Growth estimates for orbits of self-adjoint groups

Let $G$ denote a closed, connected, self-adjoint, noncompact subgroup of $GL(n,\mathbb R )$ , and let $d_{R}$ and $d_{L}$ denote respectively the right and left invariant Riemannian metrics defined by the canonical inner product on $M(n,\mathbb R ) = T_{I} GL(n,\mathbb R )$ . Let $v$ be a nonzero vector of $\mathbb R ^{n}$ such that the orbit $G(v)$ is unbounded in $\mathbb R ^{n}$ . Then the function $g \rightarrow d_{R}(g, G_{v})$ is unbounded, where $G_{v} = \{g \in G : g(v) = v \}$ , and we obtain algebraically defined upper and lower bounds $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ for the asymptotic behavior of the function $\frac{log|g(v)|}{d_{R}(g, G_{v})}$ as $d_{R}(g, G_{v}) \rightarrow \infty $ . The upper bound $\lambda ^{+}(v)$ is at most 1. The orbit $G(v)$ is closed in $\mathbb R ^{n} \Leftrightarrow \lambda ^{-}(w)$ is positive for some w $\in G(v)$ . If $G_{v}$ is compact, then $g \rightarrow |d_{R}(g,I) - d_{L}(g,I)|$ is uniformly bounded in $G$ , and the exponents $\lambda ^{+}(v)$ and $\lambda ^{-}(v)$ are sharp upper and lower asymptotic bounds for the functions $\frac{log|g(v)|}{d_{R}(g,I)}$ and $\frac{log|g(v)|}{d_{L}(g,I)}$ as $d_{R}(g,I) \rightarrow \infty $ or as $d_{L}(g,I) \rightarrow \infty $ . However, we show by example that if $G_{v}$ is noncompact, then there need not exist asymptotic upper and lower bounds for the function $\frac{log|g(v)|}{d_{L}(g, G_{v})}$ as $d_{L}(g, G_{v}) \rightarrow \infty $ . The results apply to representations of noncompact semisimple Lie groups $G$ on finite dimensional real vector spaces. We compute $\lambda ^{+}$ and $\lambda ^{-}$ for the irreducible, real representations of $SL(2,\mathbb R )$ , and we show that if the dimension of the $SL(2,\mathbb R )$ -module $V$ is odd, then $\lambda ^{+} = \lambda ^{-}$ on a nonempty open subset of $V$ . We show that the function $\lambda ^{-}$ is $K$ -invariant, where $K = O(n,\mathbb R ) \cap G$ . We do not know if $\lambda ^{-}$ is $G$ -invariant.     

Wild quotient singularities of surfaces

Let $(B,\mathcal{M }_B)$ be a noetherian regular local ring of dimension $2$ with residue field $B/\mathcal{M }_B$ of characteristic $p>0$ . Assume that $B$ is endowed with an action of a finite cyclic group $H$ whose order is divisible by $p$ . Associated with a resolution of singularities of $\mathrm{Spec}B^H$ is a resolution graph $G$ and an intersection matrix $N$ . We prove in this article three structural properties of wild quotient singularities, which suggest that in general, one should expect when $H= \mathbb{Z }/p\mathbb{Z }$ that the graph $G$ is a tree, that the Smith group $\mathbb{Z }^n/\mathrm{Im}(N)$ is killed by $p$ , and that the fundamental cycle $Z$ has self-intersection $|Z^2|\le p$ . We undertake a combinatorial study of intersection matrices $N$ with a view towards the explicit determination of the invariants $\mathbb{Z }^n/\mathrm{Im}(N)$ and $Z$ . We also exhibit explicitly the resolution graphs of an infinite set of wild $\mathbb{Z }/2\mathbb{Z }$ -singularities, using some results on elliptic curves with potentially good ordinary reduction which could be of independent interest.     

