results for au:Zhu_C in:math

- Feb 07 2018 math.RT arXiv:1802.01774v1We consider the questions of vanishing and nonvanishing in local theta correspondence and discuss two recent results which help to answer the questions. The first one is concerned with the existence of models of smooth representations and the second one with the "classical limit" of $(\g, K)$-modules, and both depend on certain correspondence of nilpotent orbits arising from a double fiberation of moment maps.
- Dec 18 2017 math.RT arXiv:1712.05552v1Let $\mathbf G$ be a complex orthogonal or complex symplectic group, and let $G$ be a real form of $\mathbf G$, namely $G$ is a real orthogonal group, a real symplectic group, a quaternionic orthogonal group, or a quaternionic symplectic group. For a fixed parity $\mathbb p\in \mathbb Z/2\mathbb Z$, we define a set $\mathrm{Nil}^{\mathbb p}_{\mathbf G}(\mathfrak g)$ of nilpotent $\mathbf G$-orbits in $\mathfrak g$ (the Lie algebra of $\mathbf G$). When $\mathbb p$ is the parity of the dimension of the standard module of $\mathbf G$, this is the set of the stably trivial special nilpotent orbits, which includes all rigid special nilpotent orbits. For each $\mathcal O \in \mathrm{Nil}^{\mathbb p}_{\mathbf G}(\mathfrak g)$, we construct all unipotent representations of $G$ (or its metaplectic cover when $G$ is a real symplectic group and $\mathbb p$ is odd) attached to $\mathcal O$ via the method of theta lifting and show in particular that they are unitary.
- Sep 25 2017 math.PR arXiv:1709.07585v1This paper considers the martingale problem for a class of weakly coupled Lévy type operators. It is shown that under some mild conditions, the martingale problem is well-posed and uniquely determines a strong Markov process $(X,\Lambda)$. The process $(X,\Lambda)$, called a regime-switching jump diffusion with Lévy type jumps, is further shown to posses Feller and strong Feller properties under non-Lipschitz conditions via the coupling method.
- Sep 13 2017 math.OC arXiv:1709.03962v4The forward-backward operator splitting algorithm is one of the most important methods for solving the optimization problem of the sum of two convex functions, where one is differentiable with a Lipschitz continuous gradient and the other is possibly nonsmooth but proximable. It is convenient to solve some optimization problems in the form of dual or primal-dual problems. Both methods are mature in theory. In this paper, we construct several efficient first-order splitting algorithms for solving a multi-block composite convex optimization problem. The objective function includes a smooth function with a Lipschitz continuous gradient, a proximable convex function that may be nonsmooth, and a finite sum of a composition of a proximable function and a bounded linear operator. To solve such an optimization problem, we transform it into the sum of three convex functions by defining an appropriate inner product space. On the basis of the dual forward-backward splitting algorithm and the primal-dual forward-backward splitting algorithm, we develop several iterative algorithms that involve only computing the gradient of the differentiable function and proximity operators of related convex functions. These iterative algorithms are matrix-inversion-free and completely splitting algorithms. Finally, we employ the proposed iterative algorithms to solve a regularized general prior image constrained compressed sensing (PICCS) model that is derived from computed tomography (CT) image reconstruction under sparse sampling of projection measurements. Numerical results show that the proposed iterative algorithms outperform other algorithms.
- Aug 31 2017 math.AP arXiv:1708.09127v1This paper is concerned with the asymptotic behavior of the solution to the Euler equations with time-depending damping on quadrant $(x,t)\in \mathbb{R}^+\times\mathbb{R}^+$, \beginequation\notag \partial_t v - \partial_x u=0, \qquad \partial_t u + \partial_x p(v) =\displaystyle -\frac\alpha(1+t)^\lambda u, \endequation with null-Dirichlet boundary condition or null-Neumann boundary condition on $u$. We show that the corresponding initial-boundary value problem admits a unique global smooth solution which tends time-asymptotically to the nonlinear diffusion wave. Compared with the previous work about Euler equations with constant coefficient damping, studied by Nishihara and Yang (1999, J. Differential Equations, 156, 439-458), and Jiang and Zhu (2009, Discrete Contin. Dyn. Syst., 23, 887-918), we obtain a general result when the initial perturbation belongs to the same space. In addition, our main novelty lies in the facts that the cut-off points of the convergence rates are different from our previous result about the Cauchy problem. Our proof is based on the classical energy method and the analyses of the nonlinear diffusion wave.
- We characterize strong type and weak type inequalities with two weights for positive operators on filtered measure spaces. These estimates are probabilistic analogues of two-weight inequalities for positive operators associated to the dyadic cubes in $\mathbb R^n$ due to Lacey, Sawyer and Uriarte-Tuero \citeLaSaUr. Several mixed bounds for the Doob maximal operator on filtered measure spaces are also obtained. In fact, Hytönen-Pérez type and Lerner-Moen type norm estimates for Doob maximal operator are established. Our approaches are mainly based on the construction of principal sets.
- This work is devoted to switching diffusions that have two components (a continuous component and a discrete component). Different from the so-called Markovian switching diffusions, in the setup, the discrete component (the switching) depends on the continuous component (the diffusion process). The objective of this paper is to provide a number of properties related to the well posedness. First, the differentiability with respect to initial data of the continuous component is established. Then, further properties including uniform continuity with respect to initial data, and smoothness of certain functionals are obtained. Moreover, Feller property is obtained under only local Lipschitz continuity. Finally, an example of Lotka-Voterra model under regime switching is provided as an illustration.
- Jun 13 2017 math.PR arXiv:1706.03684v2This work is devoted to almost sure and moment exponential stability of regime-switching jump diffusions. The Lyapunov function method is used to derive sufficient conditions for stabilities for general nonlinear systems; which further helps to derive easily verifiable conditions for linear systems. For one-dimensional linear regime-switching jump diffusions, necessary and sufficient conditions for almost sure and $p$th moment exponential stabilities are presented. Several examples are provided for illustration.
- Jun 06 2017 math.PR arXiv:1706.01393v3This paper considers multidimensional jump type stochastic differential equations with super linear growth and non-Lipschitz coefficients. After establishing a sufficient condition for nonexplosion, this paper presents sufficient non-Lipschitz conditions for pathwise uniqueness. The non confluence property for solutions is investigated. Feller and strong Feller properties under non-Lipschitz conditions are investigated via the coupling method. Sufficient conditions for irreducibility and exponential ergodicity are derived. As applications, this paper also studies multidimensional stochastic differential equations driven by Lévy processes and presents a Feynman-Kac formula for Lévy type operators.
