Lacunarity of self-similar and stochastically self-similar sets
Dimitris
Gatzouras
1953-1983
Abstract: Let $K$ be a self-similar set in $\mathbb R^d$, of Hausdorff dimension $D$, and denote by $\vert K(\epsilon)\vert$the $d$-dimensional Lebesgue measure of its $\epsilon$-neighborhood. We study the limiting behavior of the quantity $\epsilon^{-(d-D)}\vert K(\epsilon)\vert$as $\epsilon\rightarrow 0$. It turns out that this quantity does not have a limit in many interesting cases, including the usual ternary Cantor set and the Sierpinski carpet. We also study the above asymptotics for stochastically self-similar sets. The latter results then apply to zero-sets of stable bridges, which are stochastically self-similar (in the sense of the present paper), and then, more generally, to level-sets of stable processes. Specifically, it follows that, if $K_t$ is the zero-set of a real-valued stable process of index $\alpha\in (1,2]$, run up to time $t$, then $\epsilon^{-1/\alpha}\vert K_t(\epsilon)\vert$converges to a constant multiple of the local time at $0$, simultaneously for all $t\geq 0$, on a set of probability one. The asymptotics for deterministic sets are obtained via the renewal theorem. The renewal theorem also yields asymptotics for the mean $\mathbb E [\vert K(\epsilon)\vert ]$ in the random case, while the almost sure asymptotics in this case are obtained via an analogue of the renewal theorem for branching random walks.
Extending partial automorphisms and the profinite topology on free groups
Bernhard
Herwig;
Daniel
Lascar
1985-2021
Abstract: A class of structures $\mathcal{C}$ is said to have the extension property for partial automorphisms (EPPA) if, whenever $C_1$ and $C_2$ are structures in $\mathcal{C}$, $C_1$ finite, $C_1\subseteq C_2$, and $p_1,p_2,\dotsc,p_n$ are partial automorphisms of $C_1$ extending to automorphisms of $C_2$, then there exist a finite structure $C_3$ in $\mathcal{C}$ and automorphisms $\alpha _1, \alpha _2,\dotsc,\alpha _n$ of $C_3$ extending the $p_i$. We will prove that some classes of structures have the EPPA and show the equivalence of these kinds of results with problems related with the profinite topology on free groups. In particular, we will give a generalisation of the theorem, due to Ribes and Zalesskii stating that a finite product of finitely generated subgroups is closed for this topology.
The distributivity numbers of $\mathcal{P}(\omega)$/fin and its square
Saharon
Shelah;
Otmar
Spinas
2023-2047
Abstract: We show that in a model obtained by forcing with a countable support iteration of Mathias forcing of length $\omega _{2}$, the distributivity number of ${\mathcal{P}}(\omega )$/fin is $\omega _{2}$, whereas the distributivity number of r.o.$({\mathcal{P}}(\omega )$/fin)$^{2}$ is $\omega _{1}$. This answers a problem of Balcar, Pelant and Simon, and others.
Cycling in proofs and feasibility
A.
Carbone
2049-2075
Abstract: There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual but for which some very large number is defined to be not feasible. Parikh shows that sufficiently short proofs in this theory can only prove true statements of arithmetic. We pursue these topics in light of logical flow graphs of proofs (Buss, 1991) and show that Parikh's lower bound for concrete consistency reflects the presence of cycles in the logical graphs of short proofs of feasibility of large numbers. We discuss two concrete constructions which show the bound to be optimal and bring out the dynamical aspect of formal proofs. For this paper the concept of feasible numbers has two roles, as an idea with its own life and as a vehicle for exploring general principles on the dynamics and geometry of proofs. Cycles can be seen as a measure of how complicated a proof can be. We prove that short proofs must have cycles.
A note on elliptic k3 surfaces
JongHae
Keum
2077-2086
Abstract: We study the relationship between an elliptic fibration on an elliptic K3 surface and its Jacobian surface. We give an explicit description of the Picard lattice of the Jacobian surface. Then we use the description to prove that certain K3 surfaces do not admit a non-Jacobian fibration. Moreover, we obtain an inequality involving the determinant of the Picard lattice and the number of components of reducible fibres, which implies, among others, that if an elliptic K3 surface has Picard lattice with relatively small determinant, then every elliptic fibration on it must have a reducible fibre. Some examples of K3 surfaces are discussed.
