Let $F$ be a $p$-adic field of characteristic $0$, and let $M$ be an $F$-Levi subgroup of a connected reductive $F$-split group such that $\Pi_{i=1}^{r} SL_{n_i} \subseteq M \subseteq \Pi_{i=1}^{r} GL_{n_i}$ for positive integers $r$ and $n_i$. We prove that the Plancherel measure for any unitary supercuspidal representation of $M(F)$ is identically transferred under the local Jacquet-Langlands type correspondence between $M$ and its $F$-inner forms, assuming a working hypothesis that Plancherel measures are invariant on a certain set. This work extends the result of Muić and Savin (2000) for Siegel Levi subgroups of the groups $SO_{4n}$ and $Sp_{4n}$ under the local Jacquet-Langlands correspondence. It can be applied to a simply connected simple $F$-group of type $E_6$ or $E_7$, and a connected reductive $F$-group of type $A_{n}$, $B_{n}$, $C_n$ or $D_n$.

Our goal in this work is to present some function spaces on the complex plane $\mathbb C$, $X(\mathbb C)$, for which the quasiregular solutions of the Beltrami equation, $\overline\partial f (z) = \mu(z) \partial f (z)$, have first derivatives locally in $X(\mathbb C)$, provided that the Beltrami coefficient $\mu$ belongs to $X(\mathbb C)$.

We introduce the notion of families of $n$-marked smooth rational tropical curves over smooth tropical varieties and establish a one-to-one correspondence between (equivalence classes of) these families and morphisms from smooth tropical varieties into the moduli space of $n$-marked abstract rational tropical curves $\mathcal{M}_{n}$.

We fix a mistake in the proof of Theorem 1.6 in the paper in the title.

We present a general construction of Salem numbers via rational functions whose zeros and poles mostly lie on the unit circle and satisfy an interlacing condition. This extends and unifies earlier work. We then consider the ``obvious'' limit points of the set of Salem numbers produced by our theorems and show that these are all Pisot numbers, in support of a conjecture of Boyd. We then show that all Pisot numbers arise in this way. Combining this with a theorem of Boyd, we produce all Salem numbers via an interlacing construction.

We introduce the fundamental group ${\mathcal F}(A)$ of a simple $\sigma$-unital $C^*$-algebra $A$ with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of ``Fundamental Group of Simple $C^*$-algebras with Unique Trace I and II'' by Nawata and Watatani. Our definition in this paper makes sense for stably projectionless $C^*$-algebras. We show that there exist separable stably projectionless $C^*$-algebras such that their fundamental groups are equal to $\mathbb{R}_+^\times$ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case in Nawata and Watatani. This study is motivated by the work of Kishimoto and Kumjian.

Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of invariant polynomials $\mathbb C[V]^G$. We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$ over the mapping of invariants $\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$ and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass $\mathcal C \subseteq C^\infty$ satisfying some mild closedness properties that guarantee resolution of singularities in $\mathcal C$, e.g., the real analytic class, then $f$ admits a lift of the same class $\mathcal C$ after desingularization by local blow-ups and local power substitutions. As a consequence we show that $f$ itself allows for a lift that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation. If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions.

We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó-Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.

In this paper we study the notion of a convex subordination chain in several complex variables. We obtain certain necessary and sufficient conditions for a mapping to be a convex subordination chain, and we give various examples of convex subordination chains on the Euclidean unit ball in $\mathbb{C}^n$. We also obtain a sufficient condition for injectivity of $f(z/\|z\|,\|z\|)$ on $B^n\setminus\{0\}$, where $f(z,t)$ is a convex subordination chain over $(0,1)$.

For an isotropic submanifold $M^n\,(n\geqq3)$ of a space form $\widetilde{M}^{n+p}(c)$ of constant sectional curvature $c$, we show that if the mean curvature vector of $M^n$ is parallel and the sectional curvature $K$ of $M^n$ satisfies some inequality, then the second fundamental form of $M^n$ in $\widetilde{M}^{n+p}$ is parallel and our manifold $M^n$ is a space form.

Let $q$ be an algebraic integer of degree $d \geq 2$. Consider the rank of the multiplicative subgroup of $\BbC^*$ generated by the conjugates of $q$. We say $q$ is of {\em full rank} if either the rank is $d-1$ and $q$ has norm $\pm 1$, or the rank is $d$. In this paper we study some properties of $\BbZ[q]$ where $q$ is an algebraic integer of full rank. The special cases of when $q$ is a Pisot number and when $q$ is a Pisot-cyclotomic number are also studied. There are four main results. \begin{compactenum}[\rm(1)] \item If $q$ is an algebraic integer of full rank and $n$ is a fixed positive integer, then there are only finitely many $m$ such that $\disc\left(\BbZ[q^m]\right)=\disc\left(\BbZ[q^n]\right)$. \item If $q$ and $r$ are algebraic integers of degree $d$ of full rank and $\BbZ[q^n] = \BbZ[r^n]$ for infinitely many $n$, then either $q = \omega r'$ or $q={\rm Norm}(r)^{2/d}\omega/r'$, where $r'$ is some conjugate of $r$ and $\omega$ is some root of unity. \item Let $r$ be an algebraic integer of degree at most $3$. Then there are at most $40$ Pisot numbers $q$ such that $\BbZ[q] = \BbZ[r]$. \item There are only finitely many Pisot-cyclotomic numbers of any fixed order. \end{compactenum}

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 $\GL(N,F)$, 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 $\GL(N,F)$.