- May 18 2017 math.OC arXiv:1705.06164v5In this paper, we consider solving a class of convex optimization problem which minimizes the sum of three convex functions $f(x)+g(x)+h(Bx)$, where $f(x)$ is differentiable with a Lipschitz continuous gradient, $g(x)$ and $h(x)$ have a closed-form expression of their proximity operators and $B$ is a bounded linear operator. This type of optimization problem has wide application in signal recovery and image processing. To make full use of the differentiability function in the optimization problem, we take advantage of two operator splitting methods: the forward-backward splitting method and the three operator splitting method. In the iteration scheme derived from the two operator splitting methods, we need to compute the proximity operator of $g+h \circ B$ and $h \circ B$, respectively. Although these proximity operators do not have a closed-form solution in general, they can be solved very efficiently. We mainly employ two different approaches to solve these proximity operators: one is dual and the other is primal-dual. Following this way, we fortunately find that three existing iterative algorithms including Condat and Vu algorithm, primal-dual fixed point (PDFP) algorithm and primal-dual three operator (PD3O) algorithm are a special case of our proposed iterative algorithms. Moreover, we discover a new kind of iterative algorithm to solve the considered optimization problem, which is not covered by the existing ones. Under mild conditions, we prove the convergence of the proposed iterative algorithms. Numerical experiments applied on fused Lasso problem, constrained total variation regularization in computed tomography (CT) image reconstruction and low-rank total variation image super-resolution problem demonstrate the effectiveness and efficiency of the proposed iterative algorithms.
- May 16 2017 math.AP arXiv:1705.04939v1For a general dyadic grid, we give a Calderón-Zygmund type decomposition, which is the principle fact about the multilinear maximal function $\mathfrak{M}$ on the upper half-spaces. Using the decomposition, we study the boundedness of $\mathfrak{M}.$ We obtain a natural extension to the multilinear setting of Muckenhoupt's weak-type characterization. We also partially obtain characterizations of Muckenhoupt's strong-type inequalities with one weight. Assuming the reverse Hölder's condition, we get a multilinear analogue of Sawyer's two weight theorem. Moreover, we also get Hytönen-Pérez type weighted estimates.
- Descent equations play an important role in the theory of characteristic classes and find applications in theoretical physics, e.g. in the Chern-Simons field theory and in the theory of anomalies. The second Chern class (the first Pontrjagin class) is defined as $p= \langle F, F\rangle$ where $F$ is the curvature 2-form and $\langle \cdot, \cdot\rangle$ is an invariant scalar product on the corresponding Lie algebra $\mathfrak{g}$. The descent for $p$ gives rise to an element $\omega=\omega_3 + \omega_2 + \omega_1 + \omega_0$ of mixed degree. The 3-form part $\omega_3$ is the Chern-Simons form. The 2-form part $\omega_2$ is known as the Wess-Zumino action in physics. The 1-form component $\omega_1$ is related to the canonical central extension of the loop group $LG$. In this paper, we give a new interpretation of the low degree components $\omega_1$ and $\omega_0$. Our main tool is the universal differential calculus on free Lie algebras due to Kontsevich. We establish a correspondence between solutions of the first Kashiwara-Vergne equation in Lie theory and universal solutions of the descent equation for the second Chern class $p$. In more detail, we define a 1-cocycle $C$ which maps automorphisms of the free Lie algebra to one forms. A solution of the Kashiwara-Vergne equation $F$ is mapped to $\omega_1=C(F)$. Furthermore, the component $\omega_0$ is related to the associator corresponding to $F$. It is surprising that while $F$ and $\Phi$ satisfy the highly non-linear twist and pentagon equations, the elements $\omega_1$ and $\omega_0$ solve the linear descent equation.
- Feb 07 2017 math.PR arXiv:1702.01495v1This work develops Feynman-Kac formulas for a class of regime-switching jump diffusion processes, in which the jump part is driven by a Poisson random measure associated to a general Lévy process and the switching part depends on the jump diffusion processes. Under broad conditions, the connections of such stochastic processes and the corresponding partial integro-differential equations are established. Related initial, terminal, and boundary value problems are also treated. Moreover, based on weak convergence of probability measures, it is demonstrated that a sequence of random variables related to the regime-switching jump diffusion process converges in distribution to the arcsine law.
- Feb 06 2017 math.OC arXiv:1702.01041v1This paper continues the examination of inventory control in which the inventory is modelled by a diffusion process and a long-term average cost criterion is used to make decisions. The class of such models under consideration have general drift and diffusion coefficients and boundary points that are consistent with the notion that demand should tend to reduce the inventory level. The conditions on the cost functions are greatly relaxed from those in \citehelm:15b. Characterization of the cost of a general $(s,S)$ policy as a function of two variables naturally leads to a nonlinear optimization problem over the ordering levels $s$ and $S$. Existence of an optimizing pair $(s_*,S_*)$ is established for these models under very weak conditions; non-existence of an optimizing pair is also discussed. Using average expected occupation and ordering measures and weak convergence arguments, weak conditions are given for the optimality of the $(s_*,S_*)$ ordering policy in the general class of admissible policies. The analysis involves an auxiliary function that is globally $C^2$ and which, together with the infimal cost, solve a particular system of linear equations and inequalities related to but different from the long-term average Hamilton-Jacobi-Bellman equation. This approach provides an analytical solution to the problem rather than a solution involving intricate analysis of the stochastic processes. The range of applicability of these results is illustrated on a drifted Brownian motion inventory model, both unconstrained and reflected, and on a geometric Brownian motion inventory model under three different cost structures.
- Feb 06 2017 math.OC arXiv:1702.01046v1The paper \citehelm:17 studies an inventory management problem under a long-term average cost criterion using weak convergence methods applied to average expected occupation and average expected ordering measures. Under the natural condition of inf-compactness of the holding cost rate function, the average expected occupation measures are seen to be tight and hence have weak limits. However inf-compactness is not a natural assumption to impose on the ordering cost function. For example, the cost function composed of a fixed cost plus proportional (to the size of the order) cost is not inf-compact. Intuitively, it would seem that imposing a requirement that the long-term average cost be finite ought to imply tightness of the average expected ordering measures; a lack of tightness should mean that the inventory process spends large amounts of time in regions that have arbitrarily large holding costs resulting in an infinite long-term average cost. This paper demonstrates that this intuition is incorrect by identifying a model and an ordering policy for which the resulting inventory process has a finite long-term average cost, tightness of the average expected occupation measures but which lacks tightness of the average expected ordering measures.
