Isoperimetric Estimates on Sierpinski Gasket Type Fractals
Robert
S.
Strichartz
1705-1752
Abstract: For a compact Hausdorff space $F$ that is pathwise connected, we can define the connectivity dimension $\beta$ to be the infimum of all $b$ such that all points in $F$ can be connected by a path of Hausdorff dimension at most $b$. We show how to compute the connectivity dimension for a class of self-similar sets in $\mathbb{R}^{n}$ that we call point connected, meaning roughly that $F$ is generated by an iterated function system acting on a polytope $P$ such that the images of $P$ intersect at single vertices. This class includes the polygaskets, which are obtained from a regular $n$-gon in the plane by contracting equally to all $n$ vertices, provided $n$ is not divisible by 4. (The Sierpinski gasket corresponds to $n = 3$.) We also provide a separate computation for the octogasket ($n = 8$), which is not point connected. We also show, in these examples, that $\inf \mathcal{H}_{\beta }(\gamma _{x,y})^{1/\beta }$, where the infimum is taken over all paths $\gamma _{x,y}$ connecting $x$ and $y$, and $\mathcal{H}_{\beta }$ denotes Hausdorff measure, is equivalent to the original metric on $F$. Given a compact subset $F$ of the plane of Hausdorff dimension $\alpha$ and connectivity dimension $\beta$, we can define the isoperimetric profile function $h(L)$ to be the supremum of $\mathcal{H}_{\alpha }(F \cap D)$, where $D$ is a region in the plane bounded by a Jordan curve (or union of Jordan curves) $\gamma$ entirely contained in $F$, with $\mathcal{H}_{\beta }(\gamma ) \le L$. The analog of the standard isperimetric estimate is $h(L) \le cL^{\alpha /\beta }$. We are particularly interested in finding the best constant $c$ and identifying the extremal domains where we have equality. We solve this problem for polygaskets with $n = 3,5,6,8$. In addition, for $n = 5,6,8$ we find an entirely different estimate for $h(L)$ as $L \rightarrow \infty$, since the boundary of $F$ has infinite $\mathcal{H}_{\beta }$ measure. We find that the isoperimetric profile function is discontinuous, and that the extremal domains have relatively simple polygonal boundaries. We discuss briefly the properties of minimal paths for the Sierpinski gasket, and the isodiametric problem in the intrinsic metric.
The Hexagonal Honeycomb Conjecture
Frank
Morgan
1753-1763
Abstract: It is conjectured that the planar hexagonal honeycomb provides the least-perimeter way to enclose and separate infinitely many regions of unit area. Various natural formulations of the question are not known to be equivalent. We prove existence for two formulations. Many questions remain open.
The limit spaces of two-dimensional manifolds with uniformly bounded integral curvature
Takashi
Shioya
1765-1801
Abstract: We study the class of closed $2$-dimensional Riemannian manifolds with uniformly bounded diameter and total absolute curvature. Our first theorem states that this class of manifolds is precompact with respect to the Gromov-Hausdorff distance. Our goal in this paper is to completely characterize the topological structure of all the limit spaces of the class of manifolds, which are, in general, not topological manifolds and even may not be locally $2$-connected. We also study the limit of $2$-manifolds with $L^p$-curvature bound for $p \ge 1$.
Convexity properties of holomorphic mappings in $\mathbb{C}^n$
Kevin
A.
Roper;
Ted
J.
Suffridge
1803-1833
Abstract: Not many convex mappings on the unit ball in ${\mathbb C}^n$ for $n>1$ are known. We introduce two families of mappings, which we believe are actually identical, that both contain the convex mappings. These families which we have named the ``Quasi-Convex Mappings, Types A and B'' seem to be natural generalizations of the convex mappings in the plane. It is much easier to check whether a function is in one of these classes than to check for convexity. We show that the upper and lower bounds on the growth rate of such mappings is the same as for the convex mappings.