We obtain a uniform linear bound for the Chevalley function at a point in the source of an analytic mapping that is regular in the sense of Gabrielov. There is a version of Chevalley's lemma also along a fibre, or at a point of the image of a proper analytic mapping. We get a uniform linear bound for the Chevalley function of a closed Nash (or formally Nash) subanalytic set.

On associe \`a toute structure de proto-big\`ebre de Lie sur un espace vectoriel $F$ de dimension finie des structures d'alg\`ebre de Lie d'homotopie d\'efinies respectivement sur la suspension de l'alg\`ebre ext\'erieure de $F$ et celle de son dual $F^*$. Dans ces alg\`ebres, tous les crochets $n$-aires sont nuls pour $n \geq 4$ du fait qu'ils proviennent d'une structure de proto-big\`ebre de Lie. Plus g\'en\'eralement, on associe \`a un \'el\'ement de degr\'e impair de l'alg\`ebre ext\'erieure de la somme directe de $F$ et $F^*$, une collection d'applications multilin\'eaires antisym\'etriques sur l'alg\`ebre ext\'erieure de $F$ (resp.\ $F^*$), qui v\'erifient les identit\'es de Jacobi g\'en\'eralis\'ees, d\'efinissant les alg\`ebres de Lie d'homotopie, si l'\'el\'ement donn\'e est de carr\'e nul pour le grand crochet de l'alg\`ebre ext\'erieure de la somme directe de $F$ et de~$F^*$. To any proto-Lie algebra structure on a finite-dimensional vector space~$F$, we associate homotopy Lie algebra structures defined on the suspension of the exterior algebra of $F$ and that of its dual $F^*$, respectively. In these algebras, all $n$-ary brackets for $n \geq 4$ vanish because the brackets are defined by the proto-Lie algebra structure. More generally, to any element of odd degree in the exterior algebra of the direct sum of $F$ and $F^*$, we associate a set of multilinear skew-symmetric mappings on the suspension of the exterior algebra of $F$ (resp.\ $F^*$), which satisfy the generalized Jacobi identities, defining the homotopy Lie algebras, if the given element is of square zero with respect to the big bracket of the exterior algebra of the direct sum of $F$ and~$F^*$.

In previous papers, Barr and Raphael investigated the situation of a topological space $Y$ and a subspace $X$ such that the induced map $C(Y)\to C(X)$ is an epimorphism in the category $\CR$ of commutative rings (with units). We call such an embedding a $\CR$-epic embedding and we say that $X$ is absolute $\CR$-epic if every embedding of $X$ is $\CR$-epic. We continue this investigation. Our most notable result shows that a Lindel\"of space $X$ is absolute $\CR$-epic if a countable intersection of $\beta X$-neighbourhoods of $X$ is a $\beta X$-neighbourhood of $X$. This condition is stable under countable sums, the formation of closed subspaces, cozero-subspaces, and being the domain or codomain of a perfect map. A strengthening of the Lindel\"of property leads to a new class with the same closure properties that is also closed under finite products. Moreover, all \s-compact spaces and all Lindel\"of $P$-spaces satisfy this stronger condition. We get some results in the non-Lindel\"of case that are sufficient to show that the Dieudonn\'e plank and some closely related spaces are absolute $\CR$-epic.

We consider some deformations of $G_2$-structures on $7$-manifolds. We discover a canonical way to deform a $G_2$-structure by a vector field in which the associated metric gets ``twisted'' in some way by the vector cross product. We present a system of partial differential equations for an unknown vector field $w$ whose solution would yield a manifold with holonomy $G_2$. Similarly we consider analogous constructions for $\Spin(7)$-structures on $8$-manifolds. Some of the results carry over directly, while others do not because of the increased complexity of the $\Spin(7)$ case.

We consider the problem \begin{equation*} \begin{cases} \varepsilon^2 \Delta u - u + f(u) = 0, u > 0 & \mbox{in } \Omega\\ \frac{\partial u}{\partial \nu} = 0 & \mbox{on } \partial\Omega, \end{cases} \end{equation*} where $\Omega$ is a bounded smooth domain in $R^N$, $\ve>0$ is a small parameter and $f$ is a superlinear, subcritical nonlinearity. It is known that this equation possesses multiple boundary spike solutions that concentrate, as $\epsilon$ approaches zero, at multiple critical points of the mean curvature function $H(P)$, $P \in \partial \Omega$. It is also proved that this equation has multiple interior spike solutions which concentrate, as $\ep\to 0$, at {\it sphere packing\/} points in $\Om$. In this paper, we prove the existence of solutions with multiple spikes {\it both\/} on the boundary and in the interior. The main difficulty lies in the fact that the boundary spikes and the interior spikes usually have different scales of error estimation. We have to choose a special set of boundary spikes to match the scale of the interior spikes in a variational approach.

