Past Talks 歷史演講

Discussion 1, A.M. 10:00-11:00 Discussion 2, A.M. 11:00-12:00 Discussion 3, P.M. 15:00-16:00 Discussion 4, P.M. 16:00-17:00

We develop an enclosure-type method to identify penetrable and impenetrable obstacles in some anisotropic medium in R^3. The main difficulty in treating this problem lies in the fact that there are so far no complex geometrical opticssolutions available for the PDEs. Instead, we derive and use another type special solutions called the oscillating-decaying solutions. To justify the scheme, we can use the Meyers' L^p estimates for such PDEs, to compare the integrals coming from oscillating-decayingsolutions and those from the reflected solutions.

Abstract: For a given graph G, the minimum rank problem is to find the smallest possible rank among the real symmetric matrices whose off-diagonal i,j-entry is nonzero if ij is an edge and zero otherwise. The diagonal entries can be any real number. In this talk we will discuss some parameters related to the minimum rank problem and their relations. Also, a new parameter will be introduced as an lower bound for the minimum rank.

Abstract: Robertson and Seymour proved that graphs are well-quasi-ordered by the minor relation and the weak immersion relation. In other words, given infinitely many graphs, one graph contains another as a minor (or a weak immersion, respectively). Unlike the relation of minor and weak immersion, the topological minor relation does not well-quasi-order graphs in general. However, Robertson conjectured in the late 1980’s that for every positive integer k, the topological minor relation well-quasi-orders graphs that do not contain a topological minor isomorphic to the path of length k with each edge duplicated. This conjecture was proved by Thomas and I. We will sketch the idea of the proof in this talk. In addition, we will give a structure theorem for excluding a fixed graph as a topological minor. Such structure theorems were previously obtained by Grohe and Marx and by Dvorak, but we push one of the bounds in their theorems to the optimal value. This improvement is needed for our proof of Robertson’s conjecture.

Abstract: In this talk we will give a brief proof of Ito's formula. Some general forms of Ito's formula will also be given. At last, we will give some simple applications of Ito's formula.

Abstract: In this talk, we study the thermal fluid in the nozzle and the influence of the wall friction according to the shape of the nozzle. We construct the global transonic solutions to the Fanno-Rayleigh equations, which is one of the most important models in the study of the smaller engines where the heating and friction can not be ignored.

Abstract: We develop an enclosure-type method to identify penetrable and impenetrable obstacles in some anisotropic medium in R^3. The main difficulty in treating this problem lies in the fact that there are so far no complex geometrical opticssolutions available for the PDEs. Instead, we derive and use another type special solutions called the oscillating-decaying solutions. To justify the scheme, we can use the Meyers' L^p estimates for such PDEs, to compare the integrals coming from oscillating-decayingsolutions and those from the reflected solutions.

Abstract: In this talk, I will present Jimbo and Morita’s work on a two-component system of reaction-diffusion equations with mass conservation in a bounded domain with the Neumann boundary conditions. They prove the global boundedness of the solution in L^\infty-norm and the existence of a Lyapunov function. Moreover, by studying the linearized eigenvalue problem of a non-constant equilibrium solution, they provide a comparison theorem for the spectrum between the linearized operators of the system and an appropriate nonlocal scalar equation. As an application of the comparison result they obtain that any stable equilibrium solution must be monotone if the space dimension is one. Is is also shown that a modified system with a new parameter, which covers the present model, possesses a Lyapunov function.

Abstract: Compressed sensing deals with the problem of representing sparse signals in high dimensions using a small number of linear measurements. Algorithms such as basis pursuit have been extensively studied and are known to provide stable and robust methods for recovering signals from their compressed sensing measurements. An important practical consequence of robust recovery results is that compressed sensing performs well in the presence of quantization (i.e., analog-to-digital conversion). We study the class of sigma-delta algorithms as a quantization method for compressed sensing and prove error bounds to quantify the improvement over standard scalar quantization. This involves the use of Sobolev dual frames associated to Gaussian random matrices. This is joint work with Sinan Gunturk, Mark Lammers, Rayan Saab, and Ozgur Yilmaz.

