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# Euler characteristic

## Definition

### In terms of Betti numbers

The Euler characteristic of a space with finitely generated homology $X$, denoted $\chi(X)$, is defined as a signed sum of its Betti numbers, viz.:

$\chi(X) = \sum_{k=0}^\infty (-1)^k b_k(X)$

where $b_k$ is the $k^{th}$ Betti number, i.e., the rank of the torsion-free part of the $k^{th}$ homology group of $X$.

Note that by assumption, all the $b_k(X)$ are finite and only finitely many of them are nonzero.

The Euler characteristic can take any integer value, including zero, positive, and negative integers.

### In terms of Poincare polynomial

The Euler characteristic of a space with finitely generated homology $X$, denoted $\chi(X)$, is defined as the value of its Poincare polynomial at the number $-1$.

### In terms of Lefschetz number

The Euler characteristic of a space with finitely generated homology $X$ is the Lefschetz number (also called Lefschetz trace) of the identity map from $X$ to itself.

### For a CW-complex with finitely many cells

For a CW-complex with finitely many cells, the Euler characteristic can be defined as:

$\sum_i (-1)^{d_i}$

where $i$ ranges over all cells and $d_i$ isthe dimension of the cell. In particular, if there are $c_j$ cells of dimension $j$ for each $j$, this becomes:

$\sum_j (-1)^j c_j$

By assumption, all the $c_j$ are finite and only finitely many of them are nonzero.

Note that corresponding CW-space is a space with finitely generated homology and the Euler characteristic of that topological space equals the Euler characteristic of the CW-complex. However, it is possible for a CW-complex with infinitely many cells (either infinitely many cells at a given dimension, or arbitrarily large dimensions) that still has finitely generated homology. In that case, the above definition of Euler characteristic of a CW-complex will not apply, but we can still compute the Euler characteristic as a topological space.

## Particular cases

Two particular numerical values of the Euler characteristic are of significance -- $0$ and $1$:

### Euler characteristics of some typical spaces

Space Value of Euler characteristic (may depend on a parameter if the space is not specific but describes a family with a parameter) Justification
contractible space 1 All homology groups except zeroth one are zero
acyclic space 1 (similar to above)
finite discrete space with $n$ points $n$ The zeroth homology is free of rank $n$.
circle 0 Compatible with explanation for spheres, also with explanation for connected Lie groups
underlying topological space of a nontrivial compact connected Lie group 0 Euler characteristic of nontrivial compact connected Lie group is zero
sphere $S^n$ $1 + (-1)^n$, which is 2 if $n$ is even, and 0 if $n$ is odd. See homology of spheres
torus $T^n$ 0 compatible with statement for Lie groups, also from fact about circles and Euler characteristic of product is product of Euler characteristics
compact orientable genus $g$ surface $2 - 2g$ follows from homology of compact orientable surfaces
real projective space $\R\mathbb{P}^n$, $n$ odd 0 compatible with fact that Euler characteristic of covering space is degree of covering times Euler characteristic of base, and the fact that the double cover $S^n$ has Euler characteristic zero.
real projective space $\R\mathbb{P}^n$, $n$ even 1 compatible with fact that Euler characteristic of covering space is degree of covering times Euler characteristic of base, and the fact that the double cover $S^n$ has Euler characteristic 2.

## Facts

### Effect of operations

Operation Arity Brief description Effect on Euler characteristics Proof/explanation
disjoint union 2 $X \sqcup Y$ is a set-theoretic disjoint union and open subsets of this are those whose intersection with each piece is open in that. Sum of Euler characteristics: $\chi(X \sqcup Y) = \chi(X) + \chi(Y)$ Euler characteristic of disjoint union is sum of Euler characteristics
disjoint union finite number $n$ $\bigsqcup_{i=1}^n X_i$ or $X_1 \sqcup X_2 \sqcup \dots \sqcup X_n$ is a set-theoretic disjoint union and open subsets of this are those whose intersection with each piece is open in that. Sum of Euler characteristics: $\chi(\bigsqcup_{i=1}^n X_i) = \sum_{i=1}^n \chi(X_i)$ Euler characteristic of disjoint union is sum of Euler characteristics
wedge sum 2 $X \vee Y$, defined in the context of a basepoint choice for both spaces, is the quotient of the disjoint union by identification of the two basepoints $\chi(X \vee Y) = \chi(X) + \chi(Y) - 1$
product of topological spaces 2 $X \times Y$, the Cartesian product, equipped with the product topology $\chi(X \times Y) = \chi(X)\chi(Y)$ Euler characteristic of product is product of Euler characteristics
product of topological spaces finite number $n$ $X_1 \times X_2 \times \dots \times X_n$, the Cartesian product, equipped with the product topology $\chi(X_1 \times X_2 \times \dots \times X_n)$ Euler characteristic of product is product of Euler characteristics

### Covering spaces and fibrations

There is a corresponding statement for fiber bundles and fibrations. Fill this in later


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Posts

## 14S. Different Approaches to an Intuitive Problem

The fourth problem of the second season of PEN is as follows:

N17. Suppose that $a$ and $b$ are distinct real numbers such that:
$a - b, a^{2}-b^{2}, \cdots, a^k-b^k, \cdots$ are all integers. Show that $a$ and $b$ are integers.

Here is the official solution file: pen14.pdf

You can discuss the problem in MathLinks here.

format: plain


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Definitions

# Expander mixing lemma

The expander mixing lemma states that, for any two subsets $S, T$ of a regular expander graph $G$, the number of edges between $S$ and $T$ is approximately what you would expect in a random d-regular graph, i.e. $d |S| cdot |T| / n$.

## Statement

Let $G = \left(V, E\right)$ be a d-regular graph with (un-)normalized second-largest eigenvalue $lambda$. Then for any two subsets $S, T subseteq V$, let $E\left(S, T\right)$ denote the number of edges between S and T. We have

$|E\left(S, T\right) - frac\left\{d |S| cdot |T|\right\}\left\{n\right\}| leq lambda sqrt$
>

For a proof, see link in references.

## Converse

Recently, Bilu and Linial showed that the converse holds as well: if a graph satisfies the conclusion of the expander mixing lemma, that is, for any two subsets $S, T subseteq V$,

$|E\left(S, T\right) - frac\left\{d |S| cdot |T\left\{n\right\}| leq alpha sqrt$
>

then its second-largest eigenvalue is $O\left(alpha\left(1+log\left(d/alpha\right)\right)\right)$.

## References

• Notes proving the expander mixing lemma.
• Expander mixing lemma converse.

Search another word or see Expander Mixing Lemmaon Dictionary | Thesaurus |Spanish


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# Skew Field/Examples

The set $\H$ of quaternions forms a skew field under the operations of addition and multiplication.


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## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$-2\le x \le \dfrac{10}{3}$
$\bf{\text{Solution Outline:}}$ To solve the given inequality, $|3x-2|\le8 ,$ use the definition of less than (less than or equal to) absolute value inequality. Then use the properties of inequality to isolate the variable. $\bf{\text{Solution Details:}}$ Since for any $c\gt0$, $|x|\lt c$ implies $-c\lt x\lt c$ (or $|x|\le c$ implies $-c\le x\le c$), the inequality above is equivalent to \begin{array}{l}\require{cancel} -8\le 3x-2 \le8 .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} -8\le 3x-2 \le8 \\\\ -8+2\le 3x-2+2 \le8+2 \\\\ -6\le 3x \le10 \\\\ -\dfrac{6}{3}\le \dfrac{3x}{3} \le \dfrac{10}{3} \\\\ -2\le x \le \dfrac{10}{3} .\end{array} Hence, the solution set $-2\le x \le \dfrac{10}{3} .$


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# Neumann boundary condition on a boundary inside the region

I want to solve the Laplace equation in 2 different medium in 2D, each with its own dielectric constant. This is the extremely simplified geometry. The problem is the same error (which you will find out in the end!) for my more complex geometry in 3D.