Reducibility and nonreducibility between $\ell^p$ equivalence relations
Randall
Dougherty;
Greg
Hjorth
1835-1844
Abstract: We show that, for $1 \le p < q < \infty$, the relation of $\ell ^{p}$-equivalence between infinite sequences of real numbers is Borel reducible to the relation of $\ell ^{q}$-equivalence (i.e., the Borel cardinality of the quotient ${\mathbb R}^{{\mathbb N}}/\ell ^{p}$ is no larger than that of ${\mathbb R}^{{\mathbb N}}/\ell ^{q}$), but not vice versa. The Borel reduction is constructed using variants of the triadic Koch snowflake curve; the nonreducibility in the other direction is proved by taking a putative Borel reduction, refining it to a reduction map that is not only continuous but `modular,' and using this nicer map to derive a contradiction.
Criteria of algebraic independence with multiplicities and interpolation determinants
Michel
Laurent;
Damien
Roy
1845-1870
Abstract: We generalize Gel'fond's criterion of algebraic independence by taking into account the values of the derivatives of the polynomials, and show how the new criterion applies to proving results of algebraic independence using interpolation determinants. We also establish a new result of approximation of a transcendental number by algebraic numbers of bounded degree and size. It contains an earlier result of E. Wirsing and also a result announced by A. Durand.
Simple families of Thue inequalities
Günter
Lettl;
Attila
Petho;
Paul
Voutier
1871-1894
Abstract: We use the hypergeometric method to solve three families of Thue inequalities of degree 3, 4 and 6, respectively, each of which is parametrized by an integral parameter. We obtain bounds for the solutions, which are astonishingly small compared to similar results which use estimates of linear forms in logarithms.
Linking forms, reciprocity for Gauss sums and invariants of 3-manifolds
Florian
Deloup
1895-1918
Abstract: We study invariants of $3$-manifolds derived from finite abelian groups equipped with quadratic forms. These invariants arise in Turaev's theory of modular categories and generalize those of H. Murakami, T. Ohtsuki and M. Okada. The crucial algebraic tool is a new reciprocity formula for Gauss sums, generalizing classical formulas of Cauchy, Kronecker, Krazer and Siegel. We use this reciprocity formula to give an explicit formula for the invariants and to generalize them to higher dimensions.
On the number of radially symmetric solutions to Dirichlet problems with jumping nonlinearities of superlinear order
Alfonso
Castro;
Hendrik
J.
Kuiper
1919-1945
Abstract: This paper is concerned with the multiplicity of radially symmetric solutions $u(x)$ to the Dirichlet problem \begin{displaymath}\Delta u+f(u)=h(x)+c\phi(x)\end{displaymath} on the unit ball $\Omega\subset\mathbf R^N$ with boundary condition $u=0$ on $\partial\Omega$. Here $\phi(x)$ is a positive function and $f(u)$ is a function that is superlinear (but of subcritical growth) for large positive $u$, while for large negative $u$ we have that $f'(u)<\mu$, where $\mu$ is the smallest positive eigenvalue for $\Delta\psi+\mu\psi=0$ in $\Omega$ with $\psi=0$ on $\partial\Omega$. It is shown that, given any integer $k\ge 0$, the value $c$ may be chosen so large that there are $2k+1$ solutions with $k$ or less interior nodes. Existence of positive solutions is excluded for large enough values of $c$.
Local tomography with nonsmooth attenuation
A.
I.
Katsevich
1947-1974
Abstract: Local tomography for the Radon transform with nonsmooth attenuation is proposed and justified. The main theoretical tool is analysis of singularities of pseudodifferential operators with nonsmooth symbols. Results of numerical testing of local tomography are presented.
Geometry of Cantor Systems
Yunping
Jiang
1975-1987
Abstract: A Cantor system is defined. The geometry of a certain family of Cantor systems is studied. Such a family arises in dynamical systems as hyperbolicity is created. We prove that the bridge geometry of a Cantor system in such a family is uniformly bounded and that the gap geometry is regulated by the size of the leading gap.