Abstract: In this talk, I will present Jimbo and Morita’s work on a two-component system of reaction-diffusion equations with mass conservation in a bounded domain with the Neumann boundary conditions. They prove the global boundedness of the solution in L^\infty-norm and the existence of a Lyapunov function. Moreover, by studying the linearized eigenvalue problem of a non-constant equilibrium solution, they provide a comparison theorem for the spectrum between the linearized operators of the system and an appropriate nonlocal scalar equation. As an application of the comparison result they obtain that any stable equilibrium solution must be monotone if the space dimension is one. Is is also shown that a modified system with a new parameter, which covers the present model, possesses a Lyapunov function.

Abstract: In this talk, we will present the following results. Necessary and sufficient conditions for the decomposition: K_n↪αP_(n-1)+βS_3, The maximum (P_k,S_k )-packing and the minimum (P_k,S_k )-covering of 〖λK〗_n. Necessary and sufficient conditions for the spiders to be t-decomposable. Some necessary conditions and some sufficient conditions for the star forest to be antimagic. Necessary and sufficient conditions for mS_2∪S_n to be antimagic.

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Abstract: In this talk, we used mathematical modeling to some problems. Ergonomics provides human factors and includes related material in psychology, cognitive science, physiology, biomechanics, job design, human-computer interaction, safety science, human engineering, occupational health, sports science, and transportation studies.

摘要： In this seminar series, we plan to study the basic theory of stochastic differential equations. We will mainly follow the lecture note by Prevot-Rochner (2007). The approach is to consider SPDE as a partial differential equation with the noise which is a Wiener process taking values in a Hilbert space or Banach space. This approach leads to the study of martingale theory in infinite dimensional spaces (Hilbert spaces or Banach spaces), and also the stochastic integrals with respect to the Hilbert space valued martingale. We will also have chance to compare different approaches by Walsh (white noise and martingale measure) and also Da Prato-Zabczyk (in Hairer lecture note). Last year, Shang-Yuan Shiu discussed Walsh’s approach. We hope to find chance to come back to this again this year. References 1. R. C. Dalang, D. Khoshnevisan, C. Mueller, D. Nualart and Y. Xiao (2008), A Mini-course on Stochastic Partial Differential Equation, Lcture Note in Mathematics, 1962 2. C. Prevot and M. Rochner (2007), A Concise Course on Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1905 3. Martin Hairer (2009), An Introduction to Stochastic PDEs 4. J.B. Walsh (1986), An Introduction to Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1108

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: The talk will discuss most recent development on the existence and stability of two- and three-dimensional solitary waves on the surface of water of finite depth with or without surface tension using the exact governing equations. It is well-known that the motion of surface waves of this problem is determined by two non-dimensional constants: non-dimensional wave speed F and surface tension B. It will be shown that the exact governing equations have many different kinds of two- and three-dimensional waves when (F, B) is in different parameter regions. Moreover, various stability results for these waves will be addressed, such as transverse stability, spectral stability, asymptotic linear stability or conditional stability.

Abstract: First we present new gravitational field equations, derived under 1) the Einstein principle of general relativity (PGR), and 2) the principle of interaction dynamics (PID), which is required and is a direct consequence of the presence of dark matter and dark energy. With new field equations, we show that dark energy and dark matter are simply a property of gravity. Then we present a blackhole theorem and a theorem on the structure of the Universe, proved in a recently published paper. The blackhole theorem states that assume the validity of the Einstein theory of general relativity, then black holes are closed, innate and incompressible. For the Universe, assume the Einstein theory of general relativity, and the principle of cosmological principle that the Universe is homogeneous and isotropic, then the Universe is static, not originated from a Big-Bang, and the topological structure of the Universe is a 3D sphere. These two theorems are rigorously proved using astrophysical dynamical models coupling fluid dynamics and general relativity based on a symmetry-breaking principle. With the new blackhole theorem, we further demonstrate that both supernovae explosion and AGN jets, as well as many astronomical phenomena, are due to combined relativistic, magnetic and thermal effects: the radial temperature gradient causes vertical Benard type convection cells, and the relativistic viscous force (via electromagnetic, the weak and the strong interactions) gives rise to a huge explosive radial force near the Schwarzschild radius, leading e.g. to supernovae explosion and AGN jets. This is joint with Tian Ma.