- Feb 06 2017 math.PR arXiv:1702.01048v1This work focuses on a class of regime-switching jump diffusion processes, in which the switching component has countably infinite many states or regimes. The existence and uniqueness of the underlying process are obtained by an interlacing procedure. Then the Feller and strong Feller properties of such processes are derived by the coupling method and an appropriate Radon-Nikodym derivative. Finally the paper studies exponential ergodicity of regime-switching jump-diffusion processes.
- Jan 05 2017 math.DG arXiv:1701.00959v3Just like Atiyah Lie algebroids encode the infinitesimal symmetries of principal bundles, exact Courant algebroids are believed to encode the infinitesimal symmetries of $S^1$-gerbes. At the same time, transitive Courant algebroids may be viewed as the higher analogue of Atiyah Lie algebroids, and the non-commutative analogue of exact Courant algebroids. In this article, we explore what the "principal bundle" behind transitive Courant algebroids are, and they turn out to be principal 2-bundles of string groups. First, we construct the stack of principal 2-bundles of string groups with connection data. We prove a lifting theorem for the stack of string principal bundles with connections and show the multiplicity of the lifts once they exist. This is a differential geometrical refinement of what is known for string structures by Redden, Waldorf and Stolz-Teichner. We also extend the result of Bressler and Chen-Stiénon-Xu on extension obstruction involving transitive Courant algebroids to the case of transitive Courant algebroids with connections, as a lifting theorem with the description of multiplicity once liftings exist. At the end, we build a morphism between these two stacks. The morphism turns out to be neither injective nor surjective in general, which shows that the process of associating the "higher Atiyah algebroid" loses some information and at the same time, only some special transitive Courant algebroids come from string bundles.
- Lie $\infty$-groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie $\infty$-groupoids called "Lie $\infty$-groups" by integrating $L_\infty$-algebras. In order to study the compatibility between this integration procedure and the homotopy theory of $L_\infty$-algebras, we present a homotopy theory for Lie $\infty$-groupoids. Unlike Kan simplicial sets and the higher geometric groupoids of Behrend and Getzler, Lie $\infty$-groupoids do not form a category of fibrant objects (CFO), since the category of manifolds lacks pullbacks. Instead, we show that Lie $\infty$-groupoids form an "incomplete category of fibrant objects" in which the weak equivalences correspond to "stalkwise" weak equivalences of simplicial sheaves. This homotopical structure enjoys many of the same properties as a CFO, such as having, in the presence of functorial path objects, a convenient realization of its simplicial localization. We further prove that the acyclic fibrations are precisely the hypercovers, which implies that many of Behrend and Getzler's results also hold in this more general context. As an application, we show that homotopy equivalent $L_\infty$-algebras integrate to "Morita equivalent" Lie $\infty$-groups.
- Asymmetrically clipped optical orthogonal frequency division multiplexing (ACO-OFDM) is theoretically more power efficient but less spectrally efficient than DC-bias OFDM (DCO-OFDM), with less power allocating to the informationless bias component by only using odd index sub-carriers. Layered/Enhanced asymmetrically clipped optical orthogonal frequency division multiplexing (L/E-ACO-OFDM) has been proposed to increase the spectral efficiency of ACO-OFDM. In this letter, we experimentally demonstrate a 30-km single mode fiber transmission using L/E-ACO-OFDM at 4.375 Gbits/s. Using a Volterra filter based equalizer, 2-dB and 1.5-dB Q-factor improvements for L/E-ACO-OFDM comparing with DCO-OFDM can be obtained in back-to-back and 30-km fiber transmission respectively.
- In this paper, we prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space. Then we give a direct proof of iteration formulae for the Maslov-type indices of symplectic paths.
- A one-parameter generalization of the hierarchy of negative flows is introduced for integrable hierarchies of evolution equations, which yields a wider (new) class of non-evolutionary integrable nonlinear wave equations. As main results, several integrability properties of these generalized negative flow equation are established, including their symmetry structure, conservation laws, and bi-Hamiltonian formulation. (The results also apply to the hierarchy of ordinary negative flows). The first generalized negative flow equation is worked out explicitly for each of the following integrable equations: Burgers, Korteweg-de Vries, modified Korteweg-de Vries, Sawada-Kotera, Kaup-Kupershmidt, Kupershmidt.
- Motivated by the study of the interrelation between functorial and algebraic quantum field theory, we point out that on any locally trivial bundle of compact groups, representations up to homotopy are enough to separate points by means of the associated representations in cohomol- ogy. Furthermore, we observe that the derived representation category of any compact group is equivalent to the category of ordinary (finite- dimensional) representations of the group.
- Stacky Lie groupoids are generalizations of Lie groupoids in which the "space of arrows" of the groupoid is a differentiable stack. In this paper, we consider actions of stacky Lie groupoids on differentiable stacks and their associated quotients. We provide a characterization of principal actions of stacky Lie groupoids, i.e., actions whose quotients are again differentiable stacks in such a way that the projection onto the quotient is a principal bundle. As an application, we extend the notion of Morita equivalence of Lie groupoids to the realm of stacky Lie groupoids, providing examples that naturally arise from non-integrable Lie algebroids.
- Oct 23 2015 math.OC arXiv:1510.06656v1This paper establishes conditions for optimality of an $(s,S)$ ordering policy for the minimization of the long-term average cost of one-dimensional diffusion inventory models. The class of such models under consideration have general drift and diffusion coefficients and boundary points that are consistent with the notion that demand should tend to decrease the inventory level. Characterization of the cost of a general $(s,S)$ policy as a function $F$ of two variables naturally leads to a nonlinear optimization problem over the ordering levels $s$ and $S$. Existence of an optimizing pair $(s_*,S_*)$ is established for these models. Using the minimal value $F_*$ of $F$, along with $(s_*,S_*)$, a function $G$ is identified which is proven to be a solution of a quasi-variational inequality provided a simple condition holds. At this level of generality, optimality of the $(s_*,S_*)$ ordering policy is established within a large class of ordering policies such that local martingale and transversality conditions involving $G$ hold. For specific models, optimality of an $(s,S)$ policy in the general class of admissible policies can be established using comparison results. This most general optimality result is shown for the classical drifted Brownian motion inventory model with holding and fixed plus proportional ordering costs and for a geometric Brownian motion inventory model with fixed plus level-dependent ordering costs. However, for a drifted Brownian motion process with reflection at $\{0\}$, a new class of non-Markovian policies is introduced which have lower costs than the $(s,S)$ policies. In addition, interpreting reflection at $\{0\}$ as "just-in-time" ordering, a necessary and sufficient condition is given that determines when just-in-time ordering is better than traditional $(s,S)$ policies.