Needs["NDSolveFEM"]

bmesh = ToBoundaryMesh["Coordinates" -> {{0., 0.}, {.5, 0.}, {1., 0.},
{1., 1.}, {.5, 1.}, {0., 1.}},
"BoundaryElements" -> {LineElement[{{1,2}, {2, 3}, {3, 4}, {4, 5},
{5, 6}, {6, 1}, {2, 5}}]}];

bmesh["Wireframe"]

mesh = ToElementMesh[bmesh];
mesh["Wireframe"]

The result is:

You can see there is a boundary at x==0.5 and the mesh of the region is made considering that there is a boundary at x==0.5.

My attempt to solve Laplace equation with only Dirichlet boundary condition is as follows, note that I just modified the wolfram's FEM tutorial example for my problem:

\[Epsilon]r = If[x <= .5, {{1., 0.}, {0., 1.}}, {{10., 0.}, {0., 10.}}]
op = Inactive[Div][-\[Epsilon]r.Inactive[Grad][u[x, y], {x, y}], {x,y}]
\[CapitalGamma]D = {DirichletCondition[u[x, y] == 1., x == 0.],
DirichletCondition[u[x, y] == 2., x == 1.],
DirichletCondition[u[x, y] == .1, 0. <= x <= .5 && y == 0.],
DirichletCondition[u[x, y] == 0., 0.5 <= x <= 1. && y == 0.],
DirichletCondition[u[x, y] == 0., 0. <= x <= .5 && y == 1.],
DirichletCondition[u[x, y] == .1, .5 <= x <= 1 && y == 1.]}

\[CapitalGamma]N = NeumannValue[2, .45 < x < 0.55 ]; (*This is
simplified condition. I will explain the B.C. I have in mind at the
end.*)

ufun = NDSolveValue[{op == 0, \[CapitalGamma]D}, u, {x, y} \[Element] mesh]

The result without using the Neumann condition is:

Show[ContourPlot[ufun[x, y], {x, y} \[Element] mesh,
ColorFunction -> "TemperatureMap"], bmesh["Wireframe"]]

However, when I use Neumann boundary condition on the boundary at x==0.5 this happens:

ufun = NDSolveValue[{op == \[CapitalGamma]N, \[CapitalGamma]D}, u,
{x,y} \[Element] mesh]

(*NDSolveValue::bcnop: No places were found on the boundary where
x==0.5 was True, so NeumannValue[2,x==0.5] will effectively be
ignored.*)

Please note that the Neumann condition that I have in my mind for the boundary at x==0.5 is

\[Epsilon]r(*for x < 0.5*)*D[u[x,y],x] == \[Epsilon]r(*for x > 0.5*)*D[u[x,y],x]

and

D[u[x,y],y](*u[x,y] for left side of the region i.e. x < 0.5*) ==
D[u[x,y],y](*u[x,y] for right side of the region i.e. x > 0.5*)

I appreciate your help to resolve this issue and correctly implement the Neumann boundary condition at x==0.5.

P.S.

My attempt to solve the problem by solving for 2 function each with dirichlet boundary only on their own side also did not work. You can see that I made 3 types of attempts to set the Neumann conditions and non worked:

\[Epsilon]1 = 1;
\[Epsilon]2 = 2;

eq1 = Inactive[Laplacian][\[Epsilon]1 u1[x, y], {x, y}]
eq2 = Inactive[Laplacian][\[Epsilon]2 u2[x, y], {x, y}]

\[CapitalGamma]D2 = {DirichletCondition[u1[x, y] == 1., x == 0.],
DirichletCondition[u1[x, y] == 1., 0. < x < .5 && y == 0.],
DirichletCondition[u1[x, y] == 1., 0. < x < .5 && y == 1.],
DirichletCondition[u2[x, y] == 2., x == 1.],
DirichletCondition[u2[x, y] == 2., 0.5 < x < 1. && y == 0.],
DirichletCondition[u2[x, y] == 2., .5 < x < 1 && y == 1.]}

\[CapitalGamma]N21 = Inactive[NeumannValue][D[\[Epsilon]2 u2[x, y], x],
x == .5];(*type 1*)
\[CapitalGamma]N22 = Inactive[NeumannValue][D[\[Epsilon]1 u1[x, y], x],
x == .5];(*type 1*)

{u1fun, u2fun} = NDSolveValue[{eq1 == \[CapitalGamma]N21(*((\[Epsilon]2
D[u2[x,y],x])/.x\[Rule].5)(*type 2*)*),
eq2 == \[CapitalGamma]N22(*((\[Epsilon]1 D[u1[x,y],x])/.x\[Rule].5)
(*type 2*)*), \[CapitalGamma]D2
(*,((\[Epsilon]1 D[u1[x,y],x])/.x\[Rule].5)==((\[Epsilon]2
D[u2[x,y],x])/.x\[Rule].5)(*type 3*)*)}, {u1, u2}, {x, y}
\[Element]mesh]

Errors for type 1 (click on images to see them large):

and errors for type 2 (click on images to see them large):

and errors for type 3 (click on images to see them large):

[

• A few questions: 1) What is the direction of the normal of the inner boundary? 2) You expect the derivative at x+epsilon and x-epsilon to be the same but not generally at x<1/2 and x>1/2, right? – user21 Feb 5 '18 at 9:22
• Does this help you: xepsi = 10^-4; ((\[Epsilon]r*Derivative[1, 0][ufun][x, y]) /. {x -> (1/2 - xepsi), y -> 1/2}) ((\[Epsilon]r*Derivative[1, 0][ufun][x, y]) /. {x -> (1/2 + xepsi), y -> 1/2}) – user21 Feb 5 '18 at 9:22
• @user21, Numerically, yes. That is why I came up with double potential solution. One potential for left side where only boundary conditions of left side are used for it and vise versa for the right side (this is exactly how one can solve this electro magnetic problem analytically with one extra boundary condition which is potential will vanish at x->infinity for left side potential and it will vanish at x-> -infinity for the right side potential). In addition, to your second comment, the condition you are imposing works only fro one point, while I want it to be true for 0=<y=<1 – Navid Rajil Feb 5 '18 at 18:32
• Can you share a link to an example in a book, paper other FEM software where something like this has been done. I still do not quite understand what you are trying to do. – user21 Feb 8 '18 at 7:41
• @user21, Classical Electrodynamic by Jackson, look for dielectric sphere in constant electric field. – Navid Rajil Feb 8 '18 at 23:59

It turns out that, when you don't specify the internal Neumann boundary condition, NDSolve gives precisely the solution of your real problem (looks like a miracle !).

Lets look at that.