The geometry of fixed point varieties on affine flag manifolds
Daniel
S.
Sage
2087-2119
Abstract: Let $G$ be a semisimple, simply connected, algebraic group over an algebraically closed field $k$ with Lie algebra $\mathfrak{g}$. We study the spaces of parahoric subalgebras of a given type containing a fixed nil-elliptic element of $\mathfrak{g}\otimes k((\pi))$, i.e. fixed point varieties on affine flag manifolds. We define a natural class of $k^*$-actions on affine flag manifolds, generalizing actions introduced by Lusztig and Smelt. We formulate a condition on a pair $(N,f)$ consisting of $N\in \mathfrak{g}\otimes k((\pi))$ and a $k^*$-action $f$ of the specified type which guarantees that $f$ induces an action on the variety of parahoric subalgebras containing $N$. For the special linear and symplectic groups, we characterize all regular semisimple and nil-elliptic conjugacy classes containing a representative whose fixed point variety admits such an action. We then use these actions to find simple formulas for the Euler characteristics of those varieties for which the $k^*$-fixed points are finite. We also obtain a combinatorial description of the Euler characteristics of the spaces of parabolic subalgebras containing a given element of certain nilpotent conjugacy classes of $\mathfrak{g}$.
Coupled contact systems and rigidity of maximal dimensional variations of Hodge structure
Richárd
Mayer
2121-2144
Abstract: In this article we prove that locally Griffiths' horizontal distribution on the period domain is given by a generalized version of the familiar contact differential system. As a consequence of this description we obtain strong local rigidity properties of maximal dimensional variations of Hodge structure. For example, we prove that if the weight is odd (greater than one) then there is a unique germ of maximal dimensional variation of Hodge structure through every point of the period domain. Similar results hold if the weight is even with the exception of one case.
Some properties of partitions in terms of crank
A.
Bülent
Ek. in
2145 - 2156
Some Theorems on the Rogers--Ramanujan Continued Fraction in Ramanujan's Lost Notebook
Bruce
C.
Berndt;
Sen-Shan
Huang;
Jaebum
Sohn;
Seung
Hwan
Son
2157-2177
Abstract: In his first two letters to G. H. Hardy and in his notebooks, Ramanujan recorded many theorems about the Rogers-Ramanujan continued fraction. In his lost notebook, he offered several further assertions. The purpose of this paper is to provide proofs for many of the claims about the Rogers-Ramanujan and generalized Rogers-Ramanujan continued fractions found in the lost notebook. These theorems involve, among other things, modular equations, transformations, zeros, and class invariants.
Topological Hochschild homology of number rings
Ayelet
Lindenstrauss;
Ib
Madsen
2179-2204
Abstract: We calculate an explicit formula for the topological Hochschild homology of a discrete valuation ring of characteristic zero with finite residue field. From this we deduce the topological Hochschild homology of global number rings.
Relative completions of linear groups over $\mathbb{Z}[t]$ and $\mathbb{Z}[t,t^{-1}]$
Kevin
P.
Knudson
2205-2216
Abstract: We compute the completion of the groups $SL_n({\mathbb Z}[t])$ and $SL_n({\mathbb Z}[t,t^{-1}])$ relative to the obvious homomorphisms to $SL_n({\mathbb Q})$; this is a generalization of the classical Malcev completion. We also make partial computations of the rational second cohomology of these groups.
Normal forms and Hopf bifurcation for partial differential equations with delays
Teresa
Faria
2217-2238
Abstract: The paper addresses the computation of normal forms for some Partial Functional Differential Equations (PFDEs) near equilibria. The analysis is based on the theory previously developed for autonomous retarded Functional Differential Equations and on the existence of center (or other invariant) manifolds. As an illustration of this procedure, two examples of PFDEs where a Hopf singularity occurs on the center manifold are considered.
