TITLE: maximum with respect inclusion of a function whose output are sets
QUESTION [2 upvotes]: Suppose that $S(x)$ is a function from a compact space $A$ to a space
of sets $S$. Suppose that there exists a map $W: A\to A$ and
$S(x)\subseteq S(W(x)).$ Does there exist a point $x$ such
that the equality holds
$$S(x)= S(W(x))$$

REPLY [1 votes]: If $W$ admits a periodic point, then, of course, $S(x)$ is constant along its orbit. But, on the contrary, whatever nice and well-structured $A$ and $W$ you have, if $W$ has no periodic points, there is always the bad map $S(x):=$ the negative invariant set generated by $x$, that is $\cup_{k\in\mathbb{N}}W^{-k}(x),$ and this map fails to satisfy the thesis while $S(x)\subset S(W(x))$ holds for all $x$.
This is to convince you that some assumption on the map $S(x)$ is in order. Here, a mild and natural assumption, to be coupled with the compactness of $A\neq\emptyset$, is weak upper semicontinuity, that is,
$$ S:A\to2^X$$
is continuous w.r.to the product topology on $2^X$ where the two-point space $2:=\{0,1\}$ is endowed with the left-order topology (whose only proper open subset is $\{0\}$). As a consequence, the continuous image of $A$ is a compact subset of $2^X$, therefore it has a maximal element with respect to inclusion. Due to your assumption on $W$, the equality necessarily holds for any maximal set $S(x),$ proving your thesis (incidentally, note that no further assumption on $W$ is needed).
Rmk 1. The upper semicontinuity of $S$ introduced above may be equivalently stated as:

$\operatorname{graph(S)}:=\{(a,x)\in A\times X\\ : a\in S(x)\}$ is closed in $A\times X$, where $X$ has the discrete topology;
$S^* :X\to 2^A$ is a closed map, that is, for any $a\in A$ the set $S^* (x):=\{a\in A\\,:\\, x\in S(a) \}$ is a closed subset of $A$;
for any  $a\in A$ (denoting $\mathcal{N}_ a$ the family of nbd's of $a$), there holds
$$\limsup_{b\to a}  S(b) := \cup_ {U\in\mathcal{N}_ a} \cap_ {b\in U} S(b) \subset S(a).$$

Rmk 2. The fact that a compact subset $K$ of $2^X$, where $2$ has the left-order topology, admits a  maximal element, is, of course, a consequence of the Zorn lemma. Indeed, if $\Gamma$ is an infinite chain in $K$, it has a limit point in $K$, which turns out to be an upper bound of $\Gamma$.


text:
{\bf Problem.} How many distinct digits can appear as the units digit of an integral perfect-square number?
{\bf Level.} Level 3
{\bf Type.} Number Theory
{\bf Solution.} Let $d$ be the last digit of a number $n$.  Then $n^2 \equiv d^2 \pmod{10}$, so the units digit of $n^2$ is the same as the units digit of $d^2$.  Checking all the digits from 0 to 9, we find that the possible units digits of $d^2$ are 0, 1, 4, 5, 6, and 9, for a total of $\boxed{6}$.

set_name: MATH


TITLE: Partition sum for $SO(N)$ one-dimensional lattice model
QUESTION [1 upvotes]: I'm looking for derivation of explicit form of partition function for $SO(N)$ one-dimensional lattice model. The initial expression is
$$
Z = \int \limits_{-\infty}^{\infty}d\sigma_{1}...d\sigma_{N}\text{exp}\left[-\beta\sum_{x = 1}^{L}\sum_{k = 1}^{N}\sigma_{k}(x)\sigma_{k}(x+1)\right]\delta(1-\sum_{k}\sigma_{k}^{2})
$$
I've heard that it is possible to obtain explicit expression for this sum, but don't know how to do that?

REPLY [2 votes]: Yes, this can be done as an application of the transfer ``matrix'' method. You can, for example,  find an explicit computation for the XY model (N=2) in the book The theory of magnetism II: Thermodynamics and Statistical Mechanics by Daniel Mattis (p. 75 ff.). Another reference, in which the general case is treated, is Section 9.2 in the book Elements of Phase Transitions and Critical Phenomena by Nishimori and Ortiz. The result is (well, you should replace $\beta$ by $-\beta$ in your case, since you seem to be interested in the antiferromagnetic case
):
$$
Z_L^{\mathrm{free}} = \Bigl\{\bigl(\tfrac\beta2\bigr)^{1-N/2}I_{\tfrac N2-1}(\beta)\Bigr\}^{L-1}\,,
$$
where $I_{N/2-1}$ is the modified Bessel function of the first kind.

Let me illustrate an alternative approach in the case of the XY model. In that case, writing the spins $\sigma_i=(\cos\theta_i,\sin\theta_i)$, the partition function with free boundary condition can be written
$$
Z_L^{\rm free} = (2\pi)^{-L}\int_{[-\pi,\pi]^L}
\mathrm{d}\theta_1\cdots\mathrm{d}\theta_L
\prod_{i=1}^{L-1} e^{-\beta\cos(\theta_{i+1}-\theta_i)}\,.
$$
Now, making the change of variables $\tau_i=\theta_{i+1}-\theta_i$ (say, with the additional variable $\tau_L=\theta_L$), the
integrand factorizes and we get
$$
Z_L^{\rm free} = \Bigl\{
\tfrac1{2\pi}\int_{[-\pi,\pi]} \mathrm{d}\tau\, e^{-\beta\cos\tau}
\Bigr\}^{L-1}\,.
$$
Now, you need to remember the definition of the modified Bessel function of the first kind:
$$
I_0(x) = \tfrac1{2\pi} \int_{-\pi}^{\pi} e^{x\cos t} \mathrm{d} t\,.
$$
The conclusion is
$$
Z_L^{\rm free} = \bigl( I_0(-\beta) \bigr)^{L-1}\,.
$$
(Of course, you should check the above computation, I might have made some mistakes.)