We study the case of an Axiom $\text{A}$ holomorphic non-degenerate (hence non-invertible) map $f:{{\mathbb{P}}^{2}}\mathbb{C}\to {{\mathbb{P}}^{2}}\mathbb{C}$, where ${{\mathbb{P}}^{2}}\mathbb{C}$ stands for the complex projective space of dimension 2. Let $\Lambda $ denote a basic set for $f$ of unstable index 1, and $x$ an arbitrary point of $\Lambda $; we denote by ${{\delta }^{s}}(x)$ the Hausdorff dimension of $\text{W}_{r}^{s}(x) \cap \,\Lambda$, where $r$ is some fixed positive number and $\text{W}_{r}^{s}(x)$ is the local stable manifold at $x$ of size $r;{{\delta }^{s}}(x)$ is called the stable dimension at$x$. Mihailescu and Urbański introduced a notion of inverse topological pressure, denoted by ${{P}^{-}}$, which takes into consideration preimages of points. Manning and McCluskey studied the case of hyperbolic diffeomorphisms on real surfaces and give formulas for Hausdorff dimension. Our non-invertible situation is different here since the local unstable manifolds are not uniquely determined by their base point, instead they depend in general on whole prehistories of the base points. Hence our methods are different and are based on using a sequence of inverse pressures for the iterates of $f$, in order to give upper and lower estimates of the stable dimension. We obtain an estimate of the oscillation of the stable dimension on $\Lambda $. When each point $x$ from $\Lambda $ has the same number ${d}'$ of preimages in $\Lambda $, then we show that ${{\delta }^{s}}(x)$ is independent of $x$; in fact ${{\delta }^{s}}(x)$ is shown to be equal in this case with the unique zero of the map $t\to P(t{{\phi }^{s}}-\log {d}')$. We also prove the Lipschitz continuity of the stable vector spaces over $\Lambda $; this proof is again different than the one for diffeomorphisms (however, the unstable distribution is not always Lipschitz for conformal non-invertible maps). In the end we include the corresponding results for a real conformal setting.

Let $F$ be a non-archimedean local field of residue characteristic neither 2 nor 3 equipped with a galois involution with fixed field ${{F}_{0}}$, and let $G$ be a symplectic group over $F$ or an unramified unitary group over ${{F}_{0}}$. Following the methods of Bushnell–Kutzko for $\text{GL}(N,F)$, we define an analogue of a simple type attached to a certain skew simple stratum, and realize a type in $G$. In particular, we obtain an irreducible supercuspidal representation of $G$ like $\text{GL}(N,F)$.

A commutative local Cohen–Macaulay ring $R$ of finite Cohen–Macaulay type is known to be an isolated singularity; that is, $\text{Spec}(R)\backslash \{m\}$ is smooth. This paper proves a non-commutative analogue. Namely, if $A$ is a (non-commutative) graded Artin–Schelter Cohen–Macaulay algebra which is fully bounded Noetherian and has finite Cohen–Macaulay type, then the non-commutative projective scheme determined by $A$ is smooth.

The aim of this work is to provide a new approach for constructing $n$-dimensional Steinberg symbols on discrete valuation fields from $\left( n\,+\,1 \right)$-cocycles and to study reciprocity laws on curves related to these symbols.

The Kronecker modules, $\mathbb{V}\left( m,h,\alpha \right)$, where $m$ is a positive integer, $h$ is a height function, and $\alpha $ is a $K$-linear functional on the space $K(X)$ of rational functions in one variable $X$ over an algebraically closed field $K$, are models for the family of all torsion-free rank-2 modules that are extensions of finite-dimensional rank-1 modules. Every such module comes with a regulating polynomial $f$ in $K(X)[Y]$. When the endomorphism algebra of $\mathbb{V}\left( m,h,\alpha \right)$ is commutative and non-trivial, the regulator $f$ must be quadratic in $Y$. If $f$ has one repeated root in $K(X)$, the endomorphism algebra is the trivial extension $K\ltimes S$ for some vector space $S$. If $f$ has distinct roots in $K(X)$, then the endomorphisms form a structure that we call a bridge. These include the coordinate rings of some curves. Regardless of the number of roots in the regulator, those End $\mathbb{V}\left( m,h,\alpha \right)$ that are domains have zero radical. In addition, each semi-local End $\mathbb{V}\left( m,h,\alpha \right)$ must be either a trivial extension $K\ltimes S$ or the product $K\times K$.

Let $\alpha $ and $\beta $ be two Furstenberg transformations on 2-torus associated with irrational numbers ${{\theta }_{1}},\,{{\theta }_{2}}$, integers ${{d}_{1}},\,{{d}_{2}}$ and Lipschitz functions ${{f}_{1}}\,\text{and}\,{{f}_{2}}$. It is shown that $\alpha $ and $\beta $ are approximately conjugate in ameasure theoretical sense if (and only if) $\overline{{{\theta }_{1}}\,\pm \,{{\theta }_{2}}}\,=\,0\,\text{in}\,\mathbb{R}\text{/}\mathbb{Z}$. Closely related to the classification of simple amenable ${{C}^{*}}$-algebras, it is shown that $\alpha $ and $\beta $ are approximately $K$-conjugate if (and only if) $\overline{{{\theta }_{1}}\,\pm \,{{\theta }_{2}}}\,=\,0\,\text{in}\,\mathbb{R}\text{/}\mathbb{Z}$ and $|{{d}_{1}}|\,=\,|{{d}_{2}}|$. This is also shown to be equivalent to the condition that the associated crossed product ${{C}^{*}}$-algebras are isomorphic.