- Sep 02 2015 math.NT arXiv:1509.00412v1The goal of this paper is to analyze the discrete Lambert map x to xg^x modulo a power of a prime p which is important for security and verification of the ElGamal digital signature scheme. We use p-adic methods (p-adic interpolation and Hensel's Lemma) to count the number of solutions x of xg^x congruent to c modulo powers of an odd prime p and c, g are fixed integers. At the same time, we discover special patterns in the solutions.
- Aug 07 2015 math.AP arXiv:1508.01405v1This paper is concerned with the study of the nonlinear stability of the contact discontinuity of the Navier-Stokes-Poisson system with free boundary in the case where the electron background density satisfies an analogue of the Boltzmann relation. We especially allow that the electric potential can take distinct constant states at boundary. On account of the quasineutral assumption, we first construct a viscous contact wave through the quasineutral Euler equations, and then prove that such a non-trivial profile is time-asymptotically stable under small perturbations for the corresponding initial boundary value problem of the Navier-Stokes-Poisson system. The analysis is based on the techniques developed in \citeDL and an elementary $L^2$ energy method.
- Aug 07 2015 math.AP arXiv:1508.01411v1This paper is concerned with the study of nonlinear stability of superposition of boundary layer and rarefaction wave on the two-fluid Navier-Stokes-Poisson system in the half line $\mathbb{R}_{+}=:(0,+\infty)$. On account of the quasineutral assumption and the absence of the electric field for the large time behavior, we successfully construct the boundary layer and rarefaction wave, and then we give the rigorous proofs of the stability theorems on the superposition of boundary layer and rarefaction wave under small perturbations for the corresponding initial boundary value problem of the Navier-Stokes-Poisson system, only provided the strength of boundary layer is small while the strength of rarefaction wave can be arbitrarily large. The complexity of nonlinear composite wave leads to many complicated terms in the course of establishing the \it a priori estimates. The proofs are given by an elementary $L^2$ energy method.
- Jul 31 2015 math.OC arXiv:1507.08413v3Our work considers the optimization of the sum of a non-smooth convex function and a finite family of composite convex functions, each one of which is composed of a convex function and a bounded linear operator. This type of problem is associated with many interesting challenges encountered in the image restoration and image reconstruction fields. We developed a splitting primal-dual proximity algorithm to solve this problem. Further, we propose a preconditioned method, of which the iterative parameters are obtained without the need to know some particular operator norm in advance. Theoretical convergence theorems are presented. We then apply the proposed methods to solve a total variation regularization model, in which the L2 data error function is added to the L1 data error function. The main advantageous feature of this model is its capability to combine different loss functions. The numerical results obtained for computed tomography (CT) image reconstruction demonstrated the ability of the proposed algorithm to reconstruct an image with few and sparse projection views while maintaining the image quality.
- This paper is devoted to the popular Sudoku problem. We proposed several strategies for solving Sudoku puzzles based on the sparse optimization technique. Further, we defined a new difficulty level for Sudoku puzzles. The efficiency of the method is verified via Sudoku puzzles data-set, and the numerical results showed that the accurate recovery rate can be enhanced from 84%+ to 99%+ by the L1 sparse optimization method.
- Mar 12 2015 math.AP arXiv:1503.03143v1In this paper, we study the Cauchy problem of the isentropic compressible magnetohydrodynamic equations in $\mathbb{R}^{3}$. When $(\gamma-1)^{\frac{1}{6}}E_{0}^{\frac{1}{2}}$, together with the $\|H_{0}\|_{L^{2}}$, is suitably small, a result on the existence of global classical solutions is obtained. It should be pointed out that the initial energy $E_{0}$ except the $L^{2}$- norm of $H_{0}$ can be large as $\gamma$ goes to 1, and that throughout the proof of the theorem in the present paper, we make no restriction upon the initial data $(\rho_{0},u_{0})$. Our result improves the one established by Li-Xu-Zhang in \citeH.L. L, where, with small initial engergy, the existence of classical solution was proved.
- Mar 11 2015 math.AP arXiv:1503.02910v3In this paper, we investigate the existence of a global classical solution to 3D Cauchy problem of the isentropic compressible Navier-Stokes equations with large initial data and vacuum. Precisely, when the far-field density is vacuum ($\widetilde{\rho}=0$), we get the global classical solution under the assumption that $(\gamma-1)^\frac{1}{3}E_0\mu^{-1}$ is suitably small. In the case that the far-field density is away from vacuum ($\widetilde{\rho}>0$), the global classical solution is also obtained when $\left((\gamma-1)^\frac{1}{36}+\widetilde{\rho}^\frac{1}{6}\right)E_0^{\frac{1}{4}}\mu^{-\frac{1}{3}}$ is suitably small. The above results show that the initial energy $E_0$ could be large if $\gamma-1$ and $\widetilde{\rho}$ are small or the viscosity coefficient $\mu$ is taken to be large. These results improve the one obtained by Huang-Li-Xin in \citeHuang-Li-Xin, where the existence of the classical solution is proved with small initial energy. It should be noted that in the theorems obtained in this paper, no smallness restriction is put upon the initial data. It can be viewed the first result on the existence of the global classical solution to three-dimensional Navier-Stokes equations with large initial energy and vacuum when $\gamma$ is near $1$.
- This work studies the Generalized Singular Value Thresholding (GSVT) operator ${\Prox}_{g}^{\bm{\sigma}}(\cdot)$, \beginequation* \Prox_g^\bm\sigma(\B)=\arg\min\limits_\X\sum_i=1^mg(\sigma_i(\X)) + \frac12||\X-\B||_F^2, \endequation* associated with a nonconvex function $g$ defined on the singular values of $\X$. We prove that GSVT can be obtained by performing the proximal operator of $g$ (denoted as $\Prox_g(\cdot)$) on the singular values since $\Prox_g(\cdot)$ is monotone when $g$ is lower bounded. If the nonconvex $g$ satisfies some conditions (many popular nonconvex surrogate functions, e.g., $\ell_p$-norm, $0<p<1$, of $\ell_0$-norm are special cases), a general solver to find $\Prox_g(b)$ is proposed for any $b\geq0$. GSVT greatly generalizes the known Singular Value Thresholding (SVT) which is a basic subroutine in many convex low rank minimization methods. We are able to solve the nonconvex low rank minimization problem by using GSVT in place of SVT.
- Aug 25 2014 math.CT arXiv:1408.5220v2We survey the general theory of groupoids, groupoid actions, groupoid principal bundles, and various kinds of morphisms between groupoids in the framework of categories with pretopology. We study extra assumptions on pretopologies that are needed for this theory. We check these extra assumptions in several categories with pretopologies. Functors between groupoids may be localised at equivalences in two ways. One uses spans of functors, the other bibundles (commuting actions) of groupoids. We show that both approaches give equivalent bicategories. Another type of groupoid morphisms, called actors, are closely related to functors between the categories of groupoid actions. We also generalise actors using bibundles, and show that this gives another bicategory of groupoids.
