User martin - MathOverflow

Motivation for and history of pseudo-differential operators

2011-03-21T17:20:25Z

Suppose you start from partial differential equations and functional analysis (on $\mathbb R^n$ and on real manifolds, which prominent example problems lead you to work with Pseudo-differential operators?

I would appreciate any good examples, as well as some historical outlines on the topics development (Shubins classical book spends a few lines on history and motivation in the preface, but no "natural" examples. I am not aware of any historical outlines in literature.).

Application and relevance of Sobolev gradients

2013-03-26T10:29:27Z

The Sobolev gradient concept has been developed in the 1970s, with a first publication in 1985, and an introduction can be found at: Ranka

I would like to learn how strong the impact of Sobolev gradients has been in numerical partial differential equations.

Is it one of several possible tools for nonlinear pde, or a powerful approach without "competitors"?

Orthogonal projection of discontinuous piecewise polynomial space in energy scalar product

2013-03-07T20:30:59Z

Let $I = [0,1]$ be the unit interval Let $I$ be partioned into $n$ closed subintervals $(I_j)_J$, each of length $1/n$.

Let $X_{DC} = { v \in L^2[0,1] | 1 \leq j \leq n : v_{|I_j} \in \mathcal P_1( I_j ) }$,

where $\mathcal P_1$ is the space of piecewise affine functions. Let

$X_h = X_{DC} \cap C^0([0,1])$.

Given $\Theta \in X_{DC}$ with $\int_0^1\Theta = 0$, let $\theta \in X_h$ be the orthogonal projection of $\Theta$ in the energy scalar product, $(u,v) \mapsto \langle u', v' \rangle_{L^2}$. In other words,

$\forall v \in X_h : \int_0^1 \theta' v' = \int_0^1 \Theta' v'$

where $\Theta'$ is understood piecewise. Let $\|P\|$ be the linear operator that maps $\Theta$ to $\theta$.

Can you give a bound for $P$ that does not depend on $n$?

$T^\dagger T$ is an orthogonal projection. What do we know about $T^\dagger S$?

2013-02-13T13:03:22Z

Let $H$ be a Hilbert space. Suppose that $T$ and $S$ are linear bounded endomorphisms of $H$ with closed range.

Let $T^\dagger$ denote the pseudo-inverse of $T$. We know that $T^\dagger T$ is an orthogonal projection. We do not know much about $T^\dagger S$, but it should be possible to estimate its deviation from being an orthogonal projection if we know geometric information about the range or kernel of $S$ and $T$, e.g., the gaps between these two subspaces.

Do you know any starting point to achieve this?

Distributional derivative of non continuously differentiable functions

2010-07-04T17:03:59Z

Hello,

let $f$ be a continuously differentiable function on $R^n$. Then its classical derivative and its distributional derivative coincide.

It is known (cf. Rudin, Functional Analysis, Sect. 6.13) that this correspondence breaks down if $f$ is merely differentiable. Whereas in the sense of distributions $f$ is infinitely often differentiable, and obeys Schwartz' theorem, we usually can't investigate much more than the first derivatives in the classical meaning.

I want to train my intution with respect to different notions of derivation, and wonder how there is still a correspondence, or whether there is a point when both derivatives become virtually incommensurable.

Regge calculus: Questions of consistency resolved?

2011-03-22T14:12:56Z

Hello,

Regge calculus is an approximation scheme for General Relativity, which has been introduced in early-sixties and has been adopted both in numerical relativity and numerical quantum relativity. In contrast to its widespread use in computational science, there does not seem to exist much theory on whether the Regge calculus is actually consistent - i.e. whether there is some degree of exactness (like mesh width) we can adapt arbitrarly to obtain a solution with arbitrarly small error (say in some Sobolev-norm).

In fact there has been debate about this:

The question of consistency is a purely mathematical one, and therefore I do not expect it to be "debatable". I will have to work with this theory and hence I do wonder in how far there is a theoretical basis to tell apart "It works" and "It does not work".

Thank you very much!