We consider a class of fractal subsets of $\R^d$ formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion $X$ and determine its basic properties; and extend some classical Sobolev and Poincar\'e inequalities to this setting.

In this article we characterize the univalent harmonic mappings from the exterior of the unit disk, $\Delta$, onto a simply connected domain $\Omega$ containing infinity and which are solutions of the system of elliptic partial differential equations $\fzbb = a(z)f_z(z)$ where the second dilatation function $a(z)$ is a finite Blaschke product. At the end of this article, we apply our results to nonparametric minimal surfaces having the property that the image of its Gauss map is the upper half-sphere covered once or twice.

In this paper we establish conditions that guarantee, in the setting of a general Banach space, the Painlev\'e-Kuratowski convergence of the graphs of the subdifferentials of convexly composite functions. We also provide applications to the convergence of multipliers of families of constrained optimization problems and to the generalized second-order derivability of convexly composite functions.

Pieri's formula describes the intersection product of a Schubert cycle by a special Schubert cycle on a Grassmannian. We present a new geometric proof, exhibiting an explicit chain of rational equivalences from a suitable sum of distinct Schubert cycles to the intersection of a Schubert cycle with a special Schubert cycle. The geometry of these rational equivalences indicates a link to a combinatorial proof of Pieri's formula using Schensted insertion.

For a fixed algebraic number $\alpha$ we discuss how closely $\alpha$ can be approximated by a root of a $\{0,+1,-1\}$ polynomial of given degree. We show that the worst rate of approximation tends to occur for roots of unity, particularly those of small degree. For roots of unity these bounds depend on the order of vanishing, $k$, of the polynomial at $\alpha$. In particular we obtain the following. Let ${\cal B}_{N}$ denote the set of roots of all $\{0,+1,-1\}$ polynomials of degree at most $N$ and ${\cal B}_{N}(\alpha,k)$ the roots of those polynomials that have a root of order at most $k$ at $\alpha$. For a Pisot number $\alpha$ in $(1,2]$ we show that \[ \min_{\beta \in {\cal B}_{N}\setminus \{ \alpha \}} |\alpha -\beta| \asymp \frac{1}{\alpha^{N}}, \] and for a root of unity $\alpha$ that \[ \min_{\beta \in {\cal B}_{N}(\alpha,k)\setminus \{\alpha\}} |\alpha -\beta|\asymp \frac{1}{N^{(k+1) \left\lceil \frac{1}{2}\phi (d)\right\rceil +1}}. \] We study in detail the case of $\alpha=1$, where, by far, the best approximations are real. We give fairly precise bounds on the closest real root to 1. When $k=0$ or 1 we can describe the extremal polynomials explicitly.

In this paper, we consider the parabolic equation with anisotropic growth conditions, and obtain some criteria on boundedness of solutions, which generalize the corresponding results for the isotropic case.

Let $L_p=L_p(X,\mu)$, $1\leq p\leq\infty$, be the usual Banach Spaces of real valued functions on a complete non-atomic probability space. Let $(T_1,\ldots,T_{K})$ be $L_2$-contractions. Let $0<\varepsilon < \delta\leq1$. Call a function $f$ a $\delta$-spanning function if $\|f\|_2 = 1$ and if $\|T_kf-Q_{k-1}T_kf\|_2\geq\delta$ for each $k=1,\ldots,K$, where $Q_0=0$ and $Q_k$ is the orthogonal projection on the subspace spanned by $(T_1f,\ldots,T_kf)$. Call a function $h$ a $(\delta,\varepsilon)$-sweeping function if $\|h\|_\infty\leq1$, $\|h\|_1<\varepsilon$, and if $\max_{1\leq k\leq K}|T_kh|>\delta-\varepsilon$ on a set of measure greater than $1-\varepsilon$. The following is the main technical result, which is obtained by elementary estimates. There is an integer $K=K(\varepsilon,\delta)\geq1$ such that if $f$ is a $\delta$-spanning function, and if the joint distribution of $(f,T_1f,\ldots,T_Kf)$ is normal, then $h=\bigl((f\wedge M)\vee(-M)\bigr)/M$ is a $(\delta,\varepsilon)$-sweeping function, for some $M>0$. Furthermore, if $T_k$s are the averages of operators induced by the iterates of a measure preserving ergodic transformation, then a similar result is true without requiring that the joint distribution is normal. This gives the following theorem on a sequence $(T_i)$ of these averages. Assume that for each $K\geq1$ there is a subsequence $(T_{i_1},\ldots,T_{i_K})$ of length $K$, and a $\delta$-spanning function $f_K$ for this subsequence. Then for each $\varepsilon>0$ there is a function $h$, $0\leq h\leq1$, $\|h\|_1<\varepsilon$, such that $\limsup_iT_ih\geq\delta$ a.e.. Another application of the main result gives a refinement of a part of Bourgain's ``Entropy Theorem'', resulting in a different, self contained proof of that theorem.