I am going to talk about an aggregation model of marine particles in VIGRE graduate student seminar this Friday. This is one part of joint work with Adrian Burd. I’ll talk about the moments for log-normal distribution, its application on size distribution of marine particles, the multi-modes model, and some results from numerical experiments. Here is our Slides for the talk. By the way, the picture by Adrian in the slides, which I named as “sunrise or sunset” , is actually sunset, he took it from the Gulf of Mexico.

As a fan of the Phoenix Mars Lander, I recently paid close attention to the news about it since it successfully landed around the north pole of the Mars. So far the good news from Mars to human beings are the finding of iced water, friendly soil environment for some plants, and other things.

The course of development in space technology is a great history. From Newton’s invention of infinitesimal calculus in mathematics in 1665 to his three laws of motion; from the first manned aeroplane by Wright Brothers in 1903, the theoretic foundation of which is the lift equation which basically says the lift force is directly proportional to the square of velocity of object and the area of its wings, , and had appeared in Newton’s “The Mathematical Principles of Natural Philosophy”, to rockets launching and sending satellites to space; from the first manned landing on the moon in 1969 to the Phoenix Mars Lander nowadays, and so on, what great strides human have made.

Minkowski space is a vector space with Minkowski norm. The space of geodesics in a Minkowski space are affines lines, which inherits a canonical symplectic structure from the cotangent bundle of . The significant Holmes-Thompson Volumes can be derived from this symplectic structure in integral geometry way.

In MATLAB, one can obtain solutions to differential equations in symbolic or numerical manner. For simple first order ODEs, one can use the command “dsolve” to solve solvable equations analytically. In particular, it can be used to obtain anti-derivatives. Numerically, one can assign numerical initial conditions, using “ode15″ for some stiff ODEs, where the stiffness is determined by the Jacobian of the ODE system, or other similar commands, depending on the type of equation. The steps usually includes defining functions/equations in M-files, setting numerical domains and step sizes for basic variables and initial conditions, applying suitable solver codes, and graphing the numerical results, and so forth.

For instance, one can build a model for simulating the coagulation of paticles in aqua-environment. By the central limit theorem in classic probability theory, the distribution of particles in different sizes can be considered as  a log-normal distribution, or a composition of log-normal distributions

    (1)

where is the diameter of particle in ocean environment, and are parameters of different modes of log-normal distributions. By the theory from particle physics, the interaction of different particles has a coagulation kernel, such as Brownian kernel

     (2)

for two particles with diameters and . Now let us consider a single-mode model based on Koziol-Leighton’s result on aerosol dynamics, as the following ODE for this modeling,

where and likely for . Given initial data to , we can solve out by solver “ode23s” or “ode15s”, and obtain a simple simulation for particles coagulation.

A strong earthquake hit eastern Sichuan of China six days ago. Thousands of lives were lost in the quake, many of them were elementary or high school students under seventeen. Let’s pray with Dion’s song “The Prayer” for the people suffered from the disaster. Please lend a helping hand, donations can be made at UNICEF or Red Cross.

I was at a conference on geometric group theory, geometric analysis, and mapping class group in the Japan-U.S. Mathematics Institute (JAMI) of JHU at Baltimore last weekend. JAMI with a history of about twenty years holds conference every year for promoting communications and cooperations of American mathematicians and Japanese mathematical society.

Mapping class group , the group of orientation-preserving self-homeomorphisms of a topological space, becomes another interesting algebraic object in recent decades besides homology groups as algebraic invariants for topological spaces. Elements in mapping class group induce actions on homology groups of a space, the kernel called Torelli group is important for understanding mapping class group because of the following exact sequence

    (1)

where is the symplectic group of degree over , and is a genus surface. There are many interesting investigations on and the so-called Johnson homomorphism that gives the free part of the abelianization of .

Geodesic is the minimizer of the energy functional on Riemannian manifold. What is more, in fact, geodesic flow is a Hamiltonian flow, and it has a relationship with natural symplectic structure on the tangent bundle of Riemannian manifold.

Let be a -dimensional Riemannian manifold with metric , , be a curve in starting at with initial velocity , then

.

Writing in terms of coordinates , , and , and then we have

(1)

by .

On the other hand, for vertical space, we have

for all .

Here is a direct way to see that a geodesic flow is a Hamiltonian flow of energy functional.

In terms of coordinates, we know that for geodesic vector field

on by (1), we have

Thus, let be the natural symplectic form on , we then have the following proposition.

Proposition 1. .

There are some relationship between the energy functional and natural differential forms on Riemannian manifold. Let us consider the case first, it means for all by (1) if . There is a natural involution between horizontal space and vertical space which is

Thus

On the other hand for , we have

Therefore for any .

Now let us consider the general case. For any , we have , analogous to the case , an involution we can have is

Therefore,

and

Thus we have obtained the following

Theorem 2. on the vertical bundle of , where E is the energy functional on Riemannian manifold and is the natural -form on TM whose differential is the natural symplectic form on .

Remark 3. In fact, since both sides are zeroes on the horizontal bundle of , therefore on .

Please go to Terry Tao’s Blog, and give your endorsement to his petition to help support maths at University of Southern Queensland (USQ) by Signing here.