Computer package for representation theory of the symmetric group

2012-05-11T20:29:27Z

Is there a computer algebra package in which I can compute the following for representations of a specific symmetric group (e.g. $S_7$):

(1) $V \otimes W$

(2) $S_\lambda V$, where $S_\lambda$ is a Schur functor, or even just $\wedge^s V$,

where $V$ and $W$ are input as sums of irreducible representations, i.e. by partitions with coefficients, and output in the same format?

Does equality of Hodge star and symplectic star imply Kähler structure?

2012-05-16T21:48:44Z

Question

The question asked is:

On a manifold $M$ equipped with a Riemann metric $g$ and a symplectic structure $\omega$, with $\ast$ the Hodge star and $\ast_s$ the symplectic star, does $\ast=\ast_s$ iff $(M,g,\omega)$ is Kähler?

Definitions

Here the star operators $\ast$ and $\ast_s$ are defined in the usual way (see for example, equation 3.9 of Tseng and Yau Cohomology and Hodge Theory on Symplectic Manifolds: II):

$$ A\wedge \ast B := \langle A,B\rangle_{g^{-1}} dV_g$$ $$A\wedge \ast_{s} B := \langle A,B\rangle_{\omega^{-1}} dV_\omega $$

where $A$ and $B$ are $k$-forms, $\langle\cdot,\cdot\rangle_g$ is the inner product associated to $g$, $dV_g$ is the volume form associated to $g$, and $\langle\cdot,\cdot\rangle_\omega$ and $dV_\omega$ are defined analogously with respect to $\omega$.

Motivation

In seeking a mathematically natural link between Hilbert-space expositions of quantum mechanics and product space expositions, both the Hodge star operator and the symplectic star operator enter naturally (for example, in the context of Onsager theory). Moreover, in cases of practical quantum systems engineering interest it is commonly observed that:

the two star operations are identical, and
the dynamical state-manifold is Kählerian.

Does each of these observations mathematically imply the other? An engineer-friendly reference for this fact (if it is a fact) would be very welcome---it's not easy to find discussion of the practical relevance of the symplectic star operation.

about the local ring of $\mathbb{Z}_p[T]/(pT^2+T+1)$ at the prime p

2012-05-16T20:49:58Z

is the localisation of the ring $$A:=\mathbb{Z}_p[T]/(pT^2+T+1)$$ at the prime ideal (p) isomorphic to $\mathbb{Z}_p$?

If not, how to understand this ring very explicitly?

Rabin's Tree Theorem

2012-05-16T21:06:08Z

I've been reading Rabin's article on decidability in Barwise's text, and I came across Rabin's discussion of the decidability proof of his tree theory: the second-order theory with two successor functions. The text mentions that the proof is hard and very technical owing to many extensions of automata theory, and I was wondering if someone might be able to sketch it out.

Thanks!

What is a complex inner product space "really"?

2011-11-03T08:56:12Z

This is an extended re-post of a question that I have asked on MSE not a long time ago. But anyway, it seems more appropriate for MO.

To begin with, in a real inner product space we have a geometric intuition for the inner product. In the finite dimensional case, the inner product of two vectors is the product of their lengths (norms) times the cosine of the angle between them. Reverse engineering this suggests that the purely algebraic properties of an abstract inner product give it the properties of the "length of the projection" (scaled somehow). In particular, we can think of two vectors with zero inner product as orthogonal geometrically (and even call them that way) and bring all the geometric notions related to orthogonality to the abstract, possibley infinite dimensional case (with the appropriate care and restrictions of course). My question is, what is the geometric or otherwise intuition behind the abstract notion of a complex inner product space?

Here are a few thoughts:

1) This is a good mathematical structure to model some physical phenomena. An (if not the) example is quantum mechanics. This is an interesting line of thought. One problem is that I don't know enough quantum mechanics to follow it more deeply. If it is possible to explain in an elementary as possible way, What does it mean that a state of a particle is an element of a complex Hilbert space I would very much want to hear about it. I would also like to hear about other, hopefully more elementary, phenomena modeled by complex inner spaces. In particular, ia there any such phenomenon modeled by a finite-dimensional complex inner product space?

2) This is a good mathematical tool for other mathematical theories. Perhaps, unitary representations of groups or of other algebraic structures. Again, I don't have enough background in representation theory myself. explicit examples of such utility are welcome.

3) it is a good tool to investigate real structures by complexification and exploit of the good properies of the complex field (such as being algebraically closed). There are a lot of such examples in linear algebra, such as the classification of orthogonal maps, but I haven't seen such examples in the context of inner product spaces.