- We consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with continuously varying (weak) symplectic structures. Assuming vanishing index, we obtain intrinsically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions we define the curve's Maslov index by symplectic reduction to the classical finite-dimensional case. We prove the transitivity of repeated symplectic reductions and obtain the invariance of the Maslov index under symplectic reduction, while recovering all the standard properties of the Maslov index. As an application, we consider curves of elliptic operators which have varying principal symbol, varying maximal domain and are not necessarily of Dirac type. For this class of operator curves, we derive a desuspension spectral flow formula for varying well-posed boundary conditions on manifolds with boundary and obtain the splitting of the spectral flow on partitioned manifolds.
- Mar 27 2014 math.AP arXiv:1403.6595v1In this paper, we are concerned with the compressible Euler-Maxwell equations with a nonconstant background density (e.g. of ions) in three dimensional space. There exist stationary solutions when the background density is a small perturbation of a positive constant state. We first show the asymptotic stability of solutions to the Cauchy problem near the stationary state provided that the initial perturbation is sufficiently small. Moreover the convergence rates are obtained by combining the $L^p$-$L^q$ estimates for the linearized equations with time-weighted estimate.
- This paper is concerned with the large-time behavior of solutions to the Cauchy problem on the two-fluid Euler-Maxwell system with collisions when initial data are around a constant equilibrium state. The main goal is the rigorous justification of diffusion phenomena in fluid plasma at the linear level. Precisely, motivated by the classical Darcy's law for the nonconductive fluid, we first give a heuristic derivation of the asymptotic equations of the Euler-Maxwell system in large time. It turns out that both the density and the magnetic field tend time-asymptotically to the diffusion equations with diffusive coefficients explicitly determined by given physical parameters. Then, in terms of the Fourier energy method, we analyze the linear dissipative structure of the system, which implies the almost exponential time-decay property of solutions over the high-frequency domain. The key part of the paper is the spectral analysis of the linearized system, exactly capturing the diffusive feature of solutions over the low-frequency domain. Finally, under some conditions on initial data, we show the convergence of the densities and the magnetic field to the corresponding linear diffusion waves with the rate $(1+t)^{-5/4}$ in $L^2$ norm and also the convergence of the velocities and the electric field to the corresponding asymptotic profiles given in the sense of the geneneralized Darcy's law with the faster rate $(1+t)^{-7/4}$ in $L^2$ norm. Thus, this work can be also regarded as the mathematical proof of the Darcy's law in the context of collisional fluid plasma.
- Jan 02 2014 math.RA arXiv:1401.0330v3Let $A$ be a Koszul Artin-Schelter regular algebra and $\sigma$ an algebra homomorphism from $A$ to $M_{2\times 2}(A)$. We compute the Nakayama automorphisms of a trimmed double Ore extension $A_P[y_1, y_2; \sigma]$ (introduced in \citeZZ08). Using a similar method, we also obtain the Nakayama automorphism of a skew polynomial extension $A[t; \theta]$, where $\theta$ is a graded algebra automorphism of $A$. These lead to a characterization of the Calabi-Yau property of $A_P[y_1, y_2; \sigma]$, the skew Laurent extension $A[t^{\pm 1}; \theta]$ and $A[y_1^{\pm 1}, y_2^{\pm 1}; \sigma]$ with $\sigma$ a diagonal type.
- We study the stochastic solution to a Cauchy problem for a degenerate parabolic equation arising from option pricing. When the diffusion coefficient of the underlying price process is locally Hölder continuous with exponent $\delta\in (0, 1]$, the stochastic solution, which represents the price of a European option, is shown to be a classical solution to the Cauchy problem. This improves the standard requirement $\delta\ge 1/2$. Uniqueness results, including a Feynman-Kac formula and a comparison theorem, are established without assuming the usual linear growth condition on the diffusion coefficient. When the stochastic solution is not smooth, it is characterized as the limit of an approximating smooth stochastic solutions. In deriving the main results, we discover a new, probabilistic proof of Kotani's criterion for martingality of a one-dimensional diffusion in natural scale.
- May 09 2013 math.RA arXiv:1305.1748v2In this paper, we study the Poisson (co)homology of a Frobenius Poisson algebra. More precisely, we show that there exists a duality between the Poisson homology and the Poisson cohomology, similar to the duality between the Hochschild homology and the Hochschild cohomology of a Frobenius algebra. Using the non-degenerated bilinear form on a Frobenius algebra we construct a Batalin-Vilkovisky structure on the Poisson cohomology ring of a class of Frobenius Poisson algebras.
- Feb 18 2013 math.RT arXiv:1302.3744v3Let $(G,\tilde{G})$ be a reductive dual pair over a local field ${\Fontauri k}$ of characteristic 0, and denote by $V$ and $\tilde{V}$ the standard modules of $G$ and $\tilde{G}$, respectively. Consider the set $Max Hom(V,\tilde{V})$ of full rank elements in $Hom(V,\tilde{V})$, and the nilpotent orbit correspondence $\mathcal{O} \subset \mathfrak{g}$ and $\Theta (\mathcal{O})\subset \tilde{\mathfrak{g}}$ induced by elements of $Max Hom(V,\tilde{V})$ via the moment maps. Let $(\pi,\mathscr{V})$ be a smooth irreducible representation of $G$. We show that there is a correspondence of the generalized Whittaker models of $\pi $ of type $\mathcal{O}$ and of $\Theta (\pi)$ of type $\Theta (\mathcal{O})$, where $\Theta (\pi)$ is the full theta lift of $\pi $. When $(G,\tilde{G})$ is in the stable range with $G$ the smaller member, every nilpotent orbit $\mathcal{O} \subset \mathfrak{g}$ is in the image of the moment map from $Max Hom (V,\tilde{V})$. In this case, and for ${\Fontauri k}$ non-Archimedean, the result has been previously obtained by M\oeglin in a different approach.
- We recall the Chernoff-Marsden definition of weak symplectic structure and give a rigorous treatment of the functional analysis and geometry of weak symplectic Banach spaces. We define the Maslov index of a continuous path of Fredholm pairs of Lagrangian subspaces in continuously varying Banach spaces. We derive basic properties of this Maslov index and emphasize the new features appearing.
