Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Edge-isoperimetric problem for Cayley graphs and generalized Takagi
  function   Let $G$ be a finite abelian abelian group of exponent $m\ge 2$. For subsets
$A,S\subset G$, denote by $\partial_S(A)$ the number of edges from $A$ to its
complement $G\setminus A$ in the directed Cayley graph, induced by $S$ on $G$.
We show that if $S$ generates $G$, and $A$ is non-empty, then
  $$ \partial_S(A) \ge \frac{e}m\,|A|\ln\frac{|G|}{|A|}. $$
  Here the coefficient $e=2.718...$ is best possible and cannot be replaced
with a number larger than $e$.
  For homocyclic groups $G$ of exponent $m$, we find an explicit closed-form
expression for $\partial_S(A)$ in the case where $S$ is a "standard" generating
subset of $G$, and $A$ is an initial segment of $G$ with respect to the
lexicographic order, induced by $S$ on $G$. Namely, we show that in this
situation
  $$ \partial_S(A) = |G|\,\omega_m(|A|/|G|), $$ where $\omega_2$ is the Takagi
function, and $\omega_m$ for $m\ge 3$ is an appropriate generalization thereof.
This particular case is of special interest, since for $m\in\{2,3,4\}$ it is
known to yield the smallest possible value of $\partial_S(A)$, over all sets
$A\subset G$ of given size. We give this classical result a new proof, somewhat
different from the standard one.
  We also give a new, short proof of the Boros-Pales inequality
$$\omega_2(\frac{x+y}2) \le \frac{\omega_2(x) + \omega_2(y)}2 + \frac12\,|y-x|,
$$ establish an extremal characterization of the Takagi function as the
(pointwise) maximal function, satisfying this inequality and the boundary
condition $\max\{\omega_2(0),\omega_2(1)\}\le 0$, and obtain similar results
for the 3-adic analog $\omega_3$ of the Takagi function.

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The categories of this paper are: supr-con
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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