The second order pullback equation

Let $f,g$ be two closed $k$ -forms over $\mathbb{R }^{n}.$ The pullback equation studies the existence of a diffeomorphism $\varphi :\mathbb{R }^{n} \rightarrow \mathbb{R }^{n}$ such that $$\begin{aligned} \varphi ^{*}(g)=f. \end{aligned}$$ We prove two types of results. The first one sharpens some of the existing regularity results. The second one discusses the possibility of choosing the map $\varphi $ as the gradient of a function $\Phi :\mathbb{R }^{n} \rightarrow \mathbb R .$ We show that this is a very rare event unless the two forms are constant.     

Clock synchronization in symmetric stochastic networks

We consider a stochastic model of clock synchronization in a wireless network of $N$ sensors interacting with one dedicated accurate time server. For large $N$ we find an estimate of the final time sychronization error for global and relative synchronization. The main results concern the behavior of the network on different timescales $t_{N}\rightarrow \infty $ , $N\rightarrow \infty $ . We discuss the existence of phase transitions and find the exact timescales for which an effective clock synchronization of the system takes place.     

Homotopy and nonlinear boundary value problems involving singular {\phi}-Laplacians

Homotopy methods are used to find sufficient conditions for the solvability of nonlinear boundary value problems of the form $$(\phi(u^\prime))^\prime = f(t, u, u^\prime), \quad g(u(\alpha), \phi(u^\prime(\beta))) = 0,$$ where (α, β) = (0, 1), (1, 0), (0, 0) or (1, 1), ${\phi}$ is a homeomorphism from the open ball ${B(a) \subset \mathbb{R}^n}$ onto ${\mathbb{R}^n}$ , f is a Carathéodory function, ${g : \mathbb{R}^n \times \, \mathbb{R}^n \rightarrow \mathbb{R}^m}$ is continuous and m ≤ 2n.     

A note on nonlinear fourth-order elliptic equations on \mathbb R ^N

We established the existence of weak solutions of the fourth-order elliptic equation of the form $$\begin{aligned} \Delta ^2 u -\Delta u + a(x)u = \lambda b(x) f(u) + \mu g (x, u), \qquad x \in \mathbb{R }^N, u \in H^2(\mathbb{R }^N), \end{aligned}$$ where $\lambda $ is a positive parameter, $a(x)$ and $b(x)$ are positive functions, while $f : \mathbb{R }\rightarrow \mathbb{R }$ is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that the problem has at least three solutions.     

Modular categories associated with unipotent groups

Let $G$ be a unipotent algebraic group over an algebraically closed field $\mathtt{k }$ of characteristic $p>0$ and let $l\ne p$ be another prime. Let $e$ be a minimal idempotent in $\mathcal{D }_G(G)$ , the $\overline{\mathbb{Q }}_l$ -linear triangulated braided monoidal category of $G$ -equivariant (for the conjugation action) $\overline{\mathbb{Q }}_l$ -complexes on $G$ under convolution (with compact support) of complexes. Then, by a construction due to Boyarchenko and Drinfeld, we can associate to $G$ and $e$ a modular category $\mathcal{M }_{G,e}$ . In this paper, we prove that the modular categories that arise in this way from unipotent groups are precisely those in the class $\mathfrak{C }_p^{\pm }$ .     

p-Groups with few conjugacy classes of normalizers

For a group $G$ , denote by $\omega (G)$ the number of conjugacy classes of normalizers of subgroups of $G$ . Clearly, $\omega (G)=1$ if and only if $G$ is a Dedekind group. Hence if $G$ is a 2-group, then $G$ is nilpotent of class $\le 2$ and if $G$ is a $p$ -group, $p>2$ , then $G$ is abelian. We prove a generalization of this. Let $G$ be a finite $p$ -group with $\omega (G)\le p+1$ . If $p=2$ , then $G$ is of class $\le 3$ ; if $p>2$ , then $G$ is of class $\le 2$ .     

Approximate Hermitian–Einstein connections on principal bundles over a compact Riemann surface

Let $X$ be a compact connected Riemann surface and $G$ a connected reductive complex affine algebraic group. Given a holomorphic principal $G$ -bundle $E_G$ over $X$ , we construct a $C^\infty $ Hermitian structure on $E_G$ together with a $1$ -parameter family of $C^