- Dec 26 2012 math.RT arXiv:1212.6015v1Determination of quasi-invariant generalized functions is important for a variety of problems in representation theory, notably character theory and restriction problems. In this note, we review some new and easy-to-use techniques to show vanishing of quasi-invariant generalized functions, developed in the recent work of the authors (Uniqueness of Ginzburg-Rallis models: the Archimedean case, Trans. Amer. Math. Soc. 363, (2011), 2763-2802). The first two techniques involve geometric notions attached to submanifolds, which we call metrical properness and unipotent $\chi$-incompatibility. The third one is analytic in nature, and it arises from the first occurrence phenomenon in Howe correspondence. We also highlight how these techniques quickly lead to two well-known uniqueness results, on trilinear forms and Whittaker models.
- Oct 19 2012 math.AP arXiv:1210.5101v1In this paper, we study the relationship between a diffusive model and a non-diffusive model which are both derived from the well-known Keller-Segel model, as a coefficient of diffusion $\varepsilon$ goes to zero. First, we establish the global well-posedness of classical solutions to the Cauchy problem for the diffusive model with smooth initial data which is of small $L^2$ norm, together with some \it a priori estimates uniform for $t$ and $\varepsilon$. Then we investigate the zero-diffusion limit, and get the global well-posedness of classical solutions to the Cauchy problem for the non-diffusive model. Finally, we derive the convergence rate of the diffusive model toward the non-diffusive model. It is shown that the convergence rate in $L^\infty$ norm is of the order $O(\varepsilon^{1/2})$. It should be noted that the initial data is small in $L^2$-norm but can be of large oscillations with constant state at far field. As a byproduct, we improve the corresponding result on the well-posedness of the non-difussive model which requires small oscillations.
- This paper investigates a singular stochastic control problem for a multi-dimensional regime-switching diffusion process confined in an unbounded domain. The objective is to maximize the total expected discounted rewards from exerting the singular control. Such a formulation stems from application areas such as optimal harvesting multiple species and optimal dividends payments schemes in random environments. With the aid of weak dynamic programming principle and an exponential transformation, we characterize the value function to be the unique constrained viscosity solution of a certain system of coupled nonlinear quasi-variational inequalities. Several examples are analyzed in details to demonstrate the main results.
- Supporting massive device transmission is challenging in Machine-to-Machine (M2M) communications. Particularly, in event-driven M2M communications, a large number of devices activate within a short period of time, which in turn causes high radio congestions and severe access delay. To address this issue, we propose a Fast Adaptive S-ALOHA (FASA) scheme for random access control of M2M communication systems with bursty traffic. Instead of the observation in a single slot, the statistics of consecutive idle and collision slots are used in FASA to accelerate the tracking process of network status which is critical for optimizing S-ALOHA systems. Using drift analysis, we design the FASA scheme such that the estimate of the backlogged devices converges fast to the true value. Furthermore, by examining the $T$-slot drifts, we prove that the proposed FASA scheme is stable as long as the average arrival rate is smaller than $e^{-1}$, in the sense that the Markov Chain derived from the scheme is geometrically ergodic. Simulation results demonstrate that the proposed FASA scheme outperforms traditional additive schemes such as PB-ALOHA and achieves near-optimal performance in reducing access delay. Moreover, compared to multiplicative schemes, FASA shows its robustness under heavy traffic load in addition to better delay performance.
- Aug 07 2012 math.AP arXiv:1208.1199v2We consider the Cauchy problem for the full compressible Navier-Stokes equations with vanishing of density at infinity in R3. Our main purpose is to prove the existence (and uniqueness) of global strong and classical solutions and study the large-time behavior of the solutions as well as the decay rates in time. Our main results show that the strong solution exists globally in time if the initial mass is small for the fixed coefficients of viscosity and heat conduction, and can be large for the large coefficients of viscosity and heat conduction. Moreover, large-time behavior and a surprisingly exponential decay rate of the strong solution are obtained. Finally, we show that the global strong solution can become classical if the initial data is more regular. Note that the assumptions on the initial density do not exclude that the initial density may vanish in a subset of R3 and that it can be of a non trivially compact support.To our knowledge, this paper contains the first result so far for the global existence of solutions to the full compressible Navier-Stokes equations when density vanishes at infinity (in space). In addition, the exponential decay rate of the strong solution is of independent interest.
- Jul 24 2012 math.RT arXiv:1207.5308v2The main purpose of this article is to supplement the authors' results on degenerate principal series representations of real symplectic groups with the analogous results for metaplectic groups. The basic theme, as in the previous case, is that their structures are anticipated by certain natural subrepresentations constructed from Howe correspondence. This supplement is necessary as these representations play a key role in understanding the basic structure of Howe correspondence (and its complications in the archimedean case), and their global counterparts play an equally essential part in the proof of Siegel-Weil formula and its generalizations (work of Kudla-Rallis). The full results in the metaplectic case also shed light on the seeming peculiarities, when the results in the symplectic case are viewed in their isolation.
- Due to the speed limitation of the conventional bit-chosen strategy in the existing weighted bit flipping algorithms, a high-speed LDPC decoder cannot be realized. To solve this problem, we propose a fast weighted bit flipping (FWBF) algorithm. Specifically, based on the stochastic error bitmap of the received vector, a partially parallel bit-choose strategy is adopted to lower the delay of choosing the bit flipped. Because of its partially parallel structure, the novel strategy can be well incorporated into the LDPC decoder [1]. The analysis of the decoding delay demonstrates that, the decoding speed can be greatly improved by adopting the proposed FWBF algorithm. Further, simulation results verify the validity of the proposed algorithm.
- The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of \pi_2 for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated to each finite-dimensional Lie algebra. In infinite dimensions, there is an obstruction for a central extension of Lie algebras to integrate to a central extension of Lie groups. This obstruction comes from non-trivial \pi_2 for general Lie groups. We show that this obstruction may be overcome by integrating central extensions of Lie algebras not to Lie groups but to central extensions of étale Lie 2-groups. As an application, we obtain a generalization of Lie's Third Theorem to infinite-dimensional Lie algebras.
- Apr 16 2012 math.RT arXiv:1204.2969v3We prove Kudla-Rallis conjecture on first occurrences of local theta correspondence, for all type I irreducible dual pairs and all local fields of characteristic zero.
- Machine-to-Machine (M2M) communication is now playing a market-changing role in a wide range of business world. However, in event-driven M2M communications, a large number of devices activate within a short period of time, which in turn causes high radio congestions and severe access delay. To address this issue, we propose a Fast Adaptive S-ALOHA (FASA) scheme for M2M communication systems with bursty traffic. The statistics of consecutive idle and collision slots, rather than the observation in a single slot, are used in FASA to accelerate the tracking process of network status. Furthermore, the fast convergence property of FASA is guaranteed by using drift analysis. Simulation results demonstrate that the proposed FASA scheme achieves near-optimal performance in reducing access delay, which outperforms that of traditional additive schemes such as PB-ALOHA. Moreover, compared to multiplicative schemes, FASA shows its robustness even under heavy traffic load in addition to better delay performance.
