pr.probability - Kullback-Leibler divergence of scaled non-central Student's T distribution

What is the Kullback-Leibler divergence of two Student's T distributions that have been shifted and scaled? That is, $textrm{D}_{textrm{KL}}(k_aA + t_a; k_bB + t_b)$ where $A$ and $B$ are Student's T distributions.

If it makes things easier, $A$ could be a Gaussian. (That is, it could have infinite degrees of freedom.)

The motivation behind this question is that the scaled non-central Student's T distribution is the posterior predictive distribution of normally distributed data with unknown mean and variance. Thus, I would like to compare the true distribution $k_aA + t_a$ with the estimate $k_bB + t_b$.

botany - What kind of fruit is this?

Just spotted this fruit while walking to school. It's the size of a small coin.The taste is almost sour and tangy and somewhat sweet (I only tried one of them and very little of it). I admit, I've never seen it before. Does anyone know what this is?

Update: Location is two or three blocks north of UC Berkeley (Berkley, CA, USA).

ag.algebraic geometry - What classes am I missing in the Picard lattice of a Kummer K3 surface?

The lattice $L_{K3}=H^2(K3,mathbb Z)$ is $2E_8+3U$, with $E_8$ negative definite and $U$ the hyperbolic lattice for the bilinear form $xy$. It is unimodular and has signature $(3,19)$.

The 16 (-2)-curves $E_i$ form a sublattice $16A_1$ of determinant $2^{16}$. It is not primitive in $L_{K3}$. The primitive lattice $K$ containing it is computed as follows. Consider a linear combination $F=frac12sum a_i E_i$ with $a_i=0,1$. Recall that $E_i$ are labeled by the 2-torsion points of the torus $A$, i.e. the elements of the group $A[2]$.

Then $F$ is in $K$ $iff$ the function $a:A[2]to mathbb F_2$,
$imapsto a_i$, is affine-linear. You will find the proof of this statement in Barth-(Hulek-)Peters-van de Ven "Compact complex surfaces", VIII.5.
(The element $frac12sum E_i$ in your example corresponds to the constant function 1, which is affine linear).
Thus, $K$ has index $2^5$ in $16A_1$ and its determinant is $2^{16}/(2^5)^2=2^6$.

$K$ is called the Kummer lattice. By the above, it is a concrete negative-definite lattice of rank 16 with determinant $2^6$. Nikulin proved that a K3 surface is a Kummer surface iff $Pic(X)$ contains $K$.

The orthogonal complement $K^{perp}$ of $K$ in $L_{K3}$ is $H^2(A,mathbb Z)$ but with the intersection form multiplied by 2. As a lattice, it is isomorphic to $3U(2)$. It has determinant $2^6$, the same as $K$. The lattice $L_{K3}=H^2(K3,mathbb Z)$ is recovered from the primitive orthogonal summands $K$ and $K^{perp}$.

However, your question has "Picard lattice" in the title. The Picard group of $X$ is strictly smaller than $H^2(X,mathbb Z)$. To begin with, it has signature $(1,r-1)$, not $(3,19)$. For a Kummer surface, it contains Kummer lattice $K$ described above, and its intersection with $K^{perp}$ is the image of the Picard group of $A$. For a Kummer surface one has $r=17,18,19$ or 20.

For the Mori-Kleiman cone of effective curves, which you would need for Gromov-Witten theory, the description you put in a box is already the best possible.

co.combinatorics - How does this relationship between the Catalan numbers and SU(2) generalize?

This is a question, or really more like a cloud of questions, I wanted to ask awhile ago based on this SBS post and this post I wrote

As the SBS post describes, the Catalan numbers can be obtained as the moments of the trace of a random element of $SU(2)$ with respect to the Haar measure. This is equivalent to the integral identity
$$int_{0}^{1} (2 cos pi x)^{2k} (2 sin^2 pi x) , dx = C_k.$$

I can prove this identity "combinatorially" as follows: let $A_n$ denote the Dynkin diagram with $n$ vertices and $n-1$ undirected edges connecting those vertices in sequence. The adjacency matrix of $A_n$ has eigenvectors $mathbf{v}_i$ with entries $mathbf{v}_{i,j} = sin frac{pi ij}{n+1}$ with corresponding eigenvalues $2 cos frac{pi i}{n+1}$. If $k le n-1$, then a straightforward computation shows that the number of closed walks from one end of $A_n$ to itself of length $2k$ is
$$frac{1}{n+1} sum_{i=1}^{n} left( 2 cos frac{pi i}{n+1} right)^{2k} 2 sin^2 frac{pi}{n+1} = C_k$$

by the combinatorial definition of the Catalan numbers. Taking the limit as $n to infty$ gives the integral identity; in other words, the integral identity is in some sense equivalent to the combinatorial definition of the Catalan numbers in terms of closed walks on the "infinite path graph" $A_{infty}$. (Is this the correct notation? I mean the infinite path graph with one end.)

