While I know next to nothing about the philosophy of science, I would think about scientific models and explore whether there are features of these models that should be required in order to say they make a good theory or whether any model that predicts the data is as good as any other. From...
Pardon this completely naive question.
I am wondering why there are net magnetic fields on the Sun. Isn't there an equal number of protons - hydrogen nuclei - to electrons? Is there a separation of protons and electrons in different parts of the Sun so that there can be net currents? or are...
The magnetic field of a steady current in a loop is given by the Biot and savart integral which is
1/4pi Integral[((x-y)/|x-y|^3) x dy] = B(x)
What is the corresponding formula for the vector potential?
The Gauss curvature of a surface in R^3 is intrinsic i.e. it is an invariant of local isometry.
For a hyper-surface of R^n is this also true? By this I mean: The Gauss curvature is the determinant of the Gauss mapping of the surface into the unit sphere. Is the determinant of the Gauss mapping...
I question whether mathematics is not an empirical science. True it has the power of absolute proof which is denied to Physics and Biology but except for that maybe it is the same.
Mathematicians look at "empirical data" which for them are mathematical objects - say such as Riemann surfaces -...
Is there a theory of linked homotopy? I am thinking of homotopies of 3 space minus some other loops - where the loop is not allow to intersect itself during the homotopy.
This type of homotopy would preserve linking of 2 loops that have linking number zero.
Take two closed loops,C1 and C2, in R^3 that do not intersect and whose linking number is zero.
Chose two manifolds D1 and D2 whose boundaries are C1 and C2 and which intersect in their interiors transversally and do not intersect anywhere along their boundaries.
The intersection is a...
let A(x) be the area under the graph of 1/x from 1 to the number,x,where x is bigger than 1.
Can you show without using calculus or any of the properties of logarithms that A(xy) = A(x) + A(y)?
In continuous Markov processes there is an idea of sample path. Starting at a point we follow the values of the process through time and find that almost surely the paths are continuous. In Brownian motion (Wiener process) the paths are crinkly curves, continuous paths of infinite variation...
The Law of Biot and Savart Law tells us how to find a differential form that generates the first de Rham cohomology of S3- embedded loop. Run a steady current through the loop.
This form is just the dual of the induced magnetic field (using the Euclidean metric).
Ampere's Law tells us...
Take a tangent vector field with isolated singularities on a compact smooth Riemannian surface ( 2 dimensional manifold without boundary). Divide v by its norm to get a field of unit vectors with isolated discontinuities.
Around each singularity chose a small open disc. The tangent circle...
I have just started a book on GM. As motivation for the idea that gravity must affect the energy of light the following example is given.
A body of rest mass,M, falls in a gravitation field and bounces off of the ground after it has achieved velocity,v. At impact, it transforms completely...
An oriented surface with a Riemannian metric has a natural action of the unit circle on its tangent bundle. Rotate the tangent vector through the angle theta in the positively direction.
Is there a natural action of SO(n) on the tangent bundle of an oriented Riemannian n-manifold?
Same...
I know I should be able to look this up but am having trouble this morning.
I would like to know the Lie Algebra structure of the three sphere. In particular I'd like to express it in terms of the left invariant vector fields that are either tangent to the fibers of the Hopf fibration or...
In classical mechanics one can give space a metric whose geodesics are the paths of a particle in a given potential.
I wonder if this technique can be applied for gravitational potentials. The hard part would be defining the geometry in a relativistic setting (special) where measurement of...
Here is a famous problem for you to enjoy.
You are betting on the World Series and want to make bets in such a way that if the Yankees win the series you win exactly a dollar and if the Red Sox win the series you lose exactly a dollar.
You can bet what ever you want on each game. If the...
The Shroedinger equation defines the time evolution of the wave function. If we observe a region of large gravitational fields where observed time has slowed, the wave function will be observed to evolve slowly. In the limit of a Black Hole it will stop evolving altogether.
Still quantum...
The Klein bottle, though unorientable, is nevertheless the boundary of a three manifold. It is easy to see how to construct this 3 manifold.
The projective plane is not a boundary of any manifold. How can we visualize this impossibility?
Is there a nice embedding of the projective plane in...
I know zero about the characteristic classes of finite group representations and would appreciate a reference.
specifically, if I have a faithful representations of a finite group,G, in O(n) what can I say about the induced map on cohomology,
P*:H*(BO(n))-> H*(BG) ?
I am mostly interested in...
I am trying to understand why the collapse of the wave function can not be a solution to the Shroedinger equation.
Certainly there are systems that evolve into eigenstates. For instance, in a two state system with constant Hamiltonian there are initial conditions from which the amplitudes...