Abstract: In this talk I will present results from recent experiments conducted inside the MetaForum building of Eindhoven University of Technology in order to study the dynamics of low-density crowds. Our experimental technique, that will be briefly discussed, allows the recording with high spatial accuracy and at relatively high frequency of the individual positions of pedestrians within crowds. The months-long recording allowed us to obtain unprecedented statistics from which the fair tails of velocity probability distribution functions could be computed and not only the fluctuations around the averages. A model based on a generalised Langevin equation, similar to the ones employed to describe the dynamics of active colloidal particles, will be discussed and compared against experimental data.

Abstract: In this talk, I will introduce some biology problem, for examples: population dynamics, species competition, and DNA pattern searching. These problem can described by mathematics. In the end also I will introduce my recent study about chemotaxis.

摘要： We consider stochastic heat equations [SHEs] on the interval with Dirichlet boundary condition or Neumann boundary condition. Kim and Khoshnevisan prove that the growth of the energy is approximately exponential to the noise level to some power and then Foodun and Joseph improve few of their results and more delicate results are discussed in the paper of Foodun and Nualart.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: In this talk, I will introduce some biology problem, for examples: population dynamics, species competition, and DNA pattern searching. These problem can described by mathematics. In the end also I will introduce my recent study about chemotaxis.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要： In this seminar series, we plan to study the basic theory of stochastic differential equations. We will mainly follow the lecture note by Prevot-Rochner (2007). The approach is to consider SPDE as a partial differential equation with the noise which is a Wiener process taking values in a Hilbert space or Banach space. This approach leads to the study of martingale theory in infinite dimensional spaces (Hilbert spaces or Banach spaces), and also the stochastic integrals with respect to the Hilbert space valued martingale. We will also have chance to compare different approaches by Walsh (white noise and martingale measure) and also Da Prato-Zabczyk (in Hairer lecture note). Last year, Shang-Yuan Shiu discussed Walsh’s approach. We hope to find chance to come back to this again this year. References 1. R. C. Dalang, D. Khoshnevisan, C. Mueller, D. Nualart and Y. Xiao (2008), A Mini-course on Stochastic Partial Differential Equation, Lcture Note in Mathematics, 1962 2. C. Prevot and M. Rochner (2007), A Concise Course on Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1905 3. Martin Hairer (2009), An Introduction to Stochastic PDEs 4. J.B. Walsh (1986), An Introduction to Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1108

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Abstract: The hydrodynamic escape problem (HEP) is significant on the study of the evolution of planetary atmospheres. In this talk, I would like to introduction the physical phenomenon of this problem, and explain the reason why the three comparable size and mass planets, Venus, Earth, and Mars, in our solar system have different composition of atmosphere. Meanwhile, I will briefly explain our mathematical work on this problem, and introduce some future work relating to it.

摘要： We consider stochastic heat equations [SHEs] on the interval with Dirichlet boundary condition or Neumann boundary condition. Kim and Khoshnevisan prove that the growth of the energy is approximately exponential to the noise level to some power and then Foodun and Joseph improve few of their results. In this talk, we consider SHEs on the real line instead of a finite interval and use a different way to get similar results.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: The modern data sets often ''live'' in very high dimensional spaces with heavy noise of a lot of uncertainty. To deal with such problem, a manifold assumption is an attracting starting point. Based on the manifold assumption, we are interested in understanding the geometric and topological properties of the given data set. We will discuss graph connection Laplacian, a recently proposed generalization of the traditional graph Laplacian framework, and how it is related to the principal bundle structure encoded in the manifold. The synchronization property will also be discussed. While noise is inevitable, how the proposed algorithm is robust to noise will be discussed. Applications to the phase retrieval problem and puzzle solving problem will be discussed if time permits.

