Let $T$ be a sectorial operator. It is known that the existence of a bounded (suitably scaled) ${{H}^{\infty }}$ calculus for $T$, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra $\Lambda _{\infty ,1}^{\alpha }({{\mathbb{R}}^{+}})$. Such an algebra includes functions defined byMikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for $T$. In this paper, we use fractional derivation to analyse in detail the relationship between $\Lambda _{\infty ,1}^{\alpha }$ and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence.

Soit ${{F}_{0}}$ un corps local non archimédien de caractéristique nulle et de caractéristique résiduelle impaire. J. Rogawski a montré l’existence du changement de base entre le groupe unitaire en trois variables $U(2,1)({{F}_{0}})$, défini relativement à une extension quadratique $F$ de ${{F}_{0}}$, et le groupe linéaire $\text{GL}(3,F)$. Par ailleurs, nous avons décrit les représentations supercuspidales irréductibles de $U(2,1)({{F}_{0}})$ comme induites à partir d’un sous-groupe compact ouvert de $U(2,1)({{F}_{0}})$, description analogue à celle des représentations admissibles irréductibles de $\text{GL}(3,F)$ obtenue par C. Bushnell et P. Kutzko. A partir de ces descriptions, nous construisons explicitement le changement de base des représentations très cuspidales de $U(2,1)({{F}_{0}})$.

The classical Hurwitz enumeration problem has a presentation in terms of transitive factorizations in the symmetric group. This presentation suggests a generalization from type $A$ to other finite reflection groups and, in particular, to type $B$. We study this generalization both from a combinatorial and a geometric point of view, with the prospect of providing a means of understanding more of the structure of the moduli spaces of maps with an ${{\mathfrak{S}}_{2}}$-symmetry. The type $A$ case has been well studied and connects Hurwitz numbers to the moduli space of curves. We conjecture an analogous setting for the type $B$ case that is studied here.

We will study the following question: Are nilpotent conjugacy classes of reductive Lie algebras over $p$-adic fields definable? By definable, we mean definable by a formula in Pas's language. In this language, there are no field extensions and no uniformisers. Using Waldspurger's parametrization, we answer in the affirmative in the case of special orthogonal Lie algebras $\mathfrak{s}\mathfrak{o}\left( n \right)$ for $n$ odd, over $p$-adic fields.

Littlewood–Paley analysis is generalized in this article. We show that the compactness of the Fourier support imposed on the analyzing function can be removed. We also prove that the Littlewood–Paley decomposition of tempered distributions converges under a topology stronger than the weak-star topology, namely, the inductive limit topology. Finally, we construct a multiparameter Littlewood–Paley analysis and obtain the corresponding “renormalization” for the convergence of this multiparameter Littlewood–Paley analysis.

We investigate the problem of deforming $n$-dimensional mod $p$ Galois representations to characteristic zero. The existence of 2-dimensional deformations has been proven under certain conditions by allowing ramification at additional primes in order to annihilate a dual Selmer group. We use the same general methods to prove the existence of $n$-dimensional deformations.

We then examine under which conditions we may place restrictions on the shape of our deformations at $p$, with the goal of showing that under the correct conditions, the deformations may have locally geometric shape. We also use the existence of these deformations to prove the existence as Galois groups over $\mathbb{Q}$ of certain infinite subgroups of $p$-adic general linear groups.

Maximum principles for subharmonic functions in the framework of quasi-regular local semi-Dirichlet forms admitting lower bounds are presented. As applications, we give weak and strong maximum principles for (local) subsolutions of a second order elliptic differential operator on the domain of Euclidean space under conditions on coefficients, which partially generalize the results by Stampacchia.

A smooth affine surface $X$ defined over the complex field $\mathbb{C}$ is an $\text{M}{{\text{L}}_{0}}$ surface if the Makar–Limanov invariant $\text{ML(}X\text{)}$ is trivial. In this paper we study the topology and geometry of $\text{M}{{\text{L}}_{0}}$ surfaces. Of particular interest is the question: Is every curve $C$ in $X$ which is isomorphic to the affine line a fiber component of an ${{\mathbb{A}}^{1}}$-fibration on $X$? We shall show that the answer is affirmative if the Picard number $\rho (X)\,=\,0$, but negative in case $\rho (X)\,\ge \,1$. We shall also study the ascent and descent of the $\text{M}{{\text{L}}_{0}}$ property under proper maps.

We prove a characteristic free version of Weyl’s theorem on polarization. Our result is an exact analogue of Weyl’s theorem, the difference being that our statement is about separating invariants rather than generating invariants. For the special case of finite group actions we introduce the concept of cheap polarization, and show that it is enough to take cheap polarizations of invariants of just one copy of a representation to obtain separating vector invariants for any number of copies. This leads to upper bounds on the number and degrees of separating vector invariants of finite groups.