A category of manifolds that includes Polygonal domains

2012-08-02T14:17:38Z

The prime motivation to introduce the category of manifolds with corners is to have a convenient theory for the analysis on simplices that is as powerful as for smooth manifolds (with boundaries).

As far as I understand, polygonally bounded subdomains of $\mathbb R^n$ are not submanifolds with corners in general, at least for two reasons: (i) The category of manifolds with corners does not include "inward corners" (ii) The number of polygonal boundary pieces that meet at a common point can be arbitrarly large.

Is there a category of manifolds which contains arbitrary polygonal domains (where the boundary pieces may be curved)?

Norm estimate for Moore-Penrose pseudo-inverse of $i^\ast T i$

2012-07-31T08:15:31Z

Let $G$ and $H$ be Hilbert spaces, let $i : G \rightarrow H$ be an isometric inclusion (so $G$ is a subspace of $H$) and let $T : H \rightarrow H$ be a bounded linear operator with closed range.

That $T$ has closed range $R(T)$ is equivalent to the existence of a constant $C > 0$ such that

$\forall y \in R(T) : \exists x \in T^{-1}({y}) : C \| y \|_H \geq \|x\|$

Another equivalent condition is that the Moore-Penrose inverse of $T$, written $T^\dagger$, has a norm bounded by the constant $C$.

Do you know a norm bound for $(i^\ast T i)^\dagger$? You assume that $G$ is finite dimensional so that $i^\ast T i$ has closed range.

Tensor analysis/Differential forms outside physics

2012-07-16T17:40:33Z

There are many "geometric systems" like tensor analysis or differential forms calculus, which more or less different perspectives onto the same abstract relations.

Most applications are physical, like electromagnetism. I wonder whether there are applications of these geometric systems beyond physics. Can you show me some active areas of research in that direction?

A related question is whether there are partial differential equations whose origin is not directly physical and that can be meaning-fully stated in terms of tensor and vector fields.

Poincare constant for $L^2$-differential-forms on a submanifold of $\mathbb R^n$ with Lipschitz boundary

2012-04-17T00:00:46Z

Let $M \subset \mathbb R^n$ be a submanifold of euclidean space whose boundary is locally a Lipschitz graph. Let $\omega \in L^2\Lambda^k(M)$ be a differential form with square-integrable coefficients.

The Poincare constant for $k = 0$ can be estimated by $1/2 \cdot \operatorname{diam}(M)$ in case the domain is convex and bounded. I have not seen a formal statement in literature that provides a comparable statement for $k > 0$.

Could you provide me with a reference? I need this only for citation and an estimate in terms of the diameter of the domain would be completely sufficient. Thank you.

$L^2$-de-Rham complex on Lipschitz domains has smooth harmonic forms?

2012-05-07T17:24:57Z

I would like to know for which choice of boundary conditions the title statement is true.

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^n$, for which we regard the $L^2$-de-Rham complex.

We can impose on the domain of the differentials either (i) no boundary conditions (ii) tangential boundary conditions on all of $\partial\Omega$, or (iii) impose partial tangential boundary conditions on a 2reasonable" part $\Sigma_t$ of $\partial\Omega$. The last case has been investigated by e.g., in [1] and [2].

For closed smoothly bounded domains, it is known that is known that merely locally integrable differential forms $h$ with $dh = 0$ and $\star d \star = 0$ are already smooth. [3] As for the $L^2$-de-Rham complex for Lipschitz bounded domains, my impression is that the smoothness of the harmonic forms is expected or even taken for granted, but I have not found an explicit statement that clarifies this.

Unfortunately, I need comparable smoothness results only for application. Maybe it is even to simple for practioners of the field to write it down explicitly. Can provide me with some (available) resources which I can cite, or a combination of theorems?

[1] M. Mitrea: Mixed boundary-value problems for Maxwell's equation
[2] V. Gol'dshtein, I. Mitrea, M. Mitrea: Hodge decompositions with mixed boundary conditions and applications to partial differential equations on lipschitz manifolds
[3] T. Iwaniec, C. Scott, B. Stroffolini: Nonlinear Hodge Theory on Manifolds with Boundary.