Here is the code. As you can see, there no internal boundary conditions. (I have refined the mesh)

 Needs["NDSolveFEM"]

bmesh = ToBoundaryMesh[
"Coordinates" -> {{0., 0.}, {.5, 0.}, {1., 0.}, {1., 1.}, {.5, 1.}, {0., 1.}},
"BoundaryElements" -> {LineElement[{{1,2}, {2, 3}, {3, 4}, {4, 5},{5, 6}, {6, 1}, {2, 5}}]}];

gr00=bmesh["Wireframe"];

mesh = ToElementMesh[bmesh,MaxCellMeasure-> 0.0001];
gr01=mesh["Wireframe"];

Row[{gr00,gr01}] //Style[#,ImageSizeMultipliers->.7 {1,1}]&

\[Epsilon]r = If[x <= .5, {{1., 0.}, {0., 1.}}, {{10., 0.}, {0., 10.}}];
op = Inactive[Div][-\[Epsilon]r.Inactive[Grad][u[x, y], {x, y}], {x,y}];
\[CapitalGamma]D = {DirichletCondition[u[x, y] == 1., x == 0.],DirichletCondition[u[x, y] == 2., x == 1.],
DirichletCondition[u[x, y] == .1, 0. <= x <= .5 && y == 0.],DirichletCondition[u[x, y] == 0., 0.5 <= x <= 1. && y == 0.],
DirichletCondition[u[x, y] == 0., 0. <= x <= .5 && y == 1.],DirichletCondition[u[x, y] == .1, .5 <= x <= 1 && y == 1.]};

\[CapitalGamma]N = NeumannValue[2, .45 < x < 0.55 ]; (*This is simplified condition. I will explain the B.C. I have in mind at the end.*)

ufun = NDSolveValue[{op == 0, \[CapitalGamma]D}, u, {x, y} \[Element] mesh];

Show[ContourPlot[ufun[x, y], {x, y} \[Element] mesh, ColorFunction -> "TemperatureMap"], bmesh["Wireframe"],ImageSize->200]

Plot[{ufun[.499, y]-ufun[.498, y], 10(ufun[.502, y]-ufun[.501, y])},{y,0,1},
PlotLegends-> {"\[Epsilon]r D[u[x,y],x] for x = 0.498","\[Epsilon]r D[u[x,y],x] for x = 0.501"},PlotStyle->{Red,Blue}]

Plot[{ufun[.499, y]-ufun[.499, y-0.01], (ufun[.501, y]-ufun[.501, y-0.01])},{y,0,1},
PlotLegends-> {" D[u[x,y],y] for x = 0.498"," D[u[x,y],y] for x = 0.501"},PlotStyle->{Red,Blue}]

Here are :
- epsilon X the normal derivative
- the tangential derivative
just before and after the boundary x == 0.5

• Thanks Andre. Now, that I see your answer (+1) I have heard the question of OP in different variants before. I think something like this should be in the documentation perhaps. What I fail to understand is why would one want to specify a Neumann value at the inner boundary. Why does one thing that this might be necessary? Do you have an idea how some can come to that conclusion? – user21 Feb 8 '18 at 20:22
• @user21 Yes, I have a very precise idea of what's happening. The OP wants to solve the Maxwell equations for electrostatics when the dielectic constant of the material suddenly jump (= is discontinuous). Maxwell says : 1) the tangential electric field E stays continuous 2) the normal of the electric field multiplied by epsilon stays continuous (equivalently D is continuous, where D=epsilon X E) – andre314 Feb 8 '18 at 20:30
• Ah, and from there the thought is that this needs to be enforced by a internal boundary condition. – user21 Feb 8 '18 at 20:35
• Yes, it seems to me logical. – andre314 Feb 8 '18 at 20:47
• Nevertheless I don't understand the "miracle" – andre314 Feb 8 '18 at 20:47


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You are currently browsing the monthly archive for December 2007.

Einstein’s equation $E=mc^2$ describing the equivalence of mass and energy is arguably the most famous equation in physics. But his beautifully elegant derivation of this formula (here is the English translation) from previously understood laws of physics is considerably less famous. (There is an amusing Far Side cartoon in this regard, with the punchline “squared away”, which you can find on-line by searching hard enough, though I will not link to it directly.)

This topic had come up in recent discussion on this blog, so I thought I would present Einstein’s derivation here. Actually, to be precise, in the paper mentioned above, Einstein uses the postulates of special relativity and other known laws of physics to show the following:

Proposition. (Mass-energy equivalence) If a body at rest emits a total energy of E while remaining at rest, then the mass of that body decreases by $E/c^2$.

Assuming that bodies at rest with zero mass necessarily have zero energy, this implies the famous formula $E = mc^2$ – but only for bodies which are at rest. For moving bodies, there is a similar formula, but one has to first decide what the correct definition of mass is for moving bodies; I will not discuss this issue here, but see for instance the Wikipedia entry on this topic.

Broadly speaking, the derivation of the above proposition proceeds via the following five steps:

1. Using the postulates of special relativity, determine how space and time coordinates transform under changes of reference frame (i.e. derive the Lorentz transformations).
2. Using 1., determine how the temporal frequency $\nu$ (and wave number k) of photons transform under changes of reference frame (i.e. derive the formulae for relativistic Doppler shift).
3. Using Planck’s relation $E = h\nu$ (and de Broglie’s law $p = \hbar k$) and 2., determine how the energy E (and momentum p) of photons transform under changes of reference frame.
4. Using the law of conservation of energy (and momentum) and 3., determine how the energy (and momentum) of bodies transform under changes of reference frame.
5. Comparing the results of 4. with the classical Newtonian approximations $KE \approx \frac{1}{2} m|v|^2$ (and $p \approx mv$), deduce the relativistic relationship between mass and energy for bodies at rest (and more generally between mass, velocity, energy, and momentum for moving bodies).

Actually, as it turns out, Einstein’s analysis for bodies at rest only needs to understand changes of reference frame at infinitesimally low velocity, $|v| \ll c$. However, in order to see enough relativistic effects to deduce the mass-energy equivalence, one needs to obtain formulae which are accurate to second order in v (or more precisely, $v/c$), as opposed to those in Newtonian physics which are accurate to first order in v. Also, to understand the relationship between mass, velocity, energy, and momentum for moving bodies rather than bodies at rest, one needs to consider non-infinitesimal changes of reference frame.

Important note: Einstein’s argument is, of course, a physical argument rather than a mathematical one. While I will use the language and formalism of pure mathematics here, it should be emphasised that I am not exactly giving a formal proof of the above Proposition in the sense of modern mathematics; these arguments are instead more like the classical proofs of Euclid, in that numerous “self evident” assumptions about space, time, velocity, etc. will be made along the way. (Indeed, there is a very strong analogy between Euclidean geometry and the Minkowskian geometry of special relativity.) One can of course make these assumptions more explicit, and this has been done in many other places, but I will avoid doing so here in order not to overly obscure Einstein’s original argument. Read the rest of this entry »

I’m continuing my series of articles for the Princeton Companion to Mathematics through the holiday season with my article on “Differential forms and integration“. This is my attempt to explain the concept of a differential form in differential geometry and several variable calculus; which I view as an extension of the concept of the signed integral in single variable calculus. I briefly touch on the important concept of de Rham cohomology, but mostly I stick to fundamentals. [Added, Feb 24 2018: Typo on page 5: “more any other” should be “more generally, any other”.  Thanks to Michael Drumheller for the correction.]

I would also like to highlight Doron Zeilberger‘s PCM article “Enumerative and Algebraic combinatorics“. This article describes the art of how to usefully count the number of objects of a given type exactly; this subject has a rather algebraic flavour to it, in contrast with asymptotic combinatorics, which is more concerned with computing the order of magnitude of number of objects in a class. The two subjects complement each other; for instance, in my own work, I have found enumerative and other algebraic methods tend to be useful for controlling “main terms” in a given expression, while asymptotic and other analytic methods tend to be good at controlling “error terms”.

I’m continuing my series of articles for the Princeton Companion to Mathematics with my article on phase space. This brief article, which overlaps to some extent with my article on the Schrödinger equation, introduces the concept of phase space, which is used to describe both the positions and momenta of a system in both classical and quantum mechanics, although in the latter one has to accept a certain amount of ambiguity (or non-commutativity, if one prefers) in this description thanks to the uncertainty principle. (Note that positions alone are not sufficient to fully characterise the state of a system; this observation essentially goes all the way back to Zeno with his arrow paradox.)