I would like to stress that saying that this is somehow algebraically natural analogue of real inner product spaces and that it has a lot of nice properties is somehow not enough in my opinion. Also not very satisfying is saying that it has application in such and such very advanced theories without elaboration.

Thanks!

Adjoint of a Connection Using the Hodge Map?

2012-05-15T22:34:04Z

For a Riemannian manifold $(M,g)$ with exterior derivative d, the codifferential d$^\ast$ is defined to be the unique map for which $$ g(\omega,d\omega') = g(d^* \omega,\omega'), ~~~ \omega,\omega' \in \Omega^{\bullet}. $$ Now if $\ast$ is the Hodge map for $g$, then it is not too difficult to show that d$= (-1)^k\ast$ d $\ast$, when acting on $\Omega^k(M)$.

When $M$ is a complex manifold with holomorphic and anti-holomorphic partial derivatives $\partial$, and $\overline{\partial}$, we have a similarly defined $\partial^\ast$, and $\overline{\partial}^\ast$, and a similar relation between these objects and the original derivatives involving the Hodge map (well actually there's a reversal but no matter). For the Lefschetz map something similar also happens.

What I would like to know is whether the adjoint $\nabla^*$ of the Levi--Civita connection $\nabla$ has some similar re-expression? Or is this too naive?

An algorithm for checking if a nonlinear function f is always positive

2012-05-16T19:45:41Z

Is there an algorithm to check if a given (possibly nonlinear) function f is always positive?

The idea that I currently have is to find the roots of the function (using newton-raphson algorithm or similar techniques, see http://en.wikipedia.org/wiki/Root-finding_algorithm) and check for derivatives, or finding the minimum of f, but they don't seems to be the best solution to this problem, also there are a lot of convergence issues with root finding algorithms.

For example, in Maple, function verify can do this, but I need to implement it in my own program. Maple Help on verify: http://www.maplesoft.com/support/help/Maple/view.aspx?path=verify/function_shells Maple example: assume(x,'real'); verify(x^2+1,0,'greater_than' ); --> returns true, since for every x we have x^2+1 > 0

Mirror question on stack-exchange: http://stackoverflow.com/questions/10625585/an-algorithm-for-checking-if-a-nonlinear-function-f-is-always-positive

[edit] Some background on the question: The function $f$ is the right hand-side differential nonlinear model for a circuit. A nonlinear circuit can be modeled as a set of ordinary differential equations by applying modified nodal analysis (MNA), for sake of simplicity, let's consider only systems with 1 dimension, so $x' = f(x)$ where $f$ describes the circuit, for example $f$ can be $f(x) = 10x - 100x^2 + 200x^3 - 300x^4 + 100x^5$ ( A model for nonlinear tunnel-diode) or $f=10 - 2sin(4x)+ 3x$ (A model for josephson junction).

$x$ is bounded and $f$ is only defined in interval $[a,b] \in R$. $f$ is continuous. I can also make an assumption that $f$ is Lipschitz with Lipschitz constant L>0, but I don't want to unless I have to.

How to bound the sup norm of a Rademacher process or equivalently a Gaussian process?

2012-04-18T17:38:23Z

I want to know how to find an upper bound of the following expectation taken for both $t$ and $y$ as

$$\mathbb{E}\sup_{x \in D} \left|\sum_{k=1}^n t_k x^T y_k\right|,$$ where $D$ is the set of vectors defined by $$D = ( x \in \mathbb{R}^m \mid 0\leq x_i \leq 1, \forall 1\leq i\leq m ),$$

$\left(t_k\right)_{k=1}^n$ is the Rademacher sequence, that is, $t_1, \cdots, t_n$ are i.i.d. copies of a random variable $t$ taking values $\pm 1$ with $\mathbb{P}(t=1)=\mathbb{P}(t=-1)=1/2$, and $(y_k)$ are i.i.d. copies of a random vector $y \in \mathbb{R}^m$ taking values $e_1,\cdots,e_m$ with $\mathbb{P}(y = e_i)=p_i$. Here, $e_i$ denotes the vector from the standard basis with $i$-th component being 1 and the others being 0.