If $A \subset X'$ annihilates only $0$, then $A$ is dense

2012-04-30T12:14:12Z

Let $X$ be a Banach space with continuous dual space $X'$ with norm topology. Let us regard the following property of $X$:

Property: Any linear subset $A \subset X'$ that satisfies $\bigcap_{\alpha\in A} \ker\alpha = {0}$ is dense in $X'$.

Any reflexive space $X$ has this property. Can you classify the spaces that share this property? I wonder whether it is equivalent to reflexivity.

Tensor product of linear mappings versus chain complexes

2012-04-07T21:11:52Z

A chain complex of vector spaces $X_k$ is a sequence of linear mappings

$\dots \overset{d_{k-1}}{\longrightarrow} X_k \overset{d_{k}}{\longrightarrow} X_{k+1} \overset{d_{k+1}}{\longrightarrow} \dots$

such that $d_{k+1} \circ d_k = 0$. Here $k \in \mathbb N$ runs through the natural numbers. I have held the view that a chain complex $(X_k,d_k)$ is actually just a way to write down a pair of direct sums $(X,d) = ( \bigoplus X_k, \bigoplus d_k )$ that satisfy an additional property (namely, the graduation and the chain property).

However, the notions of tensor products that apply naturally to each of these points of view appear incompatible. Let $(X_k,d_k)$ and $(Y_l,D_l)$ be chain complexes. Then we define

$(X \otimes Y )_{p} := \bigoplus\limits_{k+l=p} X_k \otimes Y_l$

The tensor product of $d = \bigoplus d_k$ and $D = \bigoplus D_l$ in the sense of linear mappings is then defined by

$d \otimes D : X_k \otimes Y_l \rightarrow X_{k+1} \otimes Y_{l+1}$,

$( d \otimes D ) ( v \otimes w ) = ( d_k v \otimes D_l w )$

On the other hand, in the case of tensor products of chain complexes, we rather define

$d \otimes D : X_k \otimes Y_l \rightarrow X_{k+1} \otimes Y_{l} + X_{k} \otimes Y_{l+1}$,

$( d \otimes D ) ( v \otimes w ) = d_k v \otimes w + (-1)^k v \otimes D_l w$

The apparent incompatibility of these definitions contradicts my (likely simplified) view that chain complexes are only special notation for pairs of vector spaces and linear mappings. I expect this to become even more confusing as more structures are regarded (e.g., if the complex "is" a graded algebra).

Question: Is there an intuitive point of view, that there relates these two products? This is not discussed in literature (MacLane, Homology; Lang, Algebra).

Inner product of linear bounded operators between Hilbert spaces

2012-03-10T01:44:51Z

Let $X$ and $Y$ be Hilbert spaces, and let $L(X,Y)$ be the set of bounded linear operators between Hilbert spaces.

Can we equip $L(X,Y)$ with a natural inner product? I think it should look like

$\langle S, T \rangle = \sup_{x \in X} \dfrac{ \langle S x, T x \rangle_Y }{ \|x\|^2_X }$

where $S$ and $T$ and are from $L(X,Y)$. I have not found such a construct in standard text books on Hilbert spaces, therefore I would like to learn whether this is the way to do it.

Transpose of unbounded operators between Banach spaces.

2012-03-04T13:07:11Z

Let $X$ and $Y$ be Banach spaces, and let $L : X \rightarrow Y$ be a unbounded operator with dense domain $\operatorname{dom}(L)$. We can then talk about the transposed operator

$L' : \operatorname{dom}(L') \subset Y' \rightarrow X' : y' \rightarrow y'(T\cdot)$

whose domain is given by those functionals $y'$, such that the term $y'(T\cdot)$, initially defined on $\operatorname{dom}(L)$, has bounded extension to all of $X$. If $L$ is closed and densely defined, then it is standard to show that $L'$ is closed, too. But if what the density of the domain of transpose? The proof by Reed and Simons seems in the Hilbert space case seems to use specific Hilbert space techniques.

