave Equation with Density and Time Delay Term in $\mathbb{R}^n$

    Feng, Baowei
    A linear viscoelastic wave equation with density and time delay in the whole space $\mathbb{R}^n$ ($n \geq 3$) is considered. In order to overcome the difficulties in the non-compactness of some operators, we introduce some weighted spaces. Under suitable assumptions on the relaxation function, we establish a general decay result of solution for the initial value problem by using energy perturbation method. Our result extends earlier results.

  6. General Decay for a Viscoelastic Wave Equation with Density and Time Delay Term in $\mathbb{R}^n$

    Feng, Baowei
    A linear viscoelastic wave equation with density and time delay in the whole space $\mathbb{R}^n$ ($n \geq 3$) is considered. In order to overcome the difficulties in the non-compactness of some operators, we introduce some weighted spaces. Under suitable assumptions on the relaxation function, we establish a general decay result of solution for the initial value problem by using energy perturbation method. Our result extends earlier results.

  7. Pointwise Multipliers on BMO Spaces with Non-doubling Measures

    Li, Wei; Nakai, Eiichi; Yang, Dongyong
    Let $\mu$ be a non-negative Radon measure satisfying the polynomial growth condition. In this paper, the authors characterize the set of pointwise multipliers on a BMO type space $\operatorname{RBMO}(\mu)$ introduced by Tolsa.

  8. Pointwise Multipliers on BMO Spaces with Non-doubling Measures

    Li, Wei; Nakai, Eiichi; Yang, Dongyong
    Let $\mu$ be a non-negative Radon measure satisfying the polynomial growth condition. In this paper, the authors characterize the set of pointwise multipliers on a BMO type space $\operatorname{RBMO}(\mu)$ introduced by Tolsa.

  9. Volume Inequalities for Asymmetric Orlicz Zonotopes

    Yang, Congli; Chen, Fangwei
    In this paper, we deal with the asymmetric Orlicz zonotopes by using the method of shadow system. We establish the volume product inequality and volume ratio inequality for asymmetric Orlicz zonotopes, along with their equality cases.

  10. Volume Inequalities for Asymmetric Orlicz Zonotopes

    Yang, Congli; Chen, Fangwei
    In this paper, we deal with the asymmetric Orlicz zonotopes by using the method of shadow system. We establish the volume product inequality and volume ratio inequality for asymmetric Orlicz zonotopes, along with their equality cases.

  11. Blow-up Solution for $4$-dimensional Generalized Emden-Fowler Equation with Exponential Nonlinearity

    Ouni, Taieb
    Using some nonlinear domain decomposition method, we prove the existence of singular limits for solution of generalized Emden-Fowler equation with exponential nonlinearity in fourth-dimensional given by \[ \begin{cases} \Delta (a(x) \Delta u) - V(x) \operatorname{div}(a(x) \nabla u) = \rho^{4} a(x) e^{u} & \textrm{in $\Omega \subset \mathbb{R}^{4}$}, \\ u = \Delta u = 0 & \textrm{on $\partial \Omega$}. \end{cases} \] The leading part $\Delta$ is, usually, called Laplacian operator. The potential $V(x)$ belongs to $L^{\infty}_{\operatorname{loc}}(\mathbb{R}^{4})$ it is smooth and bounded and $a = a(x)$ is a given smooth function over $\overline{\Omega}$, called the Schrödinger wave function. Namely, we are still looking...

  12. Blow-up Solution for $4$-dimensional Generalized Emden-Fowler Equation with Exponential Nonlinearity

    Ouni, Taieb
    Using some nonlinear domain decomposition method, we prove the existence of singular limits for solution of generalized Emden-Fowler equation with exponential nonlinearity in fourth-dimensional given by \[ \begin{cases} \Delta (a(x) \Delta u) - V(x) \operatorname{div}(a(x) \nabla u) = \rho^{4} a(x) e^{u} & \textrm{in $\Omega \subset \mathbb{R}^{4}$}, \\ u = \Delta u = 0 & \textrm{on $\partial \Omega$}. \end{cases} \] The leading part $\Delta$ is, usually, called Laplacian operator. The potential $V(x)$ belongs to $L^{\infty}_{\operatorname{loc}}(\mathbb{R}^{4})$ it is smooth and bounded and $a = a(x)$ is a given smooth function over $\overline{\Omega}$, called the Schrödinger wave function. Namely, we are still looking...