I first get rid of the absolute value as \begin{align} & \mathbb{E}\sup_x \left|\sum t_k x^T y_k\right| \leq \mathbb{E}_y\left(\sqrt{\frac{\pi}{2}} \mathbb{E}_s \sup_x\left| \sum s_k x^T y_k\right|\right) \\ \leq & \sqrt{2\pi} \mathbb{E}_y\left(\mathbb{E}_s \sup_x\left(\sum s_k x^T y_k \right)\right) = \sqrt{2\pi} \mathbb{E} \sup_x\left(\sum s_k x^T y_k \right), \end{align} where $s_k$ are i.i.d copies of a standard normal random variable.

Then, how to continue? My guess is that the upper bound seems to be of order $O(\sqrt{n})$. Is that correct? Thanks!

Tensors with low spectral norm

2012-05-16T20:03:03Z

Consider a tensor $T$ with six indices, $T_{(ii')(jj')(kk')}$, where each index goes from $1$ to $n$. We can think of $T$ as a linear map from $\mathbb{R}^n \otimes \mathbb{R}^n \otimes \mathbb{R}^n$ to itself and consider its spectral norm:

$ \Vert T \Vert_{\infty} = \sup \vert \sum T_{(ii')(jj')(kk')} x_{ijk}y_{i'j'k'}\vert $

where the supremum is taken over all unit vectors $x,y \in \mathbb{R}^n \otimes \mathbb{R}^n \otimes \mathbb{R}^n$.

On the other hand $T$ can also be viewed as trilinear form on $n$-dimensional matrices, so we can define a norm:

$ \Vert T \Vert_{2,2,2} = \sup \vert \sum T_{(ii')(jj')(kk')} X_{ii'} Y_{jj'}Z_{kk'}\vert $

where the supremum is over all matrices $X,Y,Z$ of Frobenius norm $1$.

What are the examples of tensors which have high (as $n \to \infty$) spectral norm as linear maps, but low norm as trilinear forms?

Since the question is admittedly rather general, let's specialize to tensors of a more special form, namely $T_{(ii')(jj')(kk')} = g_{ijk}h_{i'j'k'}$, where $g,h \in \mathbb{R}^n \otimes \mathbb{R}^n \otimes \mathbb{R}^n$, so that $T = gh^{T}$ as a linear map. Then its spectral norm is simply $\Vert g\Vert \cdot \Vert h\Vert$. Taking $g$ and $h$, say, to be unit vectors, how should we choose them to get a low trilinear norm:

$\Vert T \Vert_{2,2,2} = \sup \vert \sum g_{ijk}h_{i'j'k'} X_{ii'} Y_{jj'}Z_{kk'}\vert $ ?

It can be shown that randomly chosen $g, h$ will have the desired property, but I'm interested in more explicit examples. Because there is no direct analog of the spectral decomposition for tensors, the intuition that the "mass" of $T$ should be "spread out" roughly in all directions (as in the case of matrices with low spectral norm, but high Frobenius norm) on the $\mathbb{R}^{n^2}$ components of the tensor product is not easy to formalize.

Min Bend Orthogonal Knots

2012-05-16T12:47:25Z

I am seeking literature on 3D orthogonal drawings of knots, especially minimum bend drawings. An orthogonal drawing employs segments parallel to the axes of a Cartesian coordinate system. A bend is a vertex at which two segments meet orthogonally. A drawing insists on simplicity in the sense that nonadjacent segments are disjoint, and adjacent segments meet only at their shared endpoint.

One can imagine first drawing a 2D projection with a minimal number of crossings and then removing the crossings. For the trefoil below, naive crossing-removal increments the $8$ bends in the 2D drawing to $8 + 3 \cdot 4 = 20$ bends, but the trefoil can be drawn with $12$ bends:

I would be especially interested in algorithmic methods to derive the right 3D drawing above from the left 2D projection. Thanks for ideas and pointers!

Integer matrices with a strange divisibility property

2012-05-16T19:23:38Z

Fix an integer $n$. What can you say about a (not necessarily square) matrix $A$ with integer entries that has the property that for any $k$, every $k\times k$ minor of $A$ is divisible by $n^{k-1}$? Have you seen such matrices before? Do they have a name? Are they in a bijection with some set whose description does not require knowing about determinants?

The question I really care about is the same but with "many-variable Laurent polynomials with integer coefficients" replacing the integers in the above (except $k$), but I suspect that it doesn't really make a difference.

The reason I care is that I have but I don't understand an amusing (I think) generalized Alexander invariant of tangles and virtual tangles with excellent composition properties and with values in such matrices as above, and I would like to understand its target space. A handout and a video are at http://www.math.toronto.edu/~drorbn/Talks/GWU-1203/ but no writeup exists at present.