- It is well-known that a Lie algebroid A is equivalently described by a degree 1 Q-manifold M. We study distributions on M, giving a characterization in terms of A. We show that involutive Q-invariant distributions on M correspond bijectively to IM-foliations on A (the infinitesimal version of Mackenzie's ideal systems). We perform reduction by such distributions, and investigate how they arise from non-strict actions of strict Lie 2-algebras on M.
- Nov 14 2011 math.AP arXiv:1111.2657v2In the paper, we establish a blow-up criterion in terms of the integrability of the density for strong solutions to the Cauchy problem of compressible isentropic Navier-Stokes equations in \mathbbR^3 with vacuum, under the assumptions on the coefficients of viscosity: \frac29\mu3>\lambda. This extends the corresponding results in [20, 36] where a blow-up criterion in terms of the upper bound of the density was obtained under the condition 7\mu>\lambda. As a byproduct, the restriction 7\mu>\lambda in [12, 37] is relaxed to \frac29\mu3>\lambda for the full compressible Navier-Stokes equations by giving a new proof of Lemma 3.1. Besides, we get a blow-up criterion in terms of the upper bound of the density and the temperature for strong solutions to the Cauchy problem of the full compressible Navier-Stokes equations in \mathbbR^3. The appearance of vacuum could be allowed. This extends the corresponding results in [37] where a blow-up criterion in terms of the upper bound of (\rho,\frac1\rho, \theta) was obtained without vacuum. The effective viscous flux plays a very important role in the proofs.
- This paper develops numerical methods for finding optimal dividend pay-out and reinsurance policies. A generalized singular control formulation of surplus and discounted payoff function are introduced, where the surplus is modeled by a regime-switching process subject to both regular and singular controls. To approximate the value function and optimal controls, Markov chain approximation techniques are used to construct a discrete-time controlled Markov chain with two components. The proofs of the convergence of the approximation sequence to the surplus process and the value function are given. Examples of proportional and excess-of-loss reinsurance are presented to illustrate the applicability of the numerical methods.
- Oct 31 2011 math.AP arXiv:1110.6248v1In this paper, we study the asymptotic behavior of solutions to a Gas-liquid model with external forces and general pressure law. Under some suitable assumptions on the initial date and $\gamma>1$, if $\theta\in(0,\frac{\gamma}{2}]\cap(0,\gamma-1]\cap(0,1-\alpha\gamma]$, we prove the weak solution $(cQ(x,t),u(x,t))$ behavior asymptotically to the stationary one by adapting and modifying the technique of weighted estimates. In addition, if $\theta\in(0,\frac{\gamma}{2}]\cap(0,\gamma-1)\cap(0,1-\alpha\gamma]$, following the same idea in \citeFang-Zhang4, we estimate the stabilization rate of the solution as time tends to infinity in the sense of $L^\infty$ norm.
- Oct 31 2011 math.AP arXiv:1110.6247v2In this paper, we investigate large amplitude solutions to a system of conservation laws which is transformed, by a change of variable, from the well-known Keller-Segel model describing cell (bacteria) movement toward the concentration gradient of the chemical that is consumed by the cells. For the Cauchy problem and initial-boundary value problem, the global unique solvability is proved based on the energy method. In particular, our main purpose is to investigate the convergence rates as the diffusion parameter $\epsilon$ goes to zero. It is shown that the convergence rates in $L^\infty$-norm are of the order $(\epsilon)$ and $O(\epsilon^{3/4})$ corresponding to the Cauchy problem and the initial-boundary value problem respectively.
- Sep 27 2011 math.AP arXiv:1109.5328v2First of all, we get the global existence of classical and strong solutions of the full compressible Navier-Stokes equations in three space dimensions with initial data which is large and spherically or cylindrically symmetric. The appearance of vacuum is allowed. In particular, if the initial data is spherically symmetric, the space dimension can be taken not less than two. The analysis is based on some delicate \it a priori estimates globally in time which depend on the assumption $\kappa=O(1+\theta^q)$ where $q>r$ ($r$ can be zero), which relaxes the condition $q\ge2+2r$ in [14,29,42]. This could be viewed as an extensive work of [18] where the equations hold in the sense of distributions in the set where the density is positive with initial data which is large, discontinuous, and spherically or cylindrically symmetric in three space dimension. Finally, with the assumptions that vacuum may appear and that the solutions are not necessarily symmetric, we establish a blow-up criterion in terms of $\|\rho\|_{L^\infty_tL_x^\infty}$ and $\|\rho\theta\|_{L^4_tL^(12/5)_x}$ for strong solutions.
- Given a strict Lie 2-algebra, we can integrate it to a strict Lie 2-group by integrating the corresponding Lie algebra crossed module. On the other hand, the integration procedure of Getzler and Henriques will also produce a 2-group. In this paper, we show that these two integration results are Morita equivalent. As an application, we integrate a non-strict morphism between Lie algebra crossed modules to a generalized morphism between their corresponding Lie group crossed modules.
- In this paper, we study Lie 2-bialgebras, with special attention to coboundary ones, with the help of the cohomology theory of $L_\infty$-algebras with coefficients in $L_\infty$-modules. We construct examples of strict Lie 2-bialgebras from left-symmetric algebras and symplectic Lie algebras.
- Rational Bézier functions are widely used as mapping functions in surface reparameterization, finite element analysis, image warping and morphing. The injectivity (one-to-one property) of a mapping function is typically necessary for these applications. Toric Bézier patches are generalizations of classical patches (triangular, tensor product) which are defined on the convex hull of a set of integer lattice points. We give a geometric condition on the control points that we show is equivalent to the injectivity of every 2D toric Bézier patch with those control points for all possible choices of weights. This condition refines that of Craciun, et al., which only implied injectivity on the interior of a patch.
- This paper considers the optimal dividend payment problem in piecewise-deterministic compound Poisson risk models. The objective is to maximize the expected discounted dividend payout up to the time of ruin. We provide a comparative study in this general framework of both restricted and unrestricted payment schemes, which were only previously treated separately in certain special cases of risk models in the literature. In the case of restricted payment scheme, the value function is shown to be a classical solution of the corresponding HJB equation, which in turn leads to an optimal restricted payment policy known as the threshold strategy. In the case of unrestricted payment scheme, by solving the associated integro-differential quasi-variational inequality, we obtain the value function as well as an optimal unrestricted dividend payment scheme known as the barrier strategy. When claim sizes are exponentially distributed, we provide easily verifiable conditions under which the threshold and barrier strategies are optimal restricted and unrestricted dividend payment policies, respectively. The main results are illustrated with several examples, including a new example concerning regressive growth rates.