Abstract: The modern data sets often ''live'' in very high dimensional spaces with heavy noise of a lot of uncertainty. To deal with such problem, a manifold assumption is an attracting starting point. Based on the manifold assumption, we are interested in understanding the geometric and topological properties of the given data set. We will discuss graph connection Laplacian, a recently proposed generalization of the traditional graph Laplacian framework, and how it is related to the principal bundle structure encoded in the manifold. The synchronization property will also be discussed. While noise is inevitable, how the proposed algorithm is robust to noise will be discussed. Applications to the phase retrieval problem and puzzle solving problem will be discussed if time permits.

摘要： In this seminar series, we plan to study the basic theory of stochastic differential equations. We will mainly follow the lecture note by Prevot-Rochner (2007). The approach is to consider SPDE as a partial differential equation with the noise which is a Wiener process taking values in a Hilbert space or Banach space. This approach leads to the study of martingale theory in infinite dimensional spaces (Hilbert spaces or Banach spaces), and also the stochastic integrals with respect to the Hilbert space valued martingale. We will also have chance to compare different approaches by Walsh (white noise and martingale measure) and also Da Prato-Zabczyk (in Hairer lecture note). Last year, Shang-Yuan Shiu discussed Walsh’s approach. We hope to find chance to come back to this again this year. References 1. R. C. Dalang, D. Khoshnevisan, C. Mueller, D. Nualart and Y. Xiao (2008), A Mini-course on Stochastic Partial Differential Equation, Lcture Note in Mathematics, 1962 2. C. Prevot and M. Rochner (2007), A Concise Course on Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1905 3. Martin Hairer (2009), An Introduction to Stochastic PDEs 4. J.B. Walsh (1986), An Introduction to Stochastic Partial Differential Equations, Lecture Note in Mathematics, 1108

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract In this talk, I will discuss neuronal responses in the brain including the consideration of weak noise input, and the mathematical tools used to handle it. This study is based on a simple low-dimensional and yet realistic neuron model which has become more popular.

Abstract Let [image: \Phi] be a bijection (not assumed affine nor continuous) from the normal state space of one quantum system, modelled on the self-adjoint part of a von Neumann algebra [image: M_1], to that of another one, modelled on another von Neumann algebra [image: M_2].As a ``generalising'' of Wigner's theorem, we show that there exists a Jordan [image: ^*]-isomorphism [image: \Theta:M_2\to M_1] with[image: \Phi =\Theta_*|_{\mathbb S(M_1)}] if and only if [image: \Phi] preserves a notion of ``asymmetric transition probabilities''. If [image: \Phi] is assumed to preserve either the Raggio or the Uhlmann transition probabilities instead, then [image: M_1] and [image: M_2] are Jordan [image: *]-isomorphic.On the other hand, in the spirit of Uhlhorn's theorem, we show that if [image: \Phi] is biorthogonality preserving, i.e. zero transition probability preserving, then [image: M_1] and [image: M_2] are Jordan [image: ^*]-isomorphic. As an application, we show that the sets of normal states equipped with some metrics are complete Jordan invariants for the underlying von Neumann algebras.

Abstract: As semiconductor devices had entered in the era of deep nanometer, what designed on the circuit layout was no longer what printed on the wafer. Design for manufacturability (DFM) started to catch the attention of the industry almost two decades ago. Computational lithography is a key area in DFM. General concepts and fundamental theories of computational lithography will be introduced. A glimpse of why this field is considered math intensive will be presented.

Abstract: Typically, the device of a microelectromechanical system (MEMS) consists of an electric membrane hanged above a rigid ground plate, connected in series with fixed voltage source and a fixed capacitor. As the potential difference applied between the membrane and the plate causes the membrane to be deformed. In this talk, I will describe how we model the deformation of the membrane to get a hyperbolic equation with damping. I also will provide some criteria for quenching and global existence of the solution. Finally, I would like to introduce some unsolved problem relating to this model.