  13. Dynamics of a Stochastic Fractional Reaction-Diffusion Equation

    Liu, Linfang; Fu, Xianlong
    In this paper, we study the dynamics of a stochastic fractional reaction-diffusion equation with multiplicative noise in three spatial dimension. By proving the well-posedness and conducting a priori estimates for the solutions of the considered equation we obtain simultaneously the existence and the regularity of random attractors of the random dynamical systems for the equation. The main approach here is to establish an abstract result on existence of bi-spatial random attractors for random dynamical systems. Moreover, we also give estimate of the Hausdorff dimension in $L^{2}$ space for the obtained random attractor.

  14. Dynamics of a Stochastic Fractional Reaction-Diffusion Equation

    Liu, Linfang; Fu, Xianlong
    In this paper, we study the dynamics of a stochastic fractional reaction-diffusion equation with multiplicative noise in three spatial dimension. By proving the well-posedness and conducting a priori estimates for the solutions of the considered equation we obtain simultaneously the existence and the regularity of random attractors of the random dynamical systems for the equation. The main approach here is to establish an abstract result on existence of bi-spatial random attractors for random dynamical systems. Moreover, we also give estimate of the Hausdorff dimension in $L^{2}$ space for the obtained random attractor.

  15. The Order Properties and Karcher Barycenters of Probability Measures on the Open Convex Cone

    Kim, Sejong
    We study the probability measures on the open convex cone of positive definite operators equipped with the Loewner ordering. We show that two crucial push-forward measures derived by the congruence transformation and power map preserve the stochastic order for probability measures. By the continuity of two push-forward measures with respect to the Wasserstein distance, we verify several interesting properties of the Karcher barycenter for probability measures with finite first moment such as the invariant properties and the inequality for unitarily invariant norms. Moreover, the characterization for the stochastic order of uniformly distributed probability measures has been shown.

  16. The Order Properties and Karcher Barycenters of Probability Measures on the Open Convex Cone

    Kim, Sejong
    We study the probability measures on the open convex cone of positive definite operators equipped with the Loewner ordering. We show that two crucial push-forward measures derived by the congruence transformation and power map preserve the stochastic order for probability measures. By the continuity of two push-forward measures with respect to the Wasserstein distance, we verify several interesting properties of the Karcher barycenter for probability measures with finite first moment such as the invariant properties and the inequality for unitarily invariant norms. Moreover, the characterization for the stochastic order of uniformly distributed probability measures has been shown.

  17. Inequalities for the Casorati Curvatures of Real Hypersurfaces in Some Grassmannians

    Park, Kwang-Soon
    In this paper we obtain two types of optimal inequalities consisting of the normalized scalar curvature and the generalized normalized $\delta$-Casorati curvatures for real hypersurfaces of complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians. We also find the conditions on which the equalities hold.

  18. Inequalities for the Casorati Curvatures of Real Hypersurfaces in Some Grassmannians

    Park, Kwang-Soon
    In this paper we obtain two types of optimal inequalities consisting of the normalized scalar curvature and the generalized normalized $\delta$-Casorati curvatures for real hypersurfaces of complex two-plane Grassmannians and complex hyperbolic two-plane Grassmannians. We also find the conditions on which the equalities hold.

  19. One Existence Theorem for non-CSC Extremal Kähler Metrics with Conical Singularities on $S^2$

    Wei, Zhiqiang; Wu, Yingyi
    We often call an extremal Kähler metric with finite singularities on a compact Riemann surface an HCMU (the Hessian of the Curvature of the Metric is Umbilical) metric. In this paper we consider the following question: if we give $N$ points $p_1, \ldots, p_N$ on $S^2$ and $N$ positive real numbers $2\pi \alpha_1, \ldots, 2\pi \alpha_N$ with $\alpha_n \neq 1$, $n = 1, \ldots, N$, what condition can guarantee the existence of a non-CSC HCMU metric which has conical singularities $p_1, \ldots, p_N$ with singular angles $2\pi \alpha_1, \ldots, 2\pi \alpha_N$ respectively. We prove that if there are at least...

  20. One Existence Theorem for non-CSC Extremal Kähler Metrics with Conical Singularities on $S^2$

    Wei, Zhiqiang; Wu, Yingyi
    We often call an extremal Kähler metric with finite singularities on a compact Riemann surface an HCMU (the Hessian of the Curvature of the Metric is Umbilical) metric. In this paper we consider the following question: if we give $N$ points $p_1, \ldots, p_N$ on $S^2$ and $N$ positive real numbers $2\pi \alpha_1, \ldots, 2\pi \alpha_N$ with $\alpha_n \neq 1$, $n = 1, \ldots, N$, what condition can guarantee the existence of a non-CSC HCMU metric which has conical singularities $p_1, \ldots, p_N$ with singular angles $2\pi \alpha_1, \ldots, 2\pi \alpha_N$ respectively. We prove that if there are at least...

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