Schauder estimates for equationswith fractional derivatives
Ph.
Clément;
G.
Gripenberg;
S-O.
Londen
2239-2260
Abstract: The equation \begin{equation*}D^\alpha_t (u-h_1) + D^\beta_x(u-h_2) =f,\quad 0< \alpha,\beta< 1, \quad t,x \geq 0,\tag{$*$} \end{equation*}where $D^\alpha_t$ and $D^\beta_x$ are fractional derivatives of order $\alpha$ and $\beta$ is studied. It is shown that if $f=f(\underline{t}, \underline{x})$, $h_1=h_1(\underline{x})$, and $h_2=h_2(\underline{t})$ are Hölder-continuous and $f(0,0) =0$, then there is a solution such that $D^\alpha_t u$ and $D^\beta_x u$ are Hölder-continuous as well. This is proved by first considering an abstract fractional evolution equation and then applying the results obtained to ($*$). Finally the solution of ($*$) with $f=1$ is studied.
An algorithmic approach to the construction of homomorphisms induced by maps in homology
Madjid
Allili;
Tomasz
Kaczynski
2261-2281
Abstract: This paper is devoted to giving the theoretical background for an algorithm for computing homomorphisms induced by maps in homology. The principal idea is to insert the graph of a given continuous map $\, f \,$ into a graph of a multi-valued representable map $\, F$. The multi-valued representable maps have well developed continuity properties and admit a finite coding that permits treating them by combinatorial methods. We provide the construction of the homomorphism $\, F_* \,$ induced by $\, F \,$ such that $\, F_* = f_*$. The presented construction does not require subsequent barycentric subdivisions and simplicial approximations of $\, f$. The main motivation for this paper comes from the project of computing the Conley Index for discrete dynamical systems.
From the restricted to the full three--body problem
Kenneth
R.
Meyer;
Dieter
S.
Schmidt
2283-2299
Abstract: The three-body problem with all the classical integrals fixed and all the symmetries removed is called the reduced three-body problem. We use the methods of symplectic scaling and reduction to show that the reduced planar or spatial three-body problem with one small mass is to the first approximation the product of the restricted three-body problem and a harmonic oscillator. This allows us to prove that many of the known results for the restricted problem have generalizations for the reduced three-body problem. For example, all the non-degenerate periodic solutions, generic bifurcations, Hamiltonian-Hopf bifurcations, bridges and natural centers known to exist in the restricted problem can be continued into the reduced three-body problem. The classic normalization calculations of Deprit and Deprit-Bartholomé show that there are two-dimensional KAM invariant tori near the Lagrange point in the restricted problem. With the above result this proves that there are three-dimensional KAM invariant tori near the Lagrange point in the reduced three-body problem.
Locating the first nodal linein the Neumann problem
David
Jerison
2301-2317
Abstract: The location of the nodal line of the first nonconstant Neumann eigenfunction of a convex planar domain is specified to within a distance comparable to the inradius. This is used to prove that the eigenvalue of the partial differential equation is approximated well by the eigenvalue of an ordinary differential equation whose coefficients are expressed solely in terms of the width of the domain. A variant of these estimates is given for domains that are thin strips and satisfy a Lipschitz condition.
A Construction of Homologically Area Minimizing Hypersurfaces with Higher Dimensional Singular Sets
Nathan
Smale
2319-2330
Abstract: We show that a large variety of singular sets can occur for homologically area minimizing codimension one surfaces in a Riemannian manifold. In particular, as a result of Theorem A, if $N$ is smooth, compact $n+1$ dimensional manifold, $n\geq 7$, and if $S$ is an embedded, orientable submanifold of dimension $n$, then we construct metrics on $N$ such that the homologically area minimizing hypersurface $M$, homologous to $S$, has a singular set equal to a prescribed number of spheres and tori of codimension less than $n-7$. Near each component $\Sigma$ of the singular set, $M$ is isometric to a product $C\times \Sigma$, where $C$ is any prescribed, strictly stable, strictly minimizing cone. In Theorem B, other singular examples are constructed.