- This paper examines the objective of optimally harvesting a single species in a stochastic environment. This problem has previously been analyzed in Alvarez (2000) using dynamic programming techniques and, due to the natural payoff structure of the price rate function (the price decreases as the population increases), no optimal harvesting policy exists. This paper establishes a relaxed formulation of the harvesting model in such a manner that existence of an optimal relaxed harvesting policy can not only be proven but also identified. The analysis embeds the harvesting problem in an infinite-dimensional linear program over a space of occupation measures in which the initial position enters as a parameter and then analyzes an auxiliary problem having fewer constraints. In this manner upper bounds are determined for the optimal value (with the given initial position); these bounds depend on the relation of the initial population size to a specific target size. The more interesting case occurs when the initial population exceeds this target size; a new argument is required to obtain a sharp upper bound. Though the initial population size only enters as a parameter, the value is determined in a closed-form functional expression of this parameter.
- Apr 08 2011 math.AP arXiv:1104.1271v1We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index $s\in \R$ over the whole space $\R^n$ for any spatial dimension $n\geq 1$. Here, the diffusion-type source term behaves as the usual diffusion term over the low frequency domain while it admits on the high frequency part a feature of regularity-gain and regularity-loss for $s< 1$ and $s>1$, respectively. For all $s\in \R$, we not only obtain the $L^p$-$L^q$ time-decay estimates on the linear solution semigroup but also establish the global existence and optimal time-decay rates of small-amplitude classical solutions to the nonlinear Cauchy problem. In the case of regularity-loss, the time-weighted energy method is introduced to overcome the weakly dissipative property of the equation. Moreover, the large-time behavior of solutions asymptotically tending to the heat diffusion waves is also studied. The current results have general applications to several concrete models arising from physics.
- We study the extension of a Lie algebroid by a representation up to homotopy, including semidirect products of a Lie algebroid with such representations. The extension results in a higher Lie algebroid. We give exact Courant algebroids and string Lie 2-algebras as examples of such extensions. We then apply this to obtain a Lie 2-groupoid integration of an exact Courant algebroid.
- Mar 09 2011 math.AP arXiv:1103.1429v3In this paper, we obtain a result on the existence and uniqueness of global spherically symmetric classical solutions to the compressible isentropic Navier-Stokes equations with vacuum in a bounded domain or exterior domain \Omega of Rn(n >= 2). Here, the initial data could be large. Besides, the regularities of the solutions are better than those obtained in [H.J. Choe and H. Kim, Math. Methods Appl. Sci., 28 (2005), pp. 1-28; Y. Cho and H. Kim, Manuscripta Math., 120 (2006), pp. 91-129; S.J. Ding, H.Y.Wen, and C.J. Zhu, J. Differential Equations, 251 (2011), pp. 1696-1725]. The analysis is based on some new mathematical techniques and some new useful energy estimates. This is an extension of the work of Choe and Kim, Cho and Kim, and Ding, Wen, and Zhu, where the global radially symmetric strong solutions, the local classical solutions in three dimensions, and the global classical solutions in one dimension were obtained, respectively. This paper can be viewed as the first result on the existence of global classical solutions with large initial data and vacuum in higher dimension
- Mar 09 2011 math.AP arXiv:1103.1418v1In this paper, using Euler's function, we give a formula of all integral solutions to linear indeterminate equation with $s$-variables $a_1x_1+a_2x_2+...+a_sx_s=n$. It is a explicit formula of the coefficients $a_1$, $a_2$,..., $a_s$ and the free term $n$.
- Mar 09 2011 math.AP arXiv:1103.1421v1In this paper, we consider the 1D Navier-Stokes equations for viscous compressible and heat conducting fluids (i.e., the full Navier-Stokes equations). We get a unique global classical solution to the equations with large initial data and vacuum. Because of the strong nonlinearity and degeneration of the equations brought by the temperature equation and by vanishing of density (i.e., appearance of vacuum) respectively, to our best knowledge, there are only two results until now about global existence of solutions to the full Navier-Stokes equations with special pressure, viscosity and heat conductivity when vacuum appears (see \citeFeireisl-book where the viscosity $ \mu=$const and the so-called \em variational solutions were obtained, and see \citeBresch-Desjardins where the viscosity $ \mu=\mu(\rho)$ degenerated when the density vanishes and the global weak solutions were got). It is open whether the global strong or classical solutions exist. By applying our ideas which were used in our former paper \citeDing-Wen-Zhu to get $H^3-$estimates of $u$ and $\theta$ (see Lemma \refnon-le:3.14, Lemma \refnon-le:3.15, Lemma \refnon-rle:3.12 and the corresponding corollaries), we get the existence and uniqueness of the global classical solutions (see Theorem \refnon-rth:1.1).
- Feb 18 2011 math.AP arXiv:1102.3518v1In this paper, we consider two classes of free boundary value problems of a viscous two-phase liquid-gas model relevant to the flow in wells and pipelines with mass-dependent viscosity coefficient. The liquid is treated as an incompressible fluid whereas the gas is assumed to be polytropic. We obtain the asymptotic behavior and decay rates of the mass functions $n(x,t)$,\$m(x,t)$ when the initial masses are assumed to be connected to vacuum both discontinuously and continuously, which improves the corresponding result about Navier-Stokes equations in \citeZhu.
- Over a closed manifold, we consider the sectorial projection of an elliptic pseudo-differential operator A of positive order with two rays of minimal growth. We show that it depends continuously on A when the space of pseudo-differential operators is equipped with a certain topology which we explicitly describe. Our main application deals with a continuous curve of arbitrary first order linear elliptic differential operators over a compact manifold with boundary. Under the additional assumption of the weak inner unique continuation property, we derive the continuity of a related curve of Calderon projections and hence of the Cauchy data spaces of the original operator curve. In the Appendix, we describe a topological obstruction against a verbatim use of R. Seeley's original argument for the complex powers, which was seemingly overlooked in previous studies of the sectorial projection.
- We consider a simple instance of action up to homotopy. More precisely, we consider strict actions of DGLAs in degrees -1 and 0 on degree 1 NQ-manifolds. In a more conventional language this means: strict actions of Lie algebra crossed modules on Lie algebroids. When the action is strict, we show that it integrates to group actions in the categories of Lie algebroids and Lie groupoids (i.e. actions of LA-groups and 2-groups). We perform the integration in the framework of Mackenzie's doubles.