Question: Suppose $L$ is a closed densely-defined operator between Banach spaces. Is it transpose a closed densely-defined operator, too?

Minimum solid angle and aspect ratio of an $n$-simplex

2012-02-28T22:58:43Z

In computational geometry and other fields, it is of interest to have degeneracy measures for shapes of simplices, which quantitatively seperate the regular simplex from degenerate simplices.

In two dimensions, two such shape measures are the minimum angle of a triangle and its aspect ratio, i.e. the quotient of the radii of insphere and circumsphere.

While many of these shape measures naturally generalize to higher dimensions, and are documented in literature for arbitrary dimension, I haven't found any source which relates the minimum solid angle of a simplex with any such shape measure in arbitrary dimensions. It is "obvious" that simplices with small solid angles at the corner vertices are degenerate, but I haven't found any source on this literature.

Question or reference request: Can you relate the minimum solid angle of a $d$-dimensional simplex with its aspect ratio for arbitrary $d$?

A possible answer would generalize Theorem 6.1 of "A. Liu and B. Joe. Relationship between tetrahedron shape measures, BIT, 34 (1994)" which states:

For any tetrahedron $T$ we have $\sqrt{3}/24 \rho^2 \leq \sigma_{\min} \leq (2/(3^{1/4})) \sqrt{\rho}$, where $\sigma_{\min}$ is the minimum solid angle of $T$ and $\rho$ denotes the aspect ratio of $T$.

Fibred manifolds with boundary

2011-12-08T23:42:45Z

A fibred manifold is a triple $(E,\pi,M)$ where $E$ and $M$ are manifolds and $\pi : E \rightarrow M$ is a surjective submersion. (Saunders, The Geometry of Jet bundles)

A special case of this is the fibre bundle which is a fibred manifold, where for each relatively open subset $U \subset M$ the space $\pi^{-1}(M)$ looks like (i.e. is diffeomorphic to) a product $U \times F$ for some smooth manifold $F$.

Even more special are linear bundles and, of course, the tangential bundle of a smooth manifold.

I do not know a generalization of these constructions to manifolds with boundary. For example, it seems intuitive that at a boundary point of a smooth manifold with boundary, we do not have a tangential space, but rather a tangential cone. Similarly one would adapt the more general classes of fibre bundles.

Unfortunately, this seems to be non-standard in differential geometry and differential topology, and while it seems natural to me to work with cones in that case, reinventing the wheel might be a waste of time. Do you know a good reference that settles the basic constructions and possible pitfalls for "fibred manifolds with boundary"?

Answer by Martin for conjugate gradient iteration

2011-09-17T22:17:33Z

A search for "conjugate gradient singular matrix" took me to this question. While the answer is obviously given by the responses, the question can be refined: Can CG still give a working algorithm if the matrix is singular, but behaves as a symmetric positive definite form on a (large) subspace?

A standard example is given by the finite element discretization of the Neumann problem on a simply connected domain. The constant functions are both the kernel and the cokernel of the Laplacian. On functions with vanishing mean, the Laplacian is still a positive definite symmetric operator, and we would like to leverage this structure.

This is non-trivial and best our numerical method is derived from a fully analytic setting, because this might provide us the convergence analysis as well. --- This appraoch is for example elaborated in

On the Finite Element Solution of the Pure Neumann Problem
Pavel Bochev and R. B. Lehoucq
SIAM Review
Vol. 47, No. 1 (Mar., 2005), pp. 50-66
Published by: Society for Industrial and Applied Mathematics
Article Stable URL: http://www.jstor.org/stable/20453601

Apart from this canon standard example for the Laplacian, system matrices with larger kernel appear in numerical methods for the de-Rham-complex, in particular if the domain is topologically non-trivial (Finite Element Exterior Calculus, Discrete Exterior Calculus). Singular system solves are still no standard material for education in computational science. As far as I may dare to give an estimate, there is still much room for a better theory building.