Now, closed walks of length $2k$ from one end of $A_n$ to itself can be put in bijection with ordered rooted trees of depth at most $n$ and $k$ non-root vertices. (Recall that the Catalan numbers also count ordered rooted trees of arbitrary depth.) The generating function $P_n$ of ordered rooted trees of depth at most $n$ satisfies the recursion
$$P_1(x) = 1, P_n(x) = frac{1}{1 - x P_{n-1}(x)}.$$

This is because an ordered rooted tree of depth $n$ is the same thing as a sequence of ordered rooted trees of depth $n-1$ together with a new root. (Recall that the generating function of the Catalan numbers satisfies $C(x) = frac{1}{1 - x C(x)}$. In other words, $C(x)$ has a continued fraction representation, and $P_n$ is its sequence of convergents.) On the other hand, since $P_n(x)$ counts walks on the graph $A_n$, it is possible to write down the generating function $P_n$ explicitly in terms of the characteristic polynomials of the corresponding adjacency matrices, and these polynomials have roots the eigenvalues $2 cos frac{pi i}{n+1}$. This implies that they must be the Chebyshev polynomials of the second kind, i.e. the ones satisfying
$$q_n(2 cos x) = frac{sin (n+1) x}{sin x}.$$

But the Chebyshev polynomials of the second kind are none other than the characters of the irreducible finite-dimensional representations of $SU(2)$! In particular, they're orthogonal with respect to the Haar measure.

The overarching question I have is: how does this sequence of computations generalize, and what conceptual framework ties it together? But I should probably ask more specific sub-questions.

Question 1: I remember hearing that the relationship between the Catalan numbers and the Chebyshev polynomials generalizes to some relation between moments, continued fractions, and orthogonal polynomials with respect to some measure. Where can I find a good reference for this?

Question 2: I believe that adding another edge and considering the family of cycle graphs gives the sequence ${2k choose k}$ and the Chebyshev polynomials of the first kind, both of which are related to $SO(2)$. According to the SBS post, this is a "type B" phenomenon, whereas the Catalan numbers are "type A." What exactly does this mean? What would happen if I repeated the above computations for other Dynkin diagrams? Do I get continued fractions for the other infinite families?

Question 3: Related to the above, in what sense is it natural to relate walks on the Dynkin diagram $A_n$ to representations of $SU(2)$? This seems to have something to do with question #16026. How do the eigenvectors of the adjacency matrices fit into the picture? I want to think of the eigenvectors as "discrete harmonics"; does this point of view make sense? Does it generalize?

As you can see, I'm very confused, so I would greatly appreciate any clarification.

ho.history overview - Good books on problem solving / math olympiad

From a review for Polya's book on Amazon, the books to be read in sequence:

• Mathematical Problem Solving by Alan Schoenfeld


• Thinking Mathematically by J. Mason et al.


• The Art and Craft of Problem Solving by Paul Zeitz


• Problem Solving Strategies by Arthur Engel


• Mathematical Olympiad Challenges by Titu Andreescu


• Problem Solving Through Problems by Loren Larson

Full text of the review below:

By Abhi:

Good aspects of this book have been said by most of the other
reviewers. The main problem with such books is that for slightly
experienced problem solvers, this book probably does not provide a
whole lot of information as to what needs to be done to get better.
For instance, for a kid who is in 10th grade struggling with math,
this is a very good book. For a kid who is in his 11th grade trying
for math Olympiad or for people looking at Putnam, this book won't
provide much help.

Most people simply say that "practice makes perfect". When it comes to
contest level problems, it is not as simple as that. There are
experienced trainers like Professor Titu Andreescu who spend a lot of
time training kids to get better. There is lot more to it than simply
trying out tough problems.

The most common situation occurs when you encounter extremely tough
questions like the Olympiad ones. Most people simply sit and stare at
the problem and don't go beyond that. Even the kids who are extremely
fast with 10th grade math miserably fail. Why?

The ONE book which explains this is titled "Mathematical Problem
Solving" written by Professor Alan Schoenfeld. It is simply amazing. A
must buy. In case you have ever wondered why, in spite of being
lightning fast in solving textbook exercises in the 10th and 11th
grade, you fail in being able to solve even a single problem from the
IMO, you have to read this book. I am surprised to see Polya's book
getting mentioned so very often bu nobody ever mentions Schoenfeld's
book. It is a must read book for ANY math enthusiast and the math
majors.