On the contribution of the Coulomb singularity of arbitrary charge to the Dirac Hamiltonian
Jingbo
Xia
1989-2023
Abstract: We study the self-adjoint extensions of the Dirac operator $\alpha \cdot (p - B) + \mu _{0}\beta - W$, where the electrical potential $W$ contains a Coulomb singularity of arbitrary charge and the magnetic potential $B$ is allowed to be unbounded at infinity. We show that if the Coulomb singularity has the form $v(r)/r$ where $v$ has a limit at 0, then, for any self-adjoint extension of the Dirac operator, removing the singularity results in a Hilbert-Schmidt perturbation of its resolvent.
Harnack inequality for the linearized parabolic Monge-Ampère equation
Qingbo
Huang
2025-2054
Abstract: In this paper we prove the Harnack inequality for nonnegative solutions of the linearized parabolic Monge-Ampère equation \begin{displaymath}u_{t}-\text{tr}((D^{2}\phi (x))^{-1}D^{2}u)=0\end{displaymath} on parabolic sections associated with $\phi (x)$, under the assumption that the Monge-Ampère measure generated by $\phi$ satisfies the doubling condition on sections and the uniform continuity condition with respect to Lebesgue measure. The theory established is invariant under the group $AT(n)\times AT(1)$, where $AT(n)$ denotes the group of all invertible affine transformations on ${\mathbf{R}}^{n}$.
A Gauss-Kusmin theorem for optimal continued fractions
Karma
Dajani;
Cor
Kraaikamp
2055-2079
Abstract: A Gauss-Kusmin theorem for the Optimal Continued Fraction (OCF) expansion is obtained. In order to do so, first a Gauss-Kusmin theorem is derived for the natural extension of the ergodic system underlying Hurwitz's Singular Continued Fraction (SCF) (and similarly for the continued fraction to the nearer integer (NICF)). Since the NICF, SCF and OCF are all examples of maximal $S$-expansions, it follows from a result of Kraaikamp that the SCF and OCF are metrically isomorphic. This isomorphism is then used to carry over the results for the SCF to any other maximal $S$-expansion, in particular to the OCF. Along the way, a Heilbronn-theorem is obtained for any maximal $S$-expansion.
Conical limit set and Poincaré exponent for iterations of rational functions
Feliks
Przytycki
2081-2099
Abstract: We contribute to the dictionary between action of Kleinian groups and iteration of rational functions on the Riemann sphere. We define the Poincaré exponent $\delta(f,z)=\inf\{\alpha\ge 0:\mathcal{P}(z,\alpha) \le 0\}$, where \begin{equation*}\mathcal{P}(z,\alpha):=\limsup _{n\to\infty}{1\over n}\log\sum _{f^n(x)=z} |(f^n)'(x)|^{-\alpha}. \end{equation*} We prove that $\delta (f,z)$ and $\mathcal{P}(z,\alpha)$ do not depend on $z$, provided $z$ is non-exceptional. $\mathcal{P}$ plays the role of pressure; we prove that it coincides with the Denker-Urbanski pressure if $\alpha\le \delta(f)$. Various notions of ``conical limit set" are considered. They all have Hausdorff dimension equal to $\delta(f)$ which is equal to the hyperbolic dimension of the Julia set and also equal to the exponent of some conformal Patterson-Sullivan measures. In an Appendix we also discuss notions of ``conical limit set" introduced recently by Urbanski and by Lyubich and Minsky.
Symplectic group lattices
Rudolf
Scharlau;
Pham
Huu
Tiep
2101-2139
Abstract: Let $p$ be an odd prime. It is known that the symplectic group $Sp_{2n}(p)$ has two (algebraically conjugate) irreducible representations of degree $(p^{n}+1)/2$ realized over $\mathbb{Q}(\sqrt{{\epsilon}p})$, where ${\epsilon}= (-1)^{(p-1)/2}$. We study the integral lattices related to these representations for the case $p^{n} \equiv 1 \bmod 4$. (The case $p^{n} \equiv 3 \bmod 4$ has been considered in a previous paper.) We show that the class of invariant lattices contains either unimodular or $p$-modular lattices. These lattices are explicitly constructed and classified. Gram matrices of the lattices are given, using a discrete analogue of Maslov index.