The infinity norm may be one of the usual models to study about in different situations in mathematics, not the continuous ones, but the discrete or singular ones.

We know that , for any , can be expressed as an integral on by spreading a measure on it, which is

(1)

for some constant because , where

(2)

satisfying and

(3)

satisfying are modified Dirac delta functions.

One might also be able to put an measure on for the complex norm , , which can be written in real norm as . A problem is what function would satisfy

(4)

if one parametrizes by Hopf coordinates

(5)

One can show that the function is actually some function independent of and by the invariance of the complex norm under action, so it can be denoted as . But it is unknown yet what exactly the function is, because the invariance yields that , but this doesn’t produce an appropriate function in the general expression for the complex infinity norm. If one looks back the case of real infinity norm, the way of expressing the infinity norm is by making use of the absolute values on sum and difference smartly to sift out the maximum from two number. But in the complex case the double integral on the torus

(6)

eliminates the feasibility of sifting out the maximal modulus.

Stokes’ theorem plays an important role in computing integrals of differential forms on manifolds, and its consequence, the Cauchy integral formula, is quite fundamental in complex analysis in one variable as well.

Suppose is a compact -dimensional submanifold with smooth boundary in the plane . From symplectic geometry, the natural symplectic form on can be written as , where . If one computes the integral,

(1)

where denotes the codisk bundle of , since is an one dimensional curve in , it makes the integral containing measuring the perturbations of base points of cotangent vectors in be zero. Hence each term of the right hand side of (1) is zero. So one has that , which is also true in general when is a compact hypersurface contained in some affine -plane in .

By the way, Gowers wrote a post, one way of looking at Cauchy theorem, which is very interesting and tells us that the Cauchy’s theorem is the natural generalization in -dimension of the statement is a constant for , that, we know, is a direct consequence of the second fundamnetal theorem of calculus which is the special case of the Stokes’ theorem in one dimension. So in this sense, Stokes’ theorem is the “generator” of all the theorems and statements of this kind.

I’ll 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 ice 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.

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 particles 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 similarly 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 attending 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 short 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 .

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.