This paper establishes general theorems which contain both moduli of continuity and large incremental results for ${{l}^{\infty }}$-valued Gaussian random fields indexed by a multidimensional parameter under explicit conditions.

Hua’s fundamental theorem of the geometry of hermitian matrices characterizes bijective maps on the space of all $n\times n$ hermitianmatrices preserving adjacency in both directions. The problem of possible improvements has been open for a while. There are three natural problems here. Do we need the bijectivity assumption? Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only? Can we obtain such a characterization formaps acting between the spaces of hermitian matrices of different sizes? We answer all three questions for the complex hermitian matrices, thus obtaining the optimal structural result for adjacency preserving maps on hermitian matrices over the complex field.

This paper is the continuation of our previous work on the explicit determination of the structure of theta lifts for dual pairs $\left( {{\text{S}}_{{{\text{p}}_{2n}},}}\,O\left( V \right) \right)$ over a non-archimedean field $F$ of characteristic different than 2, where $n$ is the split rank of ${{\text{S}}_{{{\text{p}}_{2n}}}}$ and the dimension of the space $V$ (over $F$) is even. We determine the structure of theta lifts of tempered representations in terms of theta lifts of representations in discrete series.

For $p$ a prime, a $p$-typical cover of a connected scheme on which $p\,=\,0$ is a finite étale cover whose monodromy group (i.e.,the Galois group of its normal closure) is a $p$-group. The geometry of such covers exhibits some unexpectedly pleasant behaviors; building on work of Katz, we demonstrate some of these. These include a criterion for when a morphism induces an isomorphism of the $p$-typical quotients of the étale fundamental groups, and a decomposition theorem for $p$-typical covers of polynomial rings over an algebraically closed field.

We show that a characterization of scaling functions for multiresolution analyses given by Hernández and Weiss and that a characterization of low-pass filters given by Gundy both hold for multivariable multiresolution analyses.

This is the third in a series of works devoted to spectral asymptotics for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, having a periodic classical flow. Assuming that the strength $\epsilon$ of the perturbation is in the range ${{h}^{2}}\ll \epsilon \ll {{h}^{1/2}}$ (and may sometimes reach even smaller values), we get an asymptotic description of the eigenvalues in rectangles $[-1/C,1/C]+i\epsilon [{{F}_{0}}-1/C,{{F}_{0}}+1/C],C\gg 1$, when $\epsilon {{F}_{0}}$ is a saddle point value of the flow average of the leading perturbation.

We observe that the small quantum product of the generalized flag manifold $G/B$ is a product operation $\star $ on ${{H}^{*}}(G/B)\otimes \mathbb{R}\left[ {{q}_{1,...,}}{{q}_{l}} \right]$ uniquely determined by the facts that it is a deformation of the cup product on ${{H}^{*}}(G/B)$; it is commutative, associative, and graded with respect to deg$({{q}_{i}})=4$; it satisfies a certain relation (of degree two); and the corresponding Dubrovin connection is flat. Previously, we proved that these properties alone imply the presentation of the ring $({{H}^{*}}(G/B)\otimes \mathbb{R}\left[ {{q}_{1,...,}}{{q}_{l}} \right]\star )$ in terms of generators and relations. In this paper we use the above observations to give conceptually new proofs of other fundamental results of the quantum Schubert calculus for $G/B$: the quantum Chevalley formula of D. Peterson (see also Fulton and Woodward) and the “quantization by standard monomials” formula of Fomin, Gelfand, and Postnikov for $G=SL(n,\mathbb{C})$. The main idea of the proofs is the same as in Amarzaya–Guest: from the quantum $\mathcal{D}$-module of $G/B$ one can decode all information about the quantum cohomology of this space.

We propose a new, constructive theory of moving frames for Lie pseudo-group actions on submanifolds. The moving frame provides an effective means for determining complete systems of differential invariants and invariant differential forms, classifying their syzygies and recurrence relations, and solving equivalence and symmetry problems arising in a broad range of applications.

Let $A$ be a unital commutative associative algebra over a field of characteristic zero, $\mathfrak{k}$ a Lie algebra, and $\mathfrak{z}$ a vector space, considered as a trivial module of the Lie algebra $\mathfrak{g}:=A\otimes \mathfrak{k}$. In this paper, we give a description of the cohomology space ${{H}^{2}}(\mathfrak{g},\mathfrak{z})$ in terms of easily accessible data associated with $A$ and $\mathfrak{k}$. We also discuss the topological situation, where $A$ and $\mathfrak{k}$ are locally convex algebras.

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 $\text{GL}(N,F)$.

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.