Phase space is also used in pure mathematics, where it is used to simultaneously describe position (or time) and frequency; thus the term “time-frequency analysis” is sometimes used to describe phase space-based methods in analysis. The counterpart of classical mechanics is then symplectic geometry and Hamiltonian ODE, while the counterpart of quantum mechanics is the theory of linear differential and pseudodifferential operators. The former is essentially the “high-frequency limit” of the latter; this can be made more precise using the techniques of microlocal analysis, semi-classical analysis, and geometric quantisation.

As usual, I will highlight another author’s PCM article in this post, this one being Frank Kelly‘s article “The mathematics of traffic in networks“, a subject which, as a resident of Los Angeles, I can relate to on a personal level :-) . Frank’s article also discusses in detail Braess’s paradox, which is the rather unintuitive fact that adding extra capacity to a network can sometimes increase the overall delay in the network, by inadvertently redirecting more traffic through bottlenecks! If nothing else, this paradox demonstrates that the mathematics of traffic is non-trivial.

Camil Muscalu, Christoph Thiele and I have just uploaded to the arXiv our joint paper, “Multi-linear multipliers associated to simplexes of arbitrary length“, submitted to Analysis & PDE. This paper grew out of our project from many years ago to attempt to prove the nonlinear (or “scattering”) version of Carleson’s theorem on the almost everywhere convergence of Fourier series. This version is still open; our original approach was to handle the nonlinear Carleson operator by multilinear expansions in terms of the potential function V, but while the first three terms of this expansion were well behaved, the fourth term was unfortunately divergent, due to the unhelpful location of a certain minus sign. [This survey by Michael Lacey, as well as this paper of ourselves, covers some of these topics.]

However, what we did find out in this paper was that if we modified the nonlinear Carleson operator slightly, by replacing the underlying Schrödinger equation by a more general AKNS system, then for “generic” choices of this system, the problem of the ill-placed minus sign goes away, and each term in the multilinear series is, in fact, convergent (though we did not yet verify that the series actually converged, though in view of the earlier work of Christ and Kiselev on this topic, this seems likely). The verification of this convergence (at least with regard to the scattering data, rather than the more difficult analysis of the eigenfunctions) is the main result of our current paper. It builds upon our earlier estimates of the bilinear term in the expansion (which we dubbed the “biest”, as a multilingual pun). The main new idea in our earlier paper was to decompose the relevant region of frequency space $\{ (\xi_1,\xi_2,\xi_3) \in {\Bbb R}^3: \xi_1 < \xi_2 < \xi_3 \}$ into more tractable regions, a typical one being the region in which $\xi_2$ was much closer to $\xi_1$ than to $\xi_3$. The contribution of each region can then be “parafactored” into a “paracomposition” of simpler operators, such as the bilinear Hilbert transform, which can be treated by standard time-frequency analysis methods. (Much as a paraproduct is a frequency-restricted version of a product, the paracompositions that arise here are frequency-restricted versions of composition.)

A similar analysis happens to work for the multilinear operators associated to the frequency region $S := \{ (\xi_1,\ldots,\xi_n): \xi_1 < \ldots < \xi_n \}$, but the combinatorics are more complicated; each of the component frequency regions has to be indexed by a tree (in a manner reminiscent of the well-separated pairs decomposition), and a certain key “weak Bessel inequality” becomes considerably more delicate. Our ultimate conclusion is that the multilinear operator

$T(V_1,\ldots,V_n) := \int_{(\xi_1,\ldots,\xi_n) \in S} \hat V_1(\xi_1) \ldots \hat V_n(\xi_n) e^{2i (\xi_1+\ldots+\xi_n) x}\ d\xi_1 \ldots d\xi_n$ (1)

(which generalises the bilinear Hilbert transform and the biest) obeys Hölder-type $L^p$ estimates (note that Hölder’s inequality related to the situation in which the (projective) simplex S is replaced by the entire frequency space ${\Bbb R}^n$).

For the remainder of this post, I thought I would describe the “nonlinear Carleson theorem” conjecture, which is still one of my favourite open problems, being an excellent benchmark for measuring progress in the (still nascent) field of “nonlinear Fourier analysis“, while also being of interest in its own right in scattering and spectral theory.

Next quarter, starting on Wednesday January 9, I will be teaching a graduate course entitled “Topics in Ergodic Theory“. As an experiment, I have decided to post my lecture notes on this blog as the course progresses, as it seems to be a good medium to encourage feedback and corrections. (On the other hand, I expect that my frequency of posting on non-ergodic theory topics is going to go down substantially during this quarter.) All of my class posts will be prefaced with the course number, 254A, and will be placed in their own special category.

The topics I plan to cover include

• Topological dynamics;
• Classical ergodic theorems;
• The Furstenberg-Zimmer structure theory of measure preserving systems;
• Multiple recurrence theorems, and the connections with Szemerédi-type theorems;
• Orbits in homogeneous spaces (and in particular, in nilmanifolds);
• (Special cases of) Ratner’s theorem, and applications to number theory (e.g. the Oppenheim conjecture).

If time allows I will cover some other topics in ergodic theory as well (I haven’t decided yet exactly which ones to discuss yet, and might be willing to entertain some suggestions in this regard.)

If this works out well then I plan to also do the same for my spring class, in which I will cover as much of Perelman’s proof of the Poincaré conjecture as I can manage. (Note though that this latter class will build upon a class on Ricci flow given by my colleague William Wylie in the winter quarter, which will thus be a de facto prerequisite for my spring course.)

This post will have only the most tangential connection to mathematics.

I am an Australian citizen (and permanent resident in the US), but I nevertheless take an interest in the upcoming US presidential election in 2008. I’ve recently learned of a grassroots campaign to have one of the presidential debates focused on policy issues in science, technology, health, and the environment. (See also this LA Times op-ed and Wall Street Journal op-ed.) Personally, I think this is an excellent idea, and hope that it succeeds; it seems that they are currently petitioning signatures towards this goal.

While on this topic, it is also interesting to see what the political prediction markets are currently forecasting as the outcome of the election…

[Update, Dec 20: Here is a table listing the major candidates and their positions on mostly science-related issues.]

Ciprian Demeter, Michael Lacey, Christoph Thiele and I have just uploaded our joint paper, “The Walsh model for $M_2^*$ Carleson” to the arXiv. This paper (which was recently accepted for publication in Revista Iberoamericana) establishes a simplified model for the key estimate (the “$M_2^*$ Carleson estimate”) in another (much longer) paper of ours on the return times theorem of Bourgain, in which the Fourier transform is replaced by its dyadic analogue, the Walsh-Fourier transform. This model estimate is established by the now-standard techniques of time-frequency analysis: one decomposes the expression to be estimated into a sum over tiles, and then uses combinatorial stopping time arguments into group the tiles into trees, and the trees into forests. One then uses (phase-space localised, and frequency-modulated) versions of classical Calderòn-Zygmund theory (or in this particular case, a certain maximal Fourier inequality of Bourgain) to control individual trees and forests, and sums up over the trees and forests using orthogonality methods (excluding an exceptional set if necessary).

Rather than discuss time-frequency analysis in detail here, I thought I would dwell instead on the return times theorem, and sketch how it is connected to the $M_2^*$ Carleson estimate; this is a more complicated version of the “$M_2$ Carleson estimate”, which is an estimate which is logically equivalent to Carleson’s famous theorem (and its extension by Hunt) on the almost everywhere convergence of Fourier series.