Modular representations with unequal characteristic - reference request

2012-05-16T10:09:03Z

Let $G$ be a finite group, and let $K$ be a finite field whose characteristic does not divide $|G|$. I am interested in the theory of finitely generated modules over $K[G]$. Of course many problems are not present here because $K[G]$ is semisimple and all modules are projective. My case is partly covered by Section 15.5 of Serre's book "Linear Representations of Finite Groups". However, Serre likes to assume that $K$ is "sufficiently large", meaning that it has a primitive $m$'th root of unity, where $m$ is the least common multiple of the orders of the elements of $G$. I do not want to assume this, so some Galois theory of finite extensions of $K$ will come into play. I do not think that anything desperately complicated happens, but it would be convenient if I could refer to the literature rather than having to write it out myself. Is there a good source for this?

[UPDATED]:

In particular, I would like to be able to control the dimensions over $K$ of the simple $K[G]$-modules. As pointed out in Alex Bartel's answer, these need not divide the order of $G$. I am willing to assume that $G$ is a $p$-group for some prime $p\neq\text{char}(K)$.

[UPDATED AGAIN]:

OK, here is a sharper question. Put $m=|K|$ (which is a power of a prime different from $p$) and let $t$ be the order of $m$ in $(\mathbb{Z}/p)^\times$. Let $L$ be a finite extension of $K$, let $G$ be a finite abelian $p$-group, and let $\rho:G\to L^\times$ be a homomorphism that does not factor through the unit group of any proper subfield containing $K$. Then $\rho$ makes $L$ into an irreducible $K$-linear representation of $G$, and every irreducible arises in this way. If I've got this straight, we see that the possible degrees of nontrivial irreducible $K$-linear representations of abelian $p$-groups are the numbers $tp^k$ for $k\geq 0$. I ask: if we let $G$ be a nonabelian $p$-group, does the set of possible degrees get any bigger?

How to quantify noncommutativity?

2012-05-11T17:39:14Z

If I have two operators or finite-dimensional matrices $A$ and $B$, how can I quantify the amount to which they commute or don't commute? (I would consider it a big plus if it is computable easily for finite complex-valued matrices $A, B \in \mathbb C^{n\times n}$.)

Let me try the obvious thing here: by definition if $A$ and $B$ commute, then the commutator $[A, B] = AB-BA = 0$. Naively would use some sort of functional like an operator norm to reduce this to a number that could potentially behave like a metric. The first thing I thought of was the trace, but clearly that doesn't work since $\mathrm{tr } [A, B] =\mathrm{tr } (AB-BA) = \mathrm{tr }AB - \mathrm{tr }AB = 0$ always. One could then turn to, say, the Frobenius norm of $[A, B]$. What is known about the maximal (or supremal) value of such norms?

Are there quantifiers of noncommutativity that can also account for higher-order effects, e.g. cases where $[A, B] \ne 0$ but $[A, [A,B]] = 0$? This should be "less" non-commuting than if $[A, B] \ne 0$ and $[A, [A,B]] \ne 0$ and $[B, [A,B]] \ne 0$ but, say, $[A, [B, [A, B]]] = 0$.

For those who prefer a free algebraic setting, the question can be framed as: how free is a non-free algebra? Is there a sensible way to measure proximity to a free algebra? What if I had an algebra where $AB=BA$ is the only one relation that makes it not a free algebra; is there a sense it is "less free" or "more free" than an algebra where $ABABAB=BAA$ is the only such relation, or example.

Motivation: it is sometimes said that free probability is the study of "maximally" non-commuting objects. I would like to know if this statement can be made precise in the sense of how one can define "maximally non-commuting" in a sensible fashion.

Functions holomorphic on a region minus a Cantor set

2012-05-16T16:51:11Z

Let $X$ and $Y$ be simply connected open regions of $\mathbb{C}$, and let $Z \subset X$ be a Cantor set. Assume we have a homeomorphism $f$ from $X$ to $Y$, which is holomorphic on $X \setminus Z$. Is $f$ necessarily holomorphic on $X$?

What is the difference between "up to conjugacy" and "up to conjugation" ?

2012-05-16T16:55:49Z

I see many times the words "conjugacy" and "conjugation", and I don't really get the difference between the two. Especially, when we take an element of a group and want to say that this has some property "up to conjugacy/tion", which one is better, and why?

The definition of the $K$-theory groups $K_{0}$ and $K_{1}$.