For a general vector field we exhibit two Hilbert spaces, namely the space of so called closed functions and the space of exact functions and we calculate the codimension of the space of exact functions inside the larger space of closed functions. In particular we provide a new approach for the known cases: the Glauber field and the second-order Ginzburg–Landau field and for the case of the fourth-order Ginzburg–Landau field.

We give a complete classification of mixed Tsirelson spaces $T\left[ ({{F}_{i,}}{{\theta }_{i}})_{i=1}^{r} \right]$ for finitely many pairs of given compact and hereditary families ${{F}_{i}}$ of finite sets of integers and $0<{{\theta }_{i}}<1$ in terms of the Cantor–Bendixson indices of the families ${{F}_{i}}$, and ${{\theta }_{i}}(1\le i\le r)$. We prove that there are unique countable ordinal $\alpha $ and $0<\theta <1$ such that every block sequence of $T\left[ ({{F}_{i,}}{{\theta }_{i}})_{i=1}^{r} \right]$ has a subsequence equivalent to a subsequence of the natural basis of the $T({{S}_{{{\omega }^{\alpha }}}},\theta )$. Finally, we give a complete criterion of comparison in between two of these mixed Tsirelson spaces.

In a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property $(\text{WLP})$, and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having $\text{WLP}$, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree ≥ 12 (in the worst case), but fail to have $\text{WLP}$. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has $\text{WLP}$ but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double $G$-linkage, the implementations use very different methods.

Geometric intuition suggests that the Néron–Tate height of Heegner points on a rational elliptic curve $E$ should be asymptotically governed by the degree of its modular parametrisation. In this paper, we show that this geometric intuition asymptotically holds on average over a subset of discriminants. We also study the asymptotic behaviour of traces of Heegner points on average over a subset of discriminants and find a difference according to the rank of the elliptic curve. By the Gross–Zagier formulae, such heights are related to the special value at the critical point for either the derivative of the Rankin–Selberg convolution of $E$ with a certain weight one theta series attached to the principal ideal class of an imaginary quadratic field or the twisted $L$-function of $E$ by a quadratic Dirichlet character. Asymptotic formulae for the first moments associated with these $L$-series and $L$-functions are proved, and experimental results are discussed. The appendix contains some conjectural applications of our results to the problem of the discretisation of odd quadratic twists of elliptic curves.

We study the geometry, topology and Lebesgue measure of the set of monic conjugate reciprocal polynomials of fixed degree with all roots on the unit circle. The set of such polynomials of degree $N$ is naturally associated to a subset of ${{\mathbb{R}}^{N-1}}$. We calculate the volume of this set, prove the set is homeomorphic to the $N-1$ ball and that its isometry group is isomorphic to the dihedral group of order $2N$.

Starting with a 2-dimensional mod $p$ Galois representation, we construct a deformation to a power series ring in infinitely many variables over the $p$-adics. The image of this representation is full in the sense that it contains $S{{L}_{2}}$ of this power series ring. Furthermore, all ${{\mathbb{Z}}_{p}}$ specializations of this deformation are potentially semistable at $p$.

The Jiang–Su algebra $Z$ has come to prominence in the classification program for nuclear ${{C}^{*}}$-algebras of late, due primarily to the fact that Elliott’s classification conjecture in its strongest form predicts that all simple, separable, and nuclear ${{C}^{*}}$-algebras with unperforated $\text{K}$-theory will absorb $Z$ tensorially, i.e., will be $Z$-stable. There exist counterexamples which suggest that the conjecture will only hold for simple, nuclear, separable and $Z$-stable ${{C}^{*}}$-algebras. We prove that virtually all classes of nuclear ${{C}^{*}}$-algebras for which the Elliott conjecture has been confirmed so far consist of $Z$-stable ${{C}^{*}}$-algebras. This follows in large part from the following result, also proved herein: separable and approximately divisible ${{C}^{*}}$-algebras are $Z$-stable.

L’objectif de cet article est d’étudier la notion d’amibe au sens de Favorov pour les systèmes finis de sommes d’exponentielles à fréquences réelles et de montrer que, sous des hypothèses de généricité sur les fréquences, le complémentaire de l’amibe d’un système de $(k\,+\,1)$ sommes d’exponentielles à fréquences réelles est un sous-ensemble $k$-convexe au sens d’Henriques.

We examine the fine structure of the short time behavior of solutions to various linear and nonlinear Schrödinger equations ${{u}_{t}}=i\Delta u+q(u)$ on $I\times {{\mathbb{R}}^{n}}$, with initial data $u(0,x)=f(x)$. Particular attention is paid to cases where $f$ is piecewise smooth, with jump across an $(n-1)$-dimensional surface. We give detailed analyses of Gibbs-like phenomena and also focusing effects, including analogues of the Pinsky phenomenon. We give results for general $n$ in the linear case. We also have detailed analyses for a broad class of nonlinear equations when $n=1$ and 2, with emphasis on the analysis of the first order correction to the solution of the corresponding linear equation. This work complements estimates on the error in this approximation.

In this paper, we find equations that characterize locally projectively flat Finsler metrics in the form $F\,=\,{{(\alpha \,+\,\beta )}^{2}}/\alpha $ where $\alpha \,=\,\sqrt{{{a}_{ij}}{{y}^{i}}{{y}^{j}}}$ is a Riemannian metric and $\beta \,=\,{{b}_{i}}{{y}^{i}}$ is a 1-form. Then we completely determine the local structure of those with constant flag curvature.