This is my final Milliman lecture, in which I talk about the sum-product phenomenon in arithmetic combinatorics, and some selected recent applications of this phenomenon to uniform distribution of exponentials, expander graphs, randomness extractors, and detecting (sieving) almost primes in group orbits, particularly as developed by Bourgain and his co-authors.
Read the rest of this entry »

This is my second Milliman lecture, in which I talk about recent applications of ideas from additive combinatorics (and in particular, from the inverse Littlewood-Offord problem) to the theory of discrete random matrices.
Read the rest of this entry »

This week I am visiting the University of Washington in Seattle, giving the Milliman Lecture Series for 2007-2008. My chosen theme here is “Recent developments in arithmetic combinatorics“. In my first lecture, I will speak (once again) on how methods in additive combinatorics have allowed us to detect additive patterns in the prime numbers, in particular discussing my joint work with Ben Green. In the second lecture I will discuss how additive combinatorics has made it possible to study the invertibility and spectral behaviour of random discrete matrices, in particular discussing my joint work with Van Vu; and in the third lecture I will discuss how sum-product estimates have recently led to progress in the theory of expanders relating to Lie groups, as well as to sieving over orbits of such groups, in particular presenting work of Jean Bourgain and his coauthors.


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### Session C: General Physics II

8:00 AM–10:15 AM, Saturday, April 1, 2006
Photonics Building Room: 205

Chair: Karl Ludwig, Boston University

Abstract ID: BAPS.2006.NES.C.8

### Abstract: C.00008 : Is a Brane' Electron Modeled by a Closed-loop String of Neutrinos, is time formally a periodic function, and can tornadoes be turned off''?

9:45 AM–10:00 AM

Preview Abstract MathJax On | Off   Abstract

#### Authors:

Roger McLeod
(University of Massachusetts, Lowel)

David McLeod
(Bastyr University)

Model a planar' electron by a closed string of vibrating neutrinos; displacement values are proportional to the speed of light times the square root of the mass. Spin' supplies required inward spherical fields in three spatial dimensions. Interlocked quark loops model protons or neutrons; ideas like strong and weak forces, or an uncertainty principle, arise. Subtle, longer string-like vibrating,' quasi-periodic, lighted phenomena we detect are at locations used by some of our Native American forebears, or by the Hopi or Maya -locations indicated by problematic constructions, by sacred' place-names, or by individuals with titles' identified as names. Lighted tubes,' associated with EMF, required by our model for tornado generation, imply breaking the EMF lines will kill' any tornado. `Kokopelli's hair,' is the place to construct a designated current loop.

To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2006.NES.C.8


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## An approach to data parametrization in parametric cubic spline interpolation problems.(English)Zbl 0558.41006

From the author’s summary: ”A new approach to the problem of parametrizing data in parametric cubic spline interpolation problems is discussed. Parametrizations $$0=t_ 0<t_ 1<...<t_ N=1$$ of K- dimensional data $$\{Z_ i\}^ N_{i=0}$$, $$Z_ i=(z^ 1_ i,...,z^ k_ i)$$ are chosen by minimizing $$\sum^{k}_{\ell =1}(1/\alpha \ell^ 2)\int^{1}_{0}(d^ 2\theta^{\ell}/dt^ 2)^ 2dt$$, where $$\theta^{\ell}(t)$$ is the natural cubic spline with broken points $$t_ 0$$, $$t_ 1,...,t_ N$$ satisfying $$\theta^{\ell}(t_ i)=z_ i^{\ell}$$, $$i=0,...,N$$ and $$\alpha_{\ell}$$, $$\ell =1,...,K$$, are positive numbers. This approach yields parametrizations which, by complementing the well-known smoothest interpolation property of natural cubic splines, leads to smoother component functions. The improvements are, in part, evidenced by reduced position overshoots and lower second derivatives. A closed form solution of the problem is derived for one- dimensional data. In higher dimensions the gradient projection method is used to obtain approximate numerical solutions. Geometric curve fitting problems and an example involving the design of a trajectory for a robot manipulator are used to illustrate the method.”
Reviewer: R.B.Saxena

### MSC:

 41A05 Interpolation in approximation theory 41A15 Spline approximation
Full Text:

### References:

 [1] () [2] Epstein, M.P, On the influence of parametrization in parametric interpolation, SIAM J. numer. anal., 13, 2, 261, (1976) · Zbl 0319.41005 [3] deBoor, C, () [4] Mullineux, G, Approximating shapes using parameterized curves, IMA J. appl. math., 29, 2, (1982) · Zbl 0507.68077 [5] Lin, C.-S; Chang, P.-R; Luh, J.Y.S, Formulation and optimization of cubic polynomial joint trajectories for mechanical manipulators, () [6] Bollenger, J; Duffie, N, Computer algorithms for high speed continuous-path robot manipulation, () [7] Edwall, C.W; Pottinger, H.J; Ho, C.Y, Trajectory generation and control of a robot arm using spline functions, () [8] deBoor, C; Lynch, R.E, On splines and their minimum properties, J. math. mech., 15, 6, (1966) · Zbl 0185.20501 [9] Luenberger, David G, Introduction to linear and nonlinear programming, (1973), Addison-Wesley Reading, Mass · Zbl 0297.90044 [10] Topfer, H.-J, Models for smooth curve Fitting, () [11] {\scP. W. Smith and K. Scherer}, “Parametric Curve Fitting via Splines with Free Knots.” unpublished manuscript.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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Wednesday, October 7, 2015

Inequality Problem

For $x,\,y,\,z>0$ and $xyz=1$, prove $\sqrt{x^2+1}+\sqrt{y^2+1}+\sqrt{z^2+1}\le\sqrt{2}(x+y+z)$.


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## The stochastic sandpile

At last week’s AIM conference on chip-firing I learned about a cool stochastic process which seems very simple, but whose behavior is apparently very hard to prove anything about!

It goes like this.  You start with a graph (say a finite connected graph, for simplicity) and your states are functions from the vertices of the graph to the natural numbers, which you think of as specifying the height of a pile of chips (or sand grains, if you want to keep up that metaphor) sitting on top of that vertex.  At least one of the vertices is a sink, which means that any chip landing there disappears forever.

Now the nature of the process is that you never want to have more than one chip at a vertex.  So every time there are at least two chips at a vertex v, you can pop two of them off; each chip flips a coin and decides whether to topple left or right.  And you keep on doing this until there’s at most one chip at each spot.  (You might start with more chips than vertices, but the sink helps you slough off the surplus.)  This process was apparently first studied by Manna and is sometimes called the Manna sandpile.

More math below the fold!

The first thing you might ask is:  how do you decide which order to execute the topples in?  And the beauty is that it doesn’t matter; this is an example of an abelian process where the end result is the same no matter how you order the moves.  (See Lionel Levine and Jim Propp’s excellent short survey about the abelian sandpile for more on this notion.)

The second thing you might ask is:  if you drop chips into the system at random for a long time, letting the system relax to stability after each drop, what is the stationary distribution on the configurations you get?

Let’s make things simple and take the graph to be a long path with a sink at either end.  In this case, just about everything I’m going to say below (and more!) can be found in the paper “Steady State of Stochastic Sandpile Models,” by Tridib Sadhu and Deepak Dhar.

Statistical physicists believe there’s a critical density for the stochastic sandpile, around 0.94.  If you start with a stable configuration which is below the critical density, and drop a chip, you may see a few local topples right near where you perturbed the system, but the topples will probably die out quickly, and the configuration will end up looking the same far away from where you dropped the chip.

But if you drop a chip on a configuration with supercritical density, what probably happens is that all hell breaks loose, topples propagate all over the place, and you end up slopping some chips into the sink.

So the configuration should “want” to stay pretty near the critical density.

None of this is proved, of course.

It’s pretty fun to play around with this!  One thing I found when running experiments was the following.  If you look at the configurations that typically arise, they are indeed pretty dense; so you can switch figure and ground and think of a typical configuration as a set of “holes” of density about 6% on the line.  And it looks in practice like the holes repel each other!  Is it easy to explain why this happens?  Can one prove it, even if one can’t compute the critical density?