2012-05-14T12:18:19Z

I have read few textbooks and papers about the $K$-theory groups $K_{0}$ and $K_{1}$ of (reduced) $C^{*}$-algebra and most of them didn't give a clear simple way to define these groups.

Just wondering if anybody can give me good sources for that?

Integer square $2 \times 2$ block matrix inverse

2012-05-16T18:02:46Z

Let $\mathbf{M}$ be an integer square $2 \times 2$ block matrix $$ \mathbf{M} = \left( \begin{array}{cc} \mathbf{A} & \mathbf{B} \ \mathbf{C} & \mathbf{D} \end{array} \right) , $$ where $\mathbf{A}$ and $\mathbf{D}$ are square matrices (not necessarily of the same size). Is there a way of testing if $\mathbf{M}$ is regular ($\det (\mathbf{M}) = \pm 1$) in terms of $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ and $\mathbf{D}$?

For example, is it known some expression of $\det (\mathbf{M})$ in terms of $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ and $\mathbf{D}$ without any extra assumption (such as some regularity or certain commutativity relations) on the blocks? And if you assume that $\mathbf{A}$ is an $1 \times 1$ matrix (i.e. $\mathbf{A} \in \mathbb{Z}$)?

Genericity of sets without unique mean value

2012-04-18T07:09:19Z

Following Rosenblatt and Yang, I say that a subset $A$ of $\mathbb Z$ has a unique mean value if for all invariant means $\lambda_1,\lambda_2$ on $\mathbb Z$, one has $\lambda_1(A)=\lambda_2(A)$.

Notice that the set of subsets having a unique mean value has the same cardinality as the set of subsets non-having a unique mean value, thanks to Andreas Thom's answer to http://mathoverflow.net/questions/65325/intrinsically-measurable-subsets-of-amenable-semigroups.

Nevertheless, roughly speaking, it should be clear that most subsets of $\mathbb Z$ should not have a unique mean value. Indeed, to a have a unique mean value one needs a very particular structure, as almost-periodicity.

Question: Is there any probabilistic way to formalize the intuition that generic subsets of $\mathbb Z$ does not have a unique mean value?

Thank you in advance,

Valerio

What is the "Lefschetz Principle" (examples) ?

2012-05-16T08:50:16Z

Hi there, can anyone explain to me what the "Lefschetz Principle" is by some clear "classical" examples (not relying explicitly on model theory, say). Thanks !

How to find the minimum number of hyperplanes to define a convex hull

2012-05-16T15:17:00Z

Hi everybody, I have the following problem:

I have a convex hull $\Omega$ defined by a set of n-dimensional hyperplanes $S = [(n_1,d_1), (n_2,d_2),...,(n_k,d_k)]$ such that a point $p \in \Omega$ if $n_i^T p \geq d_i \quad \forall n_i,d_i \in S $. Now I have a "joining" hyperplane $n_k,d_k$ and I want to know if this hyperplane "modifies the shape" of the convex hull and in that case, which hyperplanes of $S \bigcup n_k,d_k$ are not necessary anymore because redundant.

Trivial example with one dimension:

My convex hull is described by the inequality $ 3 \leq x \leq 5$ so

$S = [(1,3),(-1,-5)]$

The joining hyperplane is the inequality $ x \geq 4$ so the resulting convex hull should be $ 4 \leq x \leq 5$

$S = [(1,4),(-1,-5)]$

returning $(1,3)$.

Now I would like the same thing generalized for n-dimensions. I can get algorithms till 3 dimensions but they are not generalized.

Do you have any hints or pointers on how I can find a solution to this problem?

p.s. I apologize for the sloppy description, I am not a mathematician. Please feel free to ask for more details

Kind regards.

Permutation characters and regular orbits

2012-05-12T05:38:01Z

Let a group $G$ act on a finite set $\Omega$. Suppose that the corresponding permutation character has a regular component. Does it follow that $\Omega$ has a regular $G$-orbit? (The converse is obviously true.)

Correspondences on curves and their induced maps on differentials?

2012-05-07T19:59:51Z

How does a correspondence on an algebraic curve $C$ induce a map on $\Omega^1_C$? Apparently it passes through the Jacobian of $C$ but I don't quite understand it.

