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    Models versus sufficient reason

    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...
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    Interpreting a vector expression

    In R^3 I have two vectors a and b and the operator D = (d/dx,d/dy,d/dz) What is the interpretation/ picture of (aD)b - (bD)a? This is another vector.
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    Magnetic fields on the Sun

    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...
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    The Law of Biot and Savart again

    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?
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    Extrinsic intrinsic

    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...
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    Mathematics as an empirical science

    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 -...
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    Linked homotopy

    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.
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    Linking of curves

    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...
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    Cross product of magnetic fields

    Is there any physical meaning to the cross product of two magnetic fields e.g. two fields generated in two different current loops?
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    Multiple current loops

    I assume that if one has several current loops that the magnetic fields that they generate just add together linearlly. Just want to make sure.
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    Area under 1/x

    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)?
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    Help with a Riemann surface

    I am having trouble describing the Riemann surface of log(z) + log(z-a)
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    Amplitude paths and Markov processes

    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...
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    Tidal forces

    Can someone briefly describe what is meant by tidal forces?
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    Current Loops and fundamental group

    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...
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    Question whether 2 spaces are diffeomorphic

    Suppose I have two embeddings of the circle into the 3 sphere. Is S3-minus the first image diffeomorphic to S3 - the image of the second?
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    Alexander duality

    does Alexander duality commute with cup product?
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    Gluing maps from vector fields

    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...
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    Beginner's question on GM

    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...
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    SO(n) actions on vector bundles

    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...
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    Two fold coverings

    if H^1(M;Z2) has a non-zero element then one can use this to construct a 2 fold cover of M. How does this work?
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    Lie Algebra of S3

    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...
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    General relativity done another way?

    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...
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    Cubular homology cup product

    I would like to know how to define the cup product in cubular homology theory.
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    For Your Enjoyment

    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...
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    The wave equation in a Black Hole.

    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...
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    The kelin bottle and the projective plane

    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...
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    Wild metric on the 2 disc

    Can you show me a metric on the 2 dimensional disc so wild that no open subset can be embedded in R^3?
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    Characteristic classes of finite group representations

    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...
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    Colapse of the Wave Funcion and the Schroedinger Equation

    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...