Here’s another thing I like about this problem.  Suppose the graph is a path of N+2 vertices, with the first and last being sinks.  Now you can think of the set of probability distributions on configurations as a simplex sitting inside the space V of real-valued functions on the 2^N configurations.  And dropping a chip at position i, then relaxing, is nothing other than an operation on i, which we call x_i.  And the different operations commute!  What’s more, they obey a lot of relations.  If you drop two chips on i, you know you can topple those two chips; there’s a 1/4 chance they both topple left, a 1/4 chance they both topple right, and a 1/2 chance they topple in opposite directions.  In other words,

$x_i^2 = (1/4) x_{i-1}^2 + (1/2) x_{i-1} x_{i+1} + (1/4) x_{i+1}^2$.

Or, more succinctly,

$(x_i - \frac{1}{2}(x_{i-1} + x_{i+1}))(x_i +\frac{1}{2}(x_{i-1} + x_{i+1})) = 0$.

Call the left-hand side f_i; then the operators sit in the ring

$A = R[x_0,x_1, \ldots, x_N, x_{N+1}]/(f_1, \ldots, f_N, x_0 - 1, x_{N+1} - 1).$

What does this ring look like?  You can make lots of homomorphisms to the real numbers:  you just need to find real numbers a_0,a_1, … a_N,a_{N+1} which, for all i in [1..N], satisfy either

• $a_{i-1}, a_i,a_{i+1}$ are in arithmetic progression;
• $a_{i-1}, -a_i,a_{i+1}$ are in arithmetic progression

and which have a_1 = a_{N+1} = 1.  It appears that when N is not 3 mod 4, all these homomorphisms g: A -> R are distinct, giving an isomorphism between A and R^(2^N).  For instance, when N = 2 there should be 4 homomorphisms, and here they are:  they send x_0,x_1,x_2,x_3 to:

• [1,1,1,1]
• [1,1/5,-3/5,1]
• [1,3/5,-1/5,1]
• [1,-1/3,-1/3,1]

Moreover, the action of A on the 2^N-dimensional vector space V makes it a free rank-1 A-module; in fact, it is canonically identified with A by sending the empty configuration to 1.  So we can think of a probability distribution on configurations as an element of A; namely,

$\sum p(i_1, \ldots, i_r) x_{i_1} \ldots x_{i_r}$

where the coefficient is the probability assigned to the given configuration.  The stationary distribution is then the one which is unchanged when you drop a random chip, i.e. the element b of A such that

$(1/n) (x_1 + \ldots + x_n) b = b$.

Write a for this average; so

(a-1)b = 0.

But here’s the thing; all of those ring homomorphisms g: A -> R are going to send the average of the x_i to something other than 1.  (This follows from the fact that|g(x_i)| is a convex function.)   And this means they send b to 0.  Of course, there’s one exception; the augmentation homomorphism from A to R which sends every x_i to 1 does send their average to 1, and it sends the stationary distribution to 1 as well.

So if you can describe these other homomorphisms explicitly, you can really pin down the stationary distribution!

Now the bad news is that this seems kind of complicated to me.  For instance, I do not see from this how to derive the (apparent) fact that the expected density of the stable distribution converges to the critical density (or converges at all!) as N gets large.

I will add just one amusing fact.  One way to think of these 2^N homomorphisms is as follows.  The conditions on the a_i tell you that [a_i,a_{i+1}] is obtained from [a_{i-1},a_i] by one of two linear transformations in SL_2(Z):

$T = \left(\begin{array}{cc}0&1\\ -1&2\end{array}\right)$

$T' = \left(\begin{array}{cc}0&1\\ -1&-2\end{array}\right)$

So when you go from [a_0,a_1] to [a_N,a_{N+1}], the 2^N paths you take are governed by the 2^N possible paths in the random walk on SL_2(Z) attached to the two unipotent matrices T,T’!  In particular, for each length-N word in those generators, you can write a_{N+1} as a linear function in a_1, and then the condition a_{N+1} = 1 determines a_1 (whence all the other a_i.)

So there is some connection with very rich objects in number theory.  Can one use spectral properties of this walk to say anything equidistributional about the 2^N homomorphisms?  That’s the kind of thing that could in principle be informative about the stationary distribution.

## One thought on “The stochastic sandpile”

1. […] Sadhu and Dhar say in the paper I linked in the last post that it remains open whether the sandpile ring A is isomorphic to R^{2^N} when N is not congruent to 3 mod 4.  I think there’s an easy argument that shows the answer is yes.  Let’s give it a go.  We will freely use the notation from this afternoon’s post. […]


text:
zbMATH — the first resource for mathematics

Groves, Daniel P.

Compute Distance To:
 Author ID: groves.daniel-p Published as: Groves, Daniel; Groves, D.; Groves, Daniel P.
 Documents Indexed: 27 Publications since 1999, including 1 Book
all top 5

Co-Authors

 7 single-authored 7 Manning, Jason Fox 3 Wilton, Henry John Rutley 2 Agol, Ian 2 Bridson, Martin R. 2 Dahmani, François 1 Barnes, Donald W. 1 Dewhurst, C. J. 1 Einstein, Eduard 1 Eleuteri, Antonio 1 Hillman, Jonathan Arthur 1 Hull, Michael B. 1 Martin, Gaven J. 1 Sisto, Alessandro 1 Vaughan-Lee, Michael R.
all top 5

Serials

 3 Transactions of the American Mathematical Society 3 Geometry & Topology 3 Groups, Geometry, and Dynamics 2 Israel Journal of Mathematics 2 Journal of the London Mathematical Society. Second Series 2 Algebraic & Geometric Topology 1 American Journal of Mathematics 1 Bulletin of the London Mathematical Society 1 Compositio Mathematica 1 Publications Mathématiques 1 Journal of Algebra 1 Memoirs of the American Mathematical Society 1 The New York Journal of Mathematics 1 Journal of the Australian Mathematical Society 1 Computational & Mathematical Methods in Medicine 1 Journal of Topology 1 Forum of Mathematics, Pi
all top 5

Fields

 25 Group theory and generalizations (20-XX) 11 Manifolds and cell complexes (57-XX) 5 Nonassociative rings and algebras (17-XX) 1 Functional analysis (46-XX) 1 Statistics (62-XX) 1 Biology and other natural sciences (92-XX)