摘要： Monetary utility functions are -- except for the expected value -- not of von Neumann-Morgenstern type. In case the utility function has convex level sets in the set of probability measures on the real line, we can give some characterization that comes close to the vN-M form. For coherent utility functions this was solved by Ziegel. The concave case under the extra assumptions of weak compactness, was solved by Stephan Weber. Using the fact that law determined utility functions are monotone with respect to convex ordering, we could give a complete characterization. The characterization is similar to Weber's theorem except that we need concave functions that take the value $-\infty$. Having convex level sets can be seen as a weakened form of the independence axiom in the vN-M theory. It is a property that is related to elicitability as used in statistics. (joint work with Bellini, Bignozzi and Ziegel, paper accepted in Finance and Stochastics)

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

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Abstract Maximum likelihood estimator: Sample mean and PCA stein’s phenomenon and shrinkage in high dimensional statistics random projection and compressed sensing extended PCAs: robust PCA, sparse PCA, and MDS with uncertainty geometry and low rank structures: manifold learning

摘要： We consider stochastic heat equations [SHEs] on the interval with Dirichlet boundary condition or Neumann boundary condition. Kim and Khoshnevisan prove that the growth of the energy is approximately exponential to the noise level to some power and then Foodun and Joseph improve few of their results. In this talk, we consider SHEs on the real line instead of a finite interval and use a different way to get similar results.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

詳見附件

詳如附件

摘要： We consider stochastic heat equations [SHEs] on the interval with Dirichlet boundary condition or Neumann boundary condition. Kim and Khoshnevisan prove that the growth of the energy is approximately exponential to the noise level to some power and then Foodun and Joseph improve few of their results. In this talk, we consider SHEs on the real line instead of a finite interval and use a different way to get similar results.

摘要： In this talk, we propose an intuitive and computationally simple algorithm for clustering the probability density functions (pdfs). A data-driven learning mechanism is incorporated in the algorithm in order to determine the suitable widths of the clusters. The clustering results prove that the proposed algorithm is able to automatically group the pdfs and provide the optimal cluster number without any a priori information. The performance study also shows that the proposed algorithm is more efficient than existing ones. In addition, the clustering can serve as the intermediate compression tool in content-based multimedia retrieval that we apply the proposed algorithm to categorize a subset of COREL image database. And the clustering results indicate that the proposed algorithm performs well in color image categorization.

Abstract: I would like share my previous industrial experience, which is related to the hottest filed nowadays -- the big data, applied in the practical way. Most of them link with the Internet marketing. I will also give a brief survey about the current Internet marketing (of course, in my knowledge), online/offline advertising, what the roles the mathematicians play in the filed, and how mathematics involves in. Finally, I will point out the various directions to which the graduate students could develop forward after graduation.

摘要： Suppose a diffusion is drifted by some stationary incompressible flow. We say that the diffusivity is enhanced by the flow, if the action of the drift speeds up the convergence of the diffusion toward its equilibrium. Some vectorfields might have the ability to enhance diffusivity to infinity if one multiplies them with some growing parameter. In 2008 Constantin, Kislev, Ryzhik and Zlatos gave some condition for this phenomenon to take lace. The showed that this happens if the vectorfield (seen as a first order differential operator) has no H^1 eigenfunctions. In particular this is the case when the flow generated by the vectorfield is weakly mixing. In this talk we will describe the speed of this effect under the assumption that he flow satisfies decay of correlations.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: Random log-trees are trees with an average height of logarithmic order. They are of key importance, e.g., in computer science since many algorithms and data structures exhibit a log-tree behavior. Performance measures of these algorithms and data structures correspond to parameters of log-trees describing their shape. These so-called shape parameters have been extensively studied for many families of log-trees. One of them are $m$-ary search trees for which, e.g., the limit law of the size has been proved to exhibit a phase change from normal ($m\leq 26$) to non-existence ($m>26$). On the other hand, the limit law of the total path length does not undergo a phase change. In this talk, we consider the correlation between, e.g., the size and total path length which exhibits some surprises and further phase changes. Correlation between other shape parameters and other random log-trees will be also discussed. This talk is based on recent joint work with H.-H. Chern (National Taiwan Ocean University), H.-K. Hwang (Academia Sinica) and R. Neininger (Goethe University).

詳見附件

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要： We consider stochastic heat equations on the interval with Dirichlet boundary condition or Neumann boundary condition. Kim and Khoshnevisan prove that the growth of the energy is approximately exponential to the noise level to some power and then Foodun and Joseph improve few of their results. This talk bases on their papers.

