Expositiones Mathematicae最新文献

In this work we describe dicritical foliations in $({ℂ}^2,0)$ at a triple point of the resolution dual graph of an analytic plane branch $C$ using its semiroots. In particular, we obtain a constructive method to present a one-parameter family $C_u$ of separatrices for such foliations. As a by-product we relate the contact order between a special member of $C_u$ and $C$ with analytic discrete invariants of plane branches.

We study the correspondence between equivalence classes of pairs consisting of real semisimple Lie algebras and their Cartan subalgebras and involutions of the corresponding root system. This can be graphically described by introducing $S$- and $\Sigma$-diagrams, generalizing those of Satake and Vogan.

The concept of relative sectional category expands upon classical sectional category theory by incorporating the pullback of a fibration along a map. Our paper aims not only to explore this extension but also to thoroughly investigate its properties. We seek to uncover how the relative sectional category unifies several homotopic numerical invariants found in recent literature. These include the topological complexity of maps according to Murillo–Wu or Scott, relative topological complexity as defined by Farber, and homotopic distance for continuous maps in the sense of Macías-Virgós and Mosquera-Lois, among others.

Let $\ell$ and $p$ be two distinct prime numbers. We study $\ell$-isogeny graphs of ordinary elliptic curves defined over a finite field of characteristic $p$, together with a level structure. Firstly, we show that as the level varies over all $p$-powers, the graphs form an Iwasawa-theoretic abelian $p$-tower, which can be regarded as a graph-theoretical analogue of the Igusa tower of modular curves. Secondly, we study the structure of the crater of these graphs, generalizing previous results on volcano graphs. Finally, we solve an inverse problem of graphs arising from the crater of $\ell$-isogeny graphs with level structures, partially generalizing a recent result of Bambury, Campagna and Pazuki.

This paper considers sequences of points on the real line which have been randomly translated, and provides conditions under which various notions of convergence to a limiting point process are equivalent. In particular we consider convergence in correlation, convergence in distribution, and convergence of spacings between points. We also prove a simple Tauberian theorem regarding rescaled correlations. The results are applied to zeros of the Riemann zeta-function to show that several ways to state the GUE Hypothesis are equivalent. The proof relies on a moment bound of A. Fujii.

In this note, we give a simple closed formula for an arbitrary product, landing in dimension 0, of boundary classes on the Deligne–Mumford moduli space ${\overline{M}}_{0,n}$. For any such boundary strata $X_{T_1},\dots ,X_{T_{\ell }}$, we show the intersection product $\int {\prod }_{i=1}^{\ell }\left[X_{T_i}\right]$ is either a signed product of multinomial coefficients, or zero, and provide a simple criterion for determining when it is nonzero.

We do not claim originality for our product formula, but to our knowledge it does not appear elsewhere in the literature.

There are several graphs defined on groups. Among them we consider graphs whose vertex set consists conjugacy classes of a group $G$ and adjacency is defined by properties of the elements of conjugacy classes. In particular, we consider commuting/nilpotent/solvable conjugacy class graph of $G$ where two distinct conjugacy classes $a^G$ and $b^G$ are adjacent if there exist some elements $x\in a^G$ and $y\in b^G$ such that $〈x,y〉$ is abelian/nilpotent/solvable. After a section of introductory results and examples, we discuss all the available results on connectedness, graph realization, genus, various spectra and energies of certain induced subgraphs of these graphs. Proofs of the results are not included. However, many open problems for further investigation are stated.

We give two results for deducing dynamical properties of piecewise Möbius interval maps from their related planar extensions. First, eventual expansivity and the existence of an ergodic invariant probability measure equivalent to Lebesgue measure both follow from mild finiteness conditions on the planar extension along with a new property “bounded non-full range” used to relax traditional Markov conditions. Second, the “quilting” operation to appropriately nearby planar systems, introduced by Kraaikamp and co-authors, can be used to prove several key dynamical properties of a piecewise Möbius interval map. As a proof of concept, we apply these results to recover known results on the well-studied Nakada $\alpha$-continued fractions; we obtain similar results for interval maps derived from an infinite family of non-commensurable Fuchsian groups.