More specifically, I was reading a paper that said roughly the following:

Let C be a curve and $\Gamma$ (with maps $\pi_i: \Gamma \rightarrow C$ for $i=1,2$) be a correspondence on $C$. The map $\pi_2$ is a double cover and $\tau: \Gamma \rightarrow \Gamma$ is a map that switches the elements of the fibers of $\pi_2$. Then the induced map on $H^0(C, \Omega^1)$ is given by: $\omega \mapsto \pi_1^* \omega + \tau^* \pi_1^* \omega$, where the differentials of $C$ are identified with the ones of $\Gamma$ that are $\tau^*$-invariant.

I don't understand why the map on $H^0(C, \Omega^1)$ is what it is claimed to be (even assuming the mentioned identification). I guess this follows from general theory of correspondences but I don't know where I would find such a statement.

Planar sets closed under intersection of circles

2012-05-15T14:47:31Z

Let $P$ be the plane with a point at infinity. By plane, I mean the Euclidian plane, and therefore it has circles. A line is also a circle, though its center is at infinity. If $A\subset P$ has cardinal $|A|=3$, there exists a unique circle (possibly a line) containing $A$; let me denote it $\Gamma_A$.

Let me say that a subset $X$ of $P$ is circularly stable if it satisfies the following property:

For every subset $A,B\subset X$ with $|A|=|B|=3$ and $A\ne B$, the intersection of $\Gamma_A$ and $\Gamma_B$ is included in $X$.

Every non-colinear $X$ with $|X|\le4$ is circularly stable. A line is circularly stable, and $P$ itself is so too. If $X$ has a non-void interior, then $X=P$.

Q: What can look like a circularly stable subset $X$ of the plane ? For instance, is it true that if $|X|\ge5$ and $X$ is circularly stable, then $X$ is dense in $P$, or $X$ is a line ?

Motivation: In classical geometry, one may wander what are the continuous maps $f:P\rightarrow P$ which transform circles or lines into circles or lines. If the guess above is true, then $f$ is completely determined by the images of $5$ non-colinear points.

Matrix Operations Preserving Hurwitz Stability

2012-05-16T03:03:35Z

I begin with terminology I use in the question. A real square matrix $A$ is

negative-stable if for every eigenvalue $\lambda$ of $A$, ${\mathrm{Re}}(\lambda) < 0$;
$\ast$-negative-stable if for every eigenvalue $\lambda$ of $A$, either $\lambda = 0$ or ${\mathrm{Re}}(\lambda) < 0$;
nonpositive-stable if for every eigenvalue $\lambda$ of $A$, ${\mathrm{Re}}(\lambda) \leqslant 0$.

I made up the term '$\ast$-negative-stable' and I would welcome better and/or established terminology. For example, the Laplacian matrix of a nonnegatively weighted (directed or undirected) graph is $\ast$-negative-stable.

To put it broadly, I am looking for what is known about matrix operations that preserve the above stability properties.

Let $A$ be a real $n{\times}n$ matrix and let $u$, $v$, $w$ be real $n{\times}1$ vectors. Consider the real $n{\times}n$ matrices $D = \mathrm{diag}(u)$ and $B = vw^{\mathrm{T}}$, and the real number $\alpha = w^{\mathrm{T}}v$. I am particularly interested in what additional conditions on the matrix $A$ would make the following implications true. (I do not mean simultaneously true.) They concern preserving stability from $A$ to $AD$ for the first three and from $A$ to $A+B$ for the last three.

( $A$ is negative-stable and $u$ is positive ) $\Rightarrow$ ( $AD$ is negative-stable )
( $A$ is $\ast$-negative-stable and $u$ is nonnegative ) $\Rightarrow$ ( $AD$ is $\ast$-negative-stable )
( $A$ is nonpositive-stable and $u$ is nonnegative ) $\Rightarrow$ ( $AD$ is nonpositive-stable )
( $A$ is negative-stable and $\alpha < 0$ ) $\Rightarrow$ ( $A + B$ is negative-stable )
( $A$ is $\ast$-negative-stable and $\alpha \leqslant 0$ ) $\Rightarrow$ ( $A + B$ is $\ast$-negative-stable )
( $A$ is nonpositive-stable and $\alpha \leqslant 0$ ) $\Rightarrow$ ( $A + B$ is nonpositive-stable )

In implications 2 and 5 about $\ast$-negative-stability, it would be acceptable to assume that $A$ is similar to a Laplacian matrix (but Laplacian matrices should not be assumed to be symmetric). Would that be sufficient?