Citations contained in zbMATH Open

18 Publications have been cited 209 times in 146 Documents Cited by Year
Dehn filling in relatively hyperbolic groups. Zbl 1211.20038
Groves, Daniel; Manning, Jason Fox
2008
Limit groups for relatively hyperbolic groups. II: Makanin-Razborov diagrams. Zbl 1100.20032
Groves, Daniel
2005
The isomorphism problem for toral relatively hyperbolic groups. Zbl 1207.20038
Dahmani, François; Groves, Daniel
2008
Limit groups for relatively hyperbolic groups. I: The basic tools. Zbl 1231.20038
Groves, Daniel
2009
The quadratic isoperimetric inequality for mapping tori of free group automorphisms. Zbl 1201.20037
Bridson, Martin R.; Groves, Daniel
2010
Residual finiteness, QCERF and fillings of hyperbolic groups. Zbl 1229.20037
Agol, Ian; Groves, Daniel; Manning, Jason Fox
2009
Limits of (certain) CAT(0) groups. I: Compactification. Zbl 1085.20025
Groves, Daniel
2005
Enumerating limit groups. Zbl 1216.20029
Groves, Daniel; Wilton, Henry
2009
Detecting free splittings in relatively hyperbolic groups. Zbl 1196.20050
Dahmani, François; Groves, Daniel
2008
An alternate proof of Wise’s malnormal special quotient theorem. Zbl 1380.20047
Agol, Ian; Groves, Daniel; Fox Manning, Jason
2016
Fillings, finite generation and direct limits of relatively hyperbolic groups. Zbl 1169.20023
Groves, Daniel; Manning, Jason Fox
2007
Conjugacy classes of solutions to equations and inequations over hyperbolic groups. Zbl 1236.20047
Groves, Daniel; Wilton, Henry
2010
Dehn fillings and elementary splittings. Zbl 1427.20058
Groves, Daniel; Fox Manning, Jason
2018
Cofinitely Hopfian groups, open mappings and knot complements. Zbl 1259.20047
Bridson, Martin R.; Groves, Daniel; Hillman, Jonathan A.; Martin, Gaven J.
2010
Test elements in torsion-free hyperbolic groups. Zbl 1262.20047
Groves, Daniel
2012
Abelian splittings of right-angled Artin groups. Zbl 1446.20063
Groves, Daniel; Hull, Michael
2017
Quasiconvexity and Dehn filling. Zbl 1460.57020
Groves, Daniel; Manning, Jason Fox
2021
The Wielandt subalgebra of a Lie algebra. Zbl 1071.17006
Barnes, Donald W.; Groves, Daniel
2003
Quasiconvexity and Dehn filling. Zbl 1460.57020
Groves, Daniel; Manning, Jason Fox
2021
Dehn fillings and elementary splittings. Zbl 1427.20058
Groves, Daniel; Fox Manning, Jason
2018
Abelian splittings of right-angled Artin groups. Zbl 1446.20063
Groves, Daniel; Hull, Michael
2017
An alternate proof of Wise’s malnormal special quotient theorem. Zbl 1380.20047
Agol, Ian; Groves, Daniel; Fox Manning, Jason
2016
Test elements in torsion-free hyperbolic groups. Zbl 1262.20047
Groves, Daniel
2012
The quadratic isoperimetric inequality for mapping tori of free group automorphisms. Zbl 1201.20037
Bridson, Martin R.; Groves, Daniel
2010
Conjugacy classes of solutions to equations and inequations over hyperbolic groups. Zbl 1236.20047
Groves, Daniel; Wilton, Henry
2010
Cofinitely Hopfian groups, open mappings and knot complements. Zbl 1259.20047
Bridson, Martin R.; Groves, Daniel; Hillman, Jonathan A.; Martin, Gaven J.
2010
Limit groups for relatively hyperbolic groups. I: The basic tools. Zbl 1231.20038
Groves, Daniel
2009
Residual finiteness, QCERF and fillings of hyperbolic groups. Zbl 1229.20037
Agol, Ian; Groves, Daniel; Manning, Jason Fox
2009
Enumerating limit groups. Zbl 1216.20029
Groves, Daniel; Wilton, Henry
2009
Dehn filling in relatively hyperbolic groups. Zbl 1211.20038
Groves, Daniel; Manning, Jason Fox
2008
The isomorphism problem for toral relatively hyperbolic groups. Zbl 1207.20038
Dahmani, François; Groves, Daniel
2008
Detecting free splittings in relatively hyperbolic groups. Zbl 1196.20050
Dahmani, François; Groves, Daniel
2008
Fillings, finite generation and direct limits of relatively hyperbolic groups. Zbl 1169.20023
Groves, Daniel; Manning, Jason Fox
2007
Limit groups for relatively hyperbolic groups. II: Makanin-Razborov diagrams. Zbl 1100.20032
Groves, Daniel
2005
Limits of (certain) CAT(0) groups. I: Compactification. Zbl 1085.20025
Groves, Daniel
2005
The Wielandt subalgebra of a Lie algebra. Zbl 1071.17006
Barnes, Donald W.; Groves, Daniel
2003
all top 5

Cited by 140 Authors

 14 Groves, Daniel P. 13 Dahmani, François 9 Manning, Jason Fox 9 Myasnikov, Alexei G. 9 Osin, Denis V. 8 Guirardel, Vincent 7 Kharlampovich, Olga G. 7 Sisto, Alessandro 5 Martínez-Pedroza, Eduardo 4 Bridson, Martin R. 4 Kapovich, Ilya 4 Levitt, Gilbert 3 Fukaya, Tomohiro 3 Mackay, John M. 3 Oguni, Shinichi 3 Sapir, Mark Valentinovich 3 Touikan, Nicholas W. M. 3 Wilton, Henry John Rutley 2 Agol, Ian 2 Bartels, Arthur C. 2 Belegradek, Igor 2 Bestvina, Mladen 2 Bumagin, Inna 2 Calegari, Danny Matthew Cornelius 2 Casals-Ruiz, Montserrat 2 Cashen, Christopher H. 2 Chifan, Ionuţ 2 Durham, Matthew Gentry 2 Fujiwara, Koji 2 Gautero, François 2 Haulmark, Matthew 2 Hume, David 2 Kazachkov, Ilya V. 2 Kent, Curtis A. 2 Kida, Yoshikata 2 Liu, Yi 2 Louder, Larsen 2 Minasyan, Ashot 2 Pal, Abhijit 2 Reid, Alan W. 2 Zhao, Xuezhi 1 Abbott, Carolyn R. 1 Alibegović, Emina 1 Antolín, Yago 1 Arora, Shivam 1 Barquinero, Enrique Miguel 1 Behrstock, Jason A. 1 Bell, Mark Christopher 1 Berk, Gili 1 Bogopolski, Oleg 1 Bowditch, Brian H. 1 Button, Jack Oliver 1 Bux, Kai-Uwe 1 Chiodo, Maurice 1 Conder, Marston D. E. 1 Daniyarova, Evelina Yur’Evna 1 Delzant, Thomas 1 Druţu, Cornelia 1 Einstein, Eduard 1 Fleming, Joshua W. 1 Gandini, Giovanni 1 Gekhtman, Ilya 1 Genevois, Anthony 1 Gitik, Rita 1 Groff, Bradley W. 1 Hagen, Mark F. 1 Healy, Brendan Burns 1 Heusener, Michael 1 Hillman, Jonathan Arthur 1 Hruska, G. Christopher 1 Hull, Michael B. 1 Ioana, Adrian 1 Jaligot, Eric 1 Jeon, Woojin 1 Jiang, Boju 1 Kim, Heejoung 1 Kim, Sang-Hyun 1 Lackenby, Marc 1 Lafont, Jean-François 1 Lee, Donghi 1 Leininger, Christopher J. 1 Liang, Hao 1 Lohrey, Markus 1 Long, Darren D. 1 Lukyanenko, Anton 1 Lysenok, Igor’ Geront’evich 1 Mackay John, M. 1 Maher, Joseph 1 Margolis, Alexander J. 1 Markovic, Vladimir 1 Martin, Gaven J. 1 Mathéus, Frédéric 1 Mathieu, Pierre 1 Meng, Wei 1 Mj, Mahan 1 Mousley, Sarah C. 1 Neofytidis, Christoforos 1 Neumann, Walter David 1 Nguyen, Hoang Thanh 1 Ohshika, Ken’ichi ...and 40 more Authors
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Cited in 55 Serials

 13 Israel Journal of Mathematics 12 Algebraic & Geometric Topology 12 Groups, Geometry, and Dynamics 10 International Journal of Algebra and Computation 9 Transactions of the American Mathematical Society 7 Journal of Algebra 7 Geometry & Topology 5 Journal of Topology and Analysis 4 Mathematische Annalen 3 Annales de l’Institut Fourier 3 Geometriae Dedicata 3 Inventiones Mathematicae 3 Geometric and Functional Analysis. GAFA 2 Advances in Mathematics 2 Compositio Mathematica 2 Duke Mathematical Journal 2 Illinois Journal of Mathematics 2 Journal für die Reine und Angewandte Mathematik 2 Memoirs of the American Mathematical Society 2 Topology and its Applications 2 Conformal Geometry and Dynamics 2 Journal of Group Theory 2 Science China. Mathematics 2 Bulletin of Mathematical Sciences 2 Journal de l’École Polytechnique – Mathématiques 1 Communications in Algebra 1 Bulletin of the London Mathematical Society 1 Canadian Mathematical Bulletin 1 Glasgow Mathematical Journal 1 Publications Mathématiques 1 Journal of the London Mathematical Society. Second Series 1 Journal of Pure and Applied Algebra 1 Michigan Mathematical Journal 1 Proceedings of the American Mathematical Society 1 Proceedings of the London Mathematical Society. Third Series 1 Ergodic Theory and Dynamical Systems 1 Annals of Pure and Applied Logic 1 Bulletin of the Iranian Mathematical Society 1 Journal of the American Mathematical Society 1 Publicacions Matemàtiques 1 Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 1 Bulletin of the American Mathematical Society. New Series 1 Experimental Mathematics 1 Topology Proceedings 1 Journal of Mathematical Sciences (New York) 1 Annales Mathématiques Blaise Pascal 1 The New York Journal of Mathematics 1 Electronic Research Announcements of the American Mathematical Society 1 Annales Academiae Scientiarum Fennicae. Mathematica 1 Journal of the European Mathematical Society (JEMS) 1 Acta Mathematica Sinica. English Series 1 Proceedings of the Steklov Institute of Mathematics 1 Involve 1 Groups, Complexity, Cryptology 1 Forum of Mathematics, Pi
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Cited in 24 Fields

 132 Group theory and generalizations (20-XX) 59 Manifolds and cell complexes (57-XX) 11 Mathematical logic and foundations (03-XX) 7 Dynamical systems and ergodic theory (37-XX) 7 Differential geometry (53-XX) 6 Topological groups, Lie groups (22-XX) 4 Functions of a complex variable (30-XX) 4 Several complex variables and analytic spaces (32-XX) 4 Functional analysis (46-XX) 4 Probability theory and stochastic processes (60-XX) 3 Combinatorics (05-XX) 3 Associative rings and algebras (16-XX) 3 Algebraic topology (55-XX) 3 Global analysis, analysis on manifolds (58-XX) 3 Computer science (68-XX) 2 Algebraic geometry (14-XX) 2 Category theory; homological algebra (18-XX) 2 $$K$$-theory (19-XX) 1 General algebraic systems (08-XX) 1 Nonassociative rings and algebras (17-XX) 1 Measure and integration (28-XX) 1 Abstract harmonic analysis (43-XX) 1 Geometry (51-XX) 1 General topology (54-XX)


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## Higher rank or scholarship?

Share tips as you apply, write essays, interview...
##### This topic has expert replies
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Posts: 10
Joined: 09 Jul 2016

### Higher rank or scholarship?

by mqt001 » Fri Mar 17, 2017 9:42 am
Hello all,

I have a bit of a dilemma. I am currently torn between two programs: Vanderbilt (Owen) FTMBA and OSU (Fisher) FTMBA. Vanderbilt seems to be a better fit in regards to my desired career path, which is consulting (approximately 25% of Owen graduates end up in consulting per the official school stats). Vanderbilt offered me no scholarship money, whereas OSU has offered $30K. Vanderbilt is ranked higher than OSU in every poll I have seen. I suppose my question is: is it worth it to pay$30k extra to attend the school with the higher ranking (albeit the two are not far apart in the polls), or should I take the money and go with OSU? I should add that without the money, the cost would be about the same since I would pay out of state tuition for OSU.

Thanks!

Senior | Next Rank: 100 Posts
Posts: 34
Joined: 06 Feb 2017
Location: India
Thanked: 5 times
by EduSimplified » Sun Mar 19, 2017 10:16 pm
Hi mqt001,
It seems that you may have answered you question. When you mention that Vanderbilt fits better to your career goals, I would seldom recommend you to join any other school. Having said that I am a bearer of student loan and can tell you first hand that finances are not to be ignored.
The rush of getting into your dream school can overtake the rationale of financial reality. Which will only dawn once you graduate and trust me thats a big blow. To keep the post short I will advise you to look at your finances and if you are able to manage $30K with reasonable ease, go with your dream school Vanderbilt. Good Luck! Deep A. Chief Counselling Officer EduSimplified.com ### GMAT/MBA Expert Master | Next Rank: 500 Posts Posts: 344 Joined: 01 May 2015 Location: Brooklyn, NY Thanked: 44 times Followed by:8 members by MargaretStrother » Fri Mar 24, 2017 11:12 am I have to agree -- go with Vanderbilt. The difference in price would evaporate very quickly after your first year of employment, but the difference in brand and quality would last forever. By the way, don't forget to let Vanderbilt admissions know (tactfully!) that you have this dilemma. It doesn't always work, but sometimes they can scratch together a bit of$$if that's the only factor preventing you from joining their community. Good luck! Margaret Strother Margaret Strother Senior Consultant Stacy Blackman Consulting Learn more about me Junior | Next Rank: 30 Posts Posts: 10 Joined: 22 Mar 2017 Followed by:1 members by AlexBarret » Sun May 14, 2017 8:54 am mqt001 wrote:Hello all, I have a bit of a dilemma. I am currently torn between two programs: Vanderbilt (Owen) FTMBA and OSU (Fisher) FTMBA. Vanderbilt seems to be a better fit in regards to my desired career path, which is consulting (approximately 25% of Owen graduates end up in consulting per the official school stats). Vanderbilt offered me no scholarship money, whereas OSU has offered$30K. Vanderbilt is ranked higher than OSU in every poll I have seen. I suppose my question is: is it worth it to pay $30k extra to attend the school with the higher ranking (albeit the two are not far apart in the polls), or should I take the money and go with OSU? I should add that without the money, the cost would be about the same since I would pay out of state tuition for OSU. Thanks! First of all, you need to listen to your heart because there is an intuition in it. Secondly, I think it doesnt matter the ranking of the Ñ�ollege but your desire to study and to reach all your goals. And what about money, I think its a big advantage to get the scholarship money. Good luck for you! ### GMAT/MBA Expert Master | Next Rank: 500 Posts Posts: 298 Joined: 20 Jun 2016 Thanked: 26 times Followed by:11 members by [email protected] » Mon May 15, 2017 8:47 am mqt001 wrote:Hello all, I have a bit of a dilemma. I am currently torn between two programs: Vanderbilt (Owen) FTMBA and OSU (Fisher) FTMBA. Vanderbilt seems to be a better fit in regards to my desired career path, which is consulting (approximately 25% of Owen graduates end up in consulting per the official school stats). Vanderbilt offered me no scholarship money, whereas OSU has offered$30K. Vanderbilt is ranked higher than OSU in every poll I have seen. I suppose my question is: is it worth it to pay \$30k extra to attend the school with the higher ranking (albeit the two are not far apart in the polls), or should I take the money and go with OSU? I should add that without the money, the cost would be about the same since I would pay out of state tuition for OSU.

Thanks!
I would go to Vanderbilt because I think your chances of seeing more consulting firms are better there. And especially if it is the same out of pocket cost to you- I would go to Owen.
But honestly only you can answer this question since only you know all the variables- but if you are crowd sourcing here to see what the "market" says... count me as one for Owen! Good luck!

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by PrepMBA.AlexLeventhal » Tue May 23, 2017 9:59 am
There are tactful ways to approach Vanderbilt for some merit scholarship money. They will likely not match the 30K but I have worked with clients to communicate in these types of situations with schools and often there is some positive movement.

Good Luck!
Alex
Alex Leventhal
Harvard MBA, 1998