Function space between uniform continuity and Hölder continuity

2011-09-11T22:00:03Z

Can you give an example of a complete metric vector space of uniformly continuous functions that is strictly contained between the set of uniformly continuous functions on $\mathbb R^d$ and the Hölder spaces on $\mathbb R^d$?

Dual operators between Hilbert spaces : With or without riesz representation

2011-07-23T02:54:48Z

Let $X$ and $Y$ be Hilbert spaces over the real numbers (so complex conjugation plays no role, and everything will be linear in the strict sense). Let $f : X \rightarrow Y$ be a linear continuous mapping.

By the Riesz representation theorem, Hilbert spaces are isometric isomorphic to their own dual spaces. This leads to different notions of duality, which confuses me.

(i) The dual operator of $f^\ast$ is the operator $f^\ast : Y^\ast \rightarrow X^\ast$ defined by $y^\ast \mapsto ( x \mapsto y^\ast( f x ) )$

(ii) The dual operator is the adjoint, i.e. the unique operator $f^\ast : Y \rightarrow X$ such that $\forall x \in X, y \in Y : \langle fx,y \rangle_Y = \langle x,f^\ast y \rangle_X$

The transition between these two different notions is full and faithfully functiorial, to say it like that. - Nevertheless, I would like to differ between these two notions of duality; just think in analysis of the Hilbert space $H^1_0$ and its dual $H^{-1}$. But I even don't think speaking about "different dualities" is not a crime.

So, I would to know whether there are different words for this, whether these are really different concepts (except whether I use the isometry or not) and in which contexts, generally, to use these two appropiately. Although I think at least in the less algebraic parts of mathematics it would be helpful not to implicitly use the Riesz representation, this seems to be always swept under the rug.

Interpretation of the two-dimensional de-Rham complex

2011-07-12T16:29:04Z

The de-Rham complex in one dimension describes phenomena that can be described in terms of ordinary differential equations. The de-Rham complex in three dimensions can be used to describe classical results in vector analysis.

Whereas the cochain morphism from the de Rham complex to complexes build out of scalar and vector fields in dimension $1$ and $3$ is very canonical, I don't know how to "interpret" the de-Rham complex in two dimensions.

For a demonstration, let the domain be $\mathbb R^n$ and let $\mathcal S$ be the set of smooth scalar fields, $\mathcal V$ the set of smooth vector fields.

The (smooth) de Rham complex in two dimensions reads

$\lambda^0 \overset{d_0}{\longrightarrow} \lambda^1 \overset{d_1}{\longrightarrow} \lambda^2$

as usual. We want to translate this into terms of classical vector analysis. The spaces $\lambda^0$ and $\lambda^2$ are canonically isomorphic to scalar fields. How do we translate $\lambda^1$ into vector fields?

If we translate it into vector fields by the harmonic isomorphism, and set the second differential from $\mathcal V$ to $\mathcal S$ to be the divergence $\operatorname{div} = \partial_x + \partial_y$, then we must set the first isomorphism to $(\partial_y,-\partial_x)$ or $(-\partial_y,\partial_x)$.

If instead we set the first differential to be the gradient $\operatorname{grad} = (\partial_x,\partial_y)$ from $\mathcal V$ to $\mathcal S$, then the second differential can either be $-\partial_y+\partial_x$ or $\partial_y-\partial_x$.

so we have complexes

$\mathcal S \overset{(-\partial_y,\partial_x)}{\longrightarrow} \mathcal V \overset{\partial_x + \partial_y}{\longrightarrow} \mathcal S$

and

$\mathcal S \overset{(\partial_x,\partial_y)}{\longrightarrow} \mathcal V \overset{-\partial_y+\partial_x}{\longrightarrow} \mathcal S$

In either case, the concatenation $d_1 \circ d_0$ reads

$\operatorname{div} \begin{bmatrix} 0 & 1 \cr -1 & 0 \end{bmatrix} \operatorname{grad}$.