Addendum 1

Here is a way $\ast$-negative stability can be useful in studying negative/Hurwitz stability. Suppose I know that a matrix $A$ is similar to $C = \begin{pmatrix} O_{p{\times}p} & O_{p{\times}q} \\ S & T \end{pmatrix}$, where $T$ is nonsingular. Then $T$ is negative-stable if and only if $A$ is $\ast$-negative-stable, a potentially useful observation if $A$ looks easier to work with than $T$. The notion of nonpositive stability can become useful in similar (but not identical) circumstances.

Is any Morse trajectory contained in a contractible open set?

2012-04-18T12:55:16Z

Suppose $f$ is a Morse function on a Riemannian Hilbert manifold $M$. Let $p_{\pm}\in \text{Crit}(f)$ be given and fix some $u:R\rightarrow M$ which is an integral curve of $-\nabla f$ connecting $p_-$ and $p_+.$ Is it true that there is an open contractible set $U\subseteq M$ containing the image of $u$?

L_2-norm representation

2012-05-16T17:10:18Z

Let $$ f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+, $$ where $\alpha > -\frac 12$. I am wondering if one can get nice representation of $L^2$-norm of the function $f^{\alpha}(x)$, namely $$ \int_{-\infty}^{\infty}(f^{\alpha}_+(x))^2dx. $$

(Here $y^{\alpha}_+=\max(0, y)^{\alpha}$ is a one-sided power function).

Thank you.

Complexity of finding a 0-1 vector in a subspace or showing that there is none

2012-05-16T08:37:05Z

This question, is a slightly different disguise (see below), came up in discussions of this question about equitable partitions

A $0,1$ vector in $\mathbb{Z}^n$ is any vector with all entries $0$ and $1$ (at least one of each.) This excludes (for temporary convenience) the all ones vector $\mathbf{1}.$

Q: Given a set of $k+1$ vectors in $\mathbb{Z}^n$, one being $\mathbf{1}$, How difficult is it to determine if there is a (rational) linear combination which is a $0,1$ vector?

It would be equivalent to say that we have a set of $k$ integer vectors such the $n$ entries of each vector sum to $0.$ We wonder if there is a linear combination having only two distinct entries.

In the context of the earlier question the $k$ vectors might be a basis for a certain eigenspace of (the adjacency matrix of) a given regular graph. A linear combination with only two distinct entries would indicate an intersetig two cell partition of the vertices of the graph. If the graph is connected then the largest eigenvalue is the common degree. Then the $k+1$ vector form of the question deals with a basis for the span of two eigenspaces.

Video lectures of mathematics courses available online for free

2011-02-05T18:34:58Z

It can be difficult to learn mathematics on your own from textbooks, and I often wish universities videotaped their mathematics courses and distributed them for free online. Fortunately, some universities do that (albeit to a very limited extent), and I hope we can compile here a list of all the mathematics courses one can view in their entirety online.

Please only post videos of entire courses; that is, a speaker giving one lecture introducing a subject to the audience should be off-limits, but a sequence of, say, 30 hour-long videos, each of which is a lecture delivered in a class would be very much on-topic.

Optimization problem arising from the study of zeta zeros

2012-05-15T19:57:06Z

Motivation: The following problem arose in [1] while studying the vertical distribution of the zeros of the Riemann zeta-function. At the time, my collaborators and I were unable to solve it and I have never been able to derive a "satisfying answer."

Set-up: Let $r\ge 1$ and let $f \in L^2[0,1]$ be a continuous real-valued function of bounded variation on $[0,1]$, normalized so that $$ \int_0^1(1-u)^{r^2-1}f(u)^2 du = 1. $$ Further define $M(c)=M(c,f,r)$ as $$ M(c):=c+\frac{2 r}{\pi}\int_0^1 (1-u)^{r^2-1}f(u) \int_0^u \frac{\sin(\pi c v)}{v} f(u-v) \ dv \ du.$$

Question: How does one choose $r$ and $f$ optimally so that $$ M(c) >1$$ for $c$ as small as possible?

An argument of Conrey, Ghosh, and Gonek [2] can be used to show that $M(c)<1$ if $c<\frac{1}{2}$ for any such $f$ and $r$. In [1], choosing $f$ to be a polynomial of low degree ($\le 6$) and using Mathematica to numerically optimize the $r$ and the coefficients, we were able to find $f$ and $r$ such that $M(.5155)>1$.