After reading this book, you will possibly get a picture as to what is
involved in solving higher level math problems especially the
psychology of it. You need to know that as psychology is one of the
greatest hurdles to over when it comes to solving contest problems.
Then you move on to "Thinking Mathematically" written by J. Mason et
al. It has problems which are only few times too hard but most of the
times, have just enough "toughness" for the author to make the point
ONLY IF THE STUDENT TRIES THEM OUT.

The next level would be Paul Zeitz's The Art and Craft of Problem
Solving. This book also explains the mindset needed for solving
problems of the Olympiad kind. At this point, you will probably
realize what ExACTLY it means when others say that "problem solving is
all about practice". All the while you would be thinking "practice
what? I simply cannot make the first move successfully and how can I
practice when I can't even solve one problem even when I tried for
like a month". It is problem solving and not research in math that you
are trying to do. You will probably get a better picture after going
through the above three books.

Finally, you can move on to Arthur Engel's Problem Solving Strategies
and Titu Andreescu's Mathematical Olympiad Challenges if you managed
to get to this point. There is also problem solving through problems
by Loren Larson. These are helpful only if you could solve Paul
Zeitz's book successfully.

To conclude, if you are looking for guidance at the level of math
Olympiad, look for other books. This book won't be of much assistance.
On the other hand, if you are simply trying to get better at grade
school math, this book will be very useful.

ag.algebraic geometry - K3 surface of genus 8

To be able calculate the degree it is worth to read a bit of Griffiths-Harris about Grassmanians (chapter 1 section 5). To prove that $S$ is $K3$ one needs to caluclate the canonical bundle of $G$, use simple facts about Plucker embedding, use adjunction formula and finally the fact that a simply connected surface with $Kcong O$ is $K3$.

I will make the second bit of calculation, that proves that $S$ is a $K3$ (so I don't calculate the degree $14$).

First we want to calculate the canonical bundle of $G$. Denote by $E$ the trivial $6$-dimensional bundle over $G$, and by $S$ the universal (tautological) rank $2$ sub-bundle. Then the tangent bundle to $G$ is $TG=S^* otimes (E/S)$ . It follows from the properties of $c_1$ that $c_1(TG)=6c_1(S^*) $. Similarly for the canonical bundle $K_G$ we have the expression $K_Gcong (detS^*)^{otimes -6}$.

Now we will use the (simple) statement from Griffiths-Harris that under the Plucker embedding we have the isomorphism of the line bundles $det S^*=O(1)$. Using the previous calculation we see $K_Gcong O(-6)$.
Finally, the surface $S$ is an iterated (6 times) hyperplane section of $G$. So by Lefshetz theorem it has same fundamental group as $G$, i.e., it is simply-connected. It suffices now to see that its canonical bundle is trivial. This is done using the adjunction formula $K_D=K_X+D|_D$. Every time we cut $G$ by a hyper-plane we tensor the canonical by $O(1)$, but $O(-6)otimes O(6)cong O$.

Added. The calculation of the degree is done by Andrea

Added. The number 19 is obtaied in the following way. The dimension of the grassmanian of $8$-planes in $CP^{14}$ is $6cdot 9=54$. At the same time the Grassmanian of $2$-planes in $mathbb C^6$ has symmetires, given by $SL(6,mathbb C)$, whose dimension is 35. We should quotient by these symmetries and get $54-35=19$

differential equations - When is Sobolev space a subset of the continuous functions?

If we let $Omegasubsetmathbb{R}^d$ with $d=1,2,3$ and define $mathcal{H}^1(Omega)=(win L_2(Omega): frac{partial w}{partial x_i}in L_2(Omega), i=1,...,d)$. My tutor has repeated several times:

1. If $d=1$ then $mathcal{H}^1(Omega)subsetmathcal{C}^0(Omega)$.


2. If $d=2$ then $mathcal{H}^2(Omega)subsetmathcal{C}^0(Omega)$ but $mathcal{H}^1(Omega)notsubsetmathcal{C}^0(Omega)$.


3. If $d=3$ then $mathcal{H}^3(Omega)subsetmathcal{C}^0(Omega)$ but $mathcal{H}^2(Omega)notsubsetmathcal{C}^0(Omega)$.

I was interested in trying to show these relationships. Does anyone know any references that would be useful.

Thanks in advance.