Therefore I ask for help to understand the following issues

Which choice of chain morphism from the de Rham complex into a vector analytic setting is the "right" one? Note that if you dualize one of these vector-analytic complexes, the codifferentials turn out to be the morphisms of the other complex, with signs and arrow directions switched.
The matrix $\begin{bmatrix} 0 & 1 \cr -1 & 0 \end{bmatrix}$ is a prototypical example of a sympletic matrix. Does there exist an relation in terms of sympletic geometry?

Thank you very much.

Why are viscosity solutions useful solutions?

2011-03-24T17:12:06Z

I refer to definition of viscosity solution in user's guide to viscosity solutions of second order partial differential equations by Michael G. Crandall, Hitoshi Ishii and Pierre-Louis Lions.

Viscosity solutions are generalized solutions which can be implied if the Sobolev theory (or similar) doesn't provide "useful" solutions. A standard example is the problem

$|u'| = -1, u(-1) = 1, u(1)=1$

All "zig-zag" functions with appropiate boundary conditions provide a solution, but $u(x)=|x|$ is the unique viscosity solution.

But except its formal beauty why do we regard a viscosity solution as useful, and what is the 'physical' or 'intuitive' interpretation of being a viscosity solution?

General theory for p-normed spaces

2011-03-16T15:13:26Z

Hello,

in functional analysis and operator theory, you encounter several (at first glance, at least) similar constructions of normed spaces that can be indexed with some $ p \in [1,\infty]$, and which behave very similarly. These include

$\mathbb R^n$ with a $p$-norm
$l^p$ spaces
$L^p$ spaces
$p$-Trace-norms on matrices
$p$-Schatten-classes

Are you aware of an approach that handles these objects in an unified abstract way? An interesting result would be the construction of such a family of spaces from a given Hilbert space.

Appropiate models of numerical computation

2011-03-09T23:56:42Z

Hello,

in contrast to the more discrete part of computational mathematics (cryptography, combinatorial computation), numerical mathematics seems to ignore typical questions of theoretical computer science -- what does 'algorithm' or 'computation' mean, what is the model of computation.

This is far from fallacious. For example, a finite element theorist mostly investigates only approximation schemes and convergence rates, which in principle do not demand any computation at all. Algorithms are typically neat and short, and any exceptions to these are not rarely just combinatorial insertions for mesh management. The other end of the spectrum compromises very technical numerical mathematics. - In either case, the 'deep-down' part is largely abandoned as soon as possible, because it is largely irrelevant. Just like no cryptographer enjoys talking about Turing machines.

Has there been a rigourous treatment of a numerical model, a justification why (or for whom) a certain model might be appropiate?

I am aware of computable analysis and numerical mathematicians who participate in this field. But I am not aware of a numerical model like, say, a numerical random access machine. I even suppose there different models appropiate for researchers in fundamental numerical algoriths or a FEM reasearcher, depending on which level of detail is needed.

Reasonable "Random" matrices to test numerical algorithms

2010-12-31T18:37:48Z

Hello,

in numerical analysis, it is common to compare the behavior of different algorithms, and of different implementation of algorithms. This occurs not only on the theoretical level, but also on the concrete level of implementation - and not to forget, it serves the purpose of demonstration.

A prominent problem is the solution of linear systems, both general as well as various subcases.

To test, benchmark and profile numerical implementations, you run your work on several instances of the problem. However, it is difficult question to obtain a good set of these instances. You want to inspect pathological cases (diff. degrees of ill-conditionedness) as well as "real-life" examples (whatever this may mean). Ideally, you have an algorithm which puts out matrices with certain properties in a "reasonable" probability measure. A good notion of "reasonable" might be accessible, as most such LSE problems from physics or simulations have much more structure as is actually demanded by the algorithms in theory.

In so far, I wonder whether there are works in numerical analysis how to, given $n \in \mathbb N$ randomly produce

a sequence of ${n \times n}$-matrices
optionally constraint to be symmetric, positive definite, well-conditioned
which is reasonable in whatever sense

This is probably an interesting topic within the theory of numerical algorithms.

Thanks, Martin

"Exotic" Banach spaces of sequences

2011-01-22T05:29:12Z

Does there exist a linear subspace of $\mathbb C ^{\mathbb N}$ that can be endowed a Banach space topology that is not finer than the locally convex topology of pointwise convergence?

Best, Martin

Intuition for the Hardy space $H^1$ on $R^n$

2011-01-18T08:50:35Z

Hello,

the standard intuition for Lebesgue spaces $L^p(\mathbb R^n)$ for $p \in [1,\infty]$ are measurable functions with certain decay properties at infinity or at the singularities.

In particular, a typical $L^p$ function is a function like $|x|^{-\alpha}$ with $\alpha > \frac{n}{p}$ for $p$ < $\infty$, or simply $1$ for $p = \infty$. Furthermore, functions which are almost everywhere absolute-value-dominated by an $L^p$ function are elements of $L^p$, too. This is useful for approximation arguments, as the pointwise error has just to be dominated by an $\epsilon$-multiple of another $L^p$ function.

In contrast to that, hardy spaces seem to be less intuitive due to cancellation properties. Hence I wonder:

How do "typical" $H^1$ functions look like?
In particular, what can you typically "do" with Hardy functions?

For example, I guess convolution arguments take a more prominent role in approximation arguments than perturbation arguments do, but I am not sure about that.

The hardy space $H^1$ shall be defined via

$f \in H^1 $ if $f \in L^1_{\mathrm{loc}}$ and $Mf \in L^1$, where

$$Mf(x) := \sup \limits_{B_r(x_0) \ni x, \phi \in \mathcal L(B_r(x_0))} \int f \phi dx$$

$$\mathcal{L}(B_r(x_0)) = \left\{ \phi \in C(B_r(x_0)) s.t. |\phi(x)| < \dfrac{\max(r-|x-x_0|,0)}{r|B_r(x_0)|} , \mathrm{Lip}(\phi) < \frac{1}{r|B_r(x_0)|} \right\}$$

Matrices as dynamical systems

2011-01-16T05:28:52Z

Matrices can be understood in different ways, e.g.

Linear systems of equations
(rich algebraic structure of) Linear mappings
Graphs
Evolution law of discrete-time Dynamical system

Well, 1. und 2. are the most prominent ones, canonically tought and widely understood - in particular, understanding each of these nurtures understanding the other one.

Similarly, 3. and 4. are fairly close. Consider as an example probability diffusion on a graph, which is usually modeled by iterated powers of a stochastic matrix.

It would be interesting to have correlate these two aspects of matrices. Of course, I am aware of algebraic graph theory, but I am not aware of a mutual enrichment of the "evolution perspective" (4.) with the "algebraic perspective". (1. & 2.). For example I would be interesting to have interpretations of the trace and the determinant from the former perspective.

Question: Is there theoretical research into that direction? Can show a good book about this topic?

A senseful meaning of 'approximation of manifolds'?

2010-12-13T09:32:18Z

Any continuous function can be uniformly approximated by smooth functions.

I would like to have something similar - in what-ever sense - for continuous manifolds.

For example, by Whitney's theorem, any $n$-dimensional topological manifold $M$ can be continuously embedded into the larger-dimensional euclidian space $\mathbb R^{2n}$. You can construct a continuous function $f$ with image $[-1,1]$ on $\mathbb R^{2n}$, whose zero level set is exactly (the image of) $M$.

A meaning of "approximating a manifold" would be to approximate such a level set function by smooth functions. However, Whitney's theorem is non constructive, you need a metric on the manifold for the question to make sense, and there are likely to appear difficulties.

Do you where to find a elaboration on questions like the above? (Of course, different approaches are of interest as well.). Thank you.

Topology on the set of linear subspaces

2011-01-09T17:37:38Z

Hello,

let $X$ be a seperable Hilbert space. Let $(e_i)_i$ be a Hilbert basis, and for each index let $E_i = \langle e_1,\dots,e_i \rangle \subset X$ the span of the first $i$ basis vectors. For any $x \in X$, let $x_i$ be the best-approximation of $x$ in $E_i$, and it is clear that $x_i \rightarrow x$.

It seems intuitive to say, that the $(E_i)$ approximate $X$ in a certain sense. Nevertheless, I am not aware of a topology on the set of linear subspaces, which would give such a result rigorously.

A first attempt might be to identify each linear subspace with its projection-onto, and inspect these projections as a topological (no more linear) space. A next step might be to take into account the order of the basis vectors for each linear subspace (which might be crucial for stability in numerical analysis). I am not known a theory Grassmannian manifolds in infitinte-dimensional vector spaces, nor how to relate non-equidimensional Grassmannian manifolds.

Can you give me hints where to find theory into this direction?

Comment by Martin

2013-03-27T21:12:44Z

For whatever reason ... Well, thanks for the advice. It is corrected now.

Comment by Martin

2013-03-17T22:09:30Z

@timur: Can you give a source for that speculation?

Comment by Martin

2013-03-07T20:32:08Z

btw, it seems that the curly brackets in the first set are not rendered correctly, for whatever reason.

Comment by Martin

2012-08-02T17:19:12Z

yes, I meant not manifolds with corners. --- It seems that even manifolds with corners are only a very small subbranch that has barely been investigated and canonicalized.

Comment by Martin

2012-07-18T12:20:40Z

@Paul: Don't you need an inner product structure on $V$ to have canonical isomorphisms between primal and dual objects, do you?

Comment by Martin

2012-07-17T12:04:43Z

Yes, the question is about "applications" in the narrow sense, i.e., outside of pure mathematics.

Comment by Martin

2012-07-02T13:30:20Z

@fedja: $P$ seems to be polynomial in $n+1$ variables, isn't it?

Comment by Martin

2012-05-08T17:24:04Z

@timur: Thanks for your reference to Taylor, which I learned about just yesterday. Still, in the refered section he assumes a smooth boundary from the beginning. Furthermore, how ellipticity of the Hodge Laplacian enforces regularity is not clear to me - in case you mean elliptic regularity, I am aware that the solutions for the Hodge-Laplacian with right-hand side in L^2 and, say, homogenous tangential boundary conditions may gain regularity between 1/2 and 1 for Lipschitz domains. Yet, I do not know how more regular data imply more regular solutions, as in the scalar case.

Comment by Martin

2012-04-30T14:38:13Z

You can use Hahn-Banach to construct a functional in X′′ but not in the inclusion of X which "sees" that A does not generate X′. So lack of the property implies non-reflexivity. But this is not the converse.

Comment by Martin

2012-04-30T13:36:06Z

I missed some notation, i have corrected this.

Comment by Martin

2012-04-11T19:54:05Z

@Lentner. Thank you very much for your answer. I will read until weekend --- quite busy at the moment. Furthermore, thanks to Qiaochu for your answer and the other commentors. Really great this question allows so many perspectives!

Comment by Martin

2012-04-07T21:16:27Z

At least on my computer, the term $(X \otimes Y ){p} := \oplus{k+l=p} X_k \otimes Y_l$ does not seem to render as expected. If others see this as well, I am sorry about that.

Comment by Martin

2012-03-10T02:52:28Z

Thanks. The identity for dual vectors that I mentioned holds only for seperable Hilbert spaces.

Comment by Martin

2012-03-10T02:01:18Z

Corrected. Thank you very much. -- However, a similar formula does exist when we regard the Hilbert space $L(X,\mathbb R)$ of bounded linear functionals. Whence this is not necessarily the last word.

Comment by Martin

2012-03-04T01:43:55Z

Follow up questions: (i) Are these two notions of orthogonality symmetric? $\perp_2$ certainly is, but what about $\perp_1$? (ii) Are these notions invariant under scaling? For example, if $x \perp_1 y$, then $x \perp_1 \beta y$ for any $\beta \in K$. Most importantly, (iii): In which contexts are these two notions of orthogonality useful?