Conditions for the Existence of SBR Measures for ``Almost Anosov'' Diffeomorphisms
Huyi
Hu
2331-2367
Abstract: A diffeomorphism $f$ of a compact manifold $M$ is called ``almost Anosov'' if it is uniformly hyperbolic away from a finite set of points. We show that under some nondegeneracy condition, every almost Anosov diffeomorphism admits an invariant measure $\mu$ that has absolutely continuous conditional measures on unstable manifolds. The measure $\mu$ is either finite or infinite, and is called SBR measure or infinite SBR measure respectively. Therefore, $\frac{1}{n} \sum _{i=0}^{n-1}\delta _{f^{i}x}$ tends to either an SBR measure or $\delta _{p}$ for almost every $x$ with respect to Lebesgue measure. ($\delta _{x}$ is the Dirac measure at $x$.) For each case, we give sufficient conditions by using coefficients of the third order terms in the Taylor expansion of $f$ at $p$.
Statistical properties for nonhyperbolic maps with finite range structure
Michiko
Yuri
2369-2388
Abstract: We establish the central limit theorem and non-central limit theorems for maps admitting indifferent periodic points (the so-called intermittent maps). We also give a large class of Darling-Kac sets for intermittent maps admitting infinite invariant measures. The essential issue for the central limit theorem is to clarify the speed of $L^1$-convergence of iterated Perron-Frobenius operators for multi-dimensional maps which satisfy Renyi's condition but fail to satisfy the uniformly expanding property. Multi-dimensional intermittent maps typically admit such derived systems. There are examples in section 4 to which previous results on the central limit theorem are not applicable, but our extended central limit theorem does apply.
On the stable module category of a self-injective algebra
Karin
Erdmann;
Otto
Kerner
2389-2405
Abstract: Let $\Lambda$ be a finite-dimensional self-injective algebra. We study the dimensions of spaces of stable homomorphisms between indecomposable $\Lambda$-modules which belong to Auslander-Reiten components of the form $\mathbf{Z}A_\infty$ or $\mathbf{Z}A_\infty/\langle \tau^k\rangle$. The results are applied to representations of finite groups over fields of prime characteristic, especially blocks of wild representation type.
Representations over PID's with three distinguished submodules
Steve
Files;
Rüdiger
Göbel
2407-2427
Abstract: Let $R$ be a principal ideal domain. The $R$-representations with one distinguished submodule are classified by a theorem of Gaußin the case of finite rank, and by the ``Stacked Bases Theorem" of Cohen and Gluck in the case of infinite rank. Results of Hill and Megibben carry this classification even further. The $R$-representations with two distinguished pure submodules have recently been classified by Arnold and Dugas in the finite-rank case, and by the authors for countable rank. Although wild representation type prevails for $R$-representations with three distinguished pure submodules, an extensive category of such objects was recently classified by Arnold and Dugas. We carry their groundbreaking work further, simplifying the proofs of their main results and applying new machinery to study the structure of finite- and infinite-rank representations with two, three, and four distinguished submodules. We also apply these results to the classification of Butler groups, a class of torsion-free abelian groups that has been the focus of many investigations over the last sixteen years.
On the endomorphism monoids of (uniquely) complemented lattices
G.
Grätzer;
J.
Sichler
2429-2444
Abstract: Let $L$ be a lattice with $0$ and $1$. An endomorphism $\varphi$ of $L$ is a $\{0,1\}$-endomorphism, if it satisfies $0\varphi = 0$ and $1\varphi = 1$. The $\{0,1\}$-endomorphisms of $L$ form a monoid. In 1970, the authors proved that every monoid $\mathcal M$ can be represented as the $\{0,1\}$-endomorphism monoid of a suitable lattice $L$ with $0$ and $1$. In this paper, we prove the stronger result that the lattice $L$ with a given $\{0,1\}$-endomorphism monoid $\mathcal M$ can be constructed as a uniquely complemented lattice; moreover, if $\mathcal M$ is finite, then $L$ can be chosen as a finite complemented lattice.