詳見附檔

Abstract: In this talk, I will introduce some basic ideas in the calculus of variation such as the derivation of Euler-Lagrange equations and the direct method through examples in classical mechanics, electromagnetism and thermodynamics. In particular, I will introduce the gradient theory of phase transition in this talk.

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詳見附檔

摘要： We consider stochastic heat equations on the interval with Dirichlet boundary condition or Neumann boundary condition. Kim and Khoshnevisan prove that the growth of the energy is approximately exponential to the noise level to some power and then Foodun and Joseph improve few of their results. This talk bases on their papers.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: In this talk , I will introduce a parallel domain decomposition method for general shape optimization problems constrained by the incompressible Navier–Stokes equations. The optimization problem is first discretized with a finite element method on an unstructured moving mesh that is implicitly defined without assuming that the computational domain is known and then solved by some one-shot Lagrange–Newton–Krylov–Schwarz algorithms. In this approach, the shape of the domain, its corresponding finite element mesh, the flow fields and their corresponding Lagrange multipliers are all obtained computationally in a single solve of a nonlinear system of equations. Highly scalable parallel algorithms are absolutely necessary to solve such an expensive system. Aiming for machines with a large number of processors and robust nonlinear convergence, we introduce a two-level inexact Newton method with a hybrid two-level overlapping Schwarz preconditioner for shape optimization problems. As applications, we consider the shape optimization of a cannula problem and an artery bypass problem in 2D. Numerical experiments show that our algorithm performs well on a supercomputer with over 1000 processors for problems with millions of unknowns.

Abstract : In this work, we consider a three species diffusive food web model with one basal resource and two predators feeding by the same resource. It is well known that these two predators cannot coexist due to the Competitive Exclusion Principle. However, there are three non-trivial boundary equilibria. One the the saturated state of the prey without any predator $E_K$. Another two equilibria, $E_1$ and $E_2$, are the coexistence of the prey and one predator, respectively. With a generic assumption, $E_2$ is globally asymptotically stable in the corresponding kinetic equation. Biologically, the assumption means one predator is superior in the resource exploitation. By a high dimensional shooting method, the Wazewski' principle, we show the existence of traveling wave solutions from $E_K$ to $E_2$ and $E_1$ to $E_2$. Finally, some numerical simulations are presented and some biological interpretations are given.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: Noninvasive phase-contrast magnetic resonance imaging (PC-MRI) has been used to derive several reliable hemodynamic parameters. In this talk, two topics based on the technique of PC-MRI for quantifying cardiovascular function of Marfan syndrome (MFS) patients will be presented. The first topic will be focused on the analysis of blood flow hemodynamic quantification. In particular, the index of wall shear stress (WSS), a dragging force of flowing blood acting on vascular wall, will be discussed. The analyzed regional WSS along the aorta of MFS patients will be compared with normal controls so as to establish potential indices to predict specific sites in aorta with high risk of dissection or rupture. With dark-blood preparation, PC-MRI can be used to evaluate the motion velocity of myocardium. Thus, the second topic of this talk will be focused on the technique of tissue phase mapping (TPM). Since myocardium is also consisted of elastic fibers, a scheme to evaluate myocardial function in MFS may also play an important role on surveillance of the progress of MFS. The three-directional wall motion velocities and time-to-peak (TTP) will be computed. The indices will be compared with age-matched normal controls to show the myocardial abnormality of MFS.

Abstract: A simple criterion for determining rank deficit of matrices is proposed. It is derived by reconsidering the theoretical rank deficit from a view point of multiple precision. Its numerical computation is performed in multiple precision. Singular value decomposition(SVD), Gaussian elimination with complete pivoting(GECP) and GECP with equilibration (GECPE) are compared. Numerical results show that in developing a numerical method its practical application should be taken into consideration.

摘要： We consider the stochastic fractional heat equations [SFHE]. We like to show that when the spatial variables are far apart, the solutions at those points are “independence.” This helps to find the right scale of the chaotic behaviors of the SFHE.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: 隻身前往紐西蘭，印尼創業，分別打下第二大的網路印刷平台與第一的餐廳訂位。我準備了什麼？不管發生什麼事情，我都已經準備好，就是為了變化快速的未來做好準備。

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