In the special case when $r=1$, Montgomery and Odlyzko [3] observed that this optimization problem had already been solved using prolate spheroidal wave functions (see comments below). Here is how they reduced the problem to one that was already solved. In this case, we have $$ M(c) = c+ \int_0^1\int_0^1 f(u) f(v) \frac{\sin(\pi c(u-v))}{\pi(u-v)} \ dv \ du. $$ The double integral on the right-hand side is $$ \int_{-c/2}^{c/2} \left| \int_0^1 f(v) e^{2\pi i v} dv \right|^2 dt :=I(c),$$ say. They then observed that choosing $$ f(x) = aR_{00}^{(1)}[\pi c/2,2x-1]$$ maximizes $I(c)$, where $R_{mn}^{(1)}[c,x]$ is the radial prolate spheroidal wave function of the first kind of order m and degree n, and a is a constant to be chosen according to our above normalization.

These wave functions are very hard to study numerically, and Montgomery and Odlyzko approximated them using modified Bessel functions. In [1], when $r=1$, we recovered their results to four decimal places using polynomials of degree four. So in this case it seems that polynomials of small degree work (almost) as well as more sophisticated techniques.

References:

[1] H. M. Bui, M. B. Milinovich, and N. C. Ng, A note on the gaps between consecutive zeros of the Riemann zeta-function, Proc. Amer. Math. Soc. 138 (2010), no. 12, pp. 4167-4175.

[2] J. B. Conrey, A. Ghosh, and S. M. Gonek, A note on gaps between zeros of the zeta function, Bull. London Math. Soc. 16 (1984), 421–424.

[3] H. L. Montgomery and A. M. Odlyzko, Gaps between zeros of the zeta function, Colloq. Math. Soc. Janos Bolyai, 34. Topics in Classical Number Theory (Budapest, 1981), North-Holland, Amsterdam, 1984.

Edit/Additional Comments: The solution optimization problem when $r=1$ should probably be attributed to Slepian and Pollak and to Landau and Pollack in Prolate spheroidal wave functions, Fourier analysis and uncertainty I and II, Bell System Tech. J. (1961) 40, pp. 43-61 (I) and pp. 65-84 (II). Among other things, they prove the following results.

Theorem: Let $\alpha(c)$ be the least number such that $$ \int_{-c/2}^{c/2} \left|\int_0^1 f(x) e^{2\pi i x} dx \right|^2 dt \le \alpha(c) \int_0^1 |f(x)|^2 dx $$ for all $f\in L^2[0,1]$. Then $\alpha(c)$ is strictly increasing, $\alpha(c)\lt c$ for $c \gt 0$, $\alpha(c)\sim c$ as $c\to 0^+$, and $\alpha(c)\to 1$ as $c\to \infty$. Moreover, equality is achieved in the above inequality if $f(x) = R_{00}^{(1)}[\pi c/2,2x-1]$.

One of the possible hang-ups in solving the optimization problem when $r>1$ is that I have not been able to "complete" the double integral $$\int_0^1 (1-u)^{r^2-1}f(u) \int_0^u \frac{\sin(\pi c v)}{\pi v} f(u-v) \ dv \ du$$ into a double integral of the form $$ \int_0^1 \int_0^1 [\text{nice integrand}] \ dv \ du,$$ which is the first step in Montgomery & Odlyzko's argument.

Sum of exponential functions

2012-05-16T15:32:51Z

Suppose that we have $q$ positive integers $a_1, \ldots, a_q$ satisfying $a_1 \leq \cdots \leq a_q$. I'm interested in the possible types of behaviors for the function given by $$f(x) = (a_1^{x-1} + \cdots + a_q^{x-1})^{1/x},$$ where $x \in [2,\infty)$. In particular, I'm interested in the behavior at integer values of $x$ in that range, but I think that the continuous version of $f$ might be easier to handle.

It isn't hard to see that $\lim_{x \to \infty} f(x) = a_q$. I can show that if there is an $x_0 \in (2,\infty)$ with $f(x_0) > a_q$, then $f$ is decreasing at $x_0$, and furthermore I can show that $f$ is either eventually decreasing, increasing, or constant (this almost entirely comes from the number of $i$ with $a_i = a_q$).

Each example $(a_1,\ldots,a_q)$ that I've used also has the nice property that it is either always decreasing, always constant (this in fact only occurs if $q=a_q$ and $a_i = a_q$ for each $i$), or decreasing on $[2,x_0)$ and increasing on $(x_0,\infty)$ for some $x_0$ which depends on the $a_i$. Does anyone know of either a proof or counterexample of this property? Is any other behavior possible? Simply computing the derivative of $f$ doesn't seem to be too helpful, but perhaps I'm missing the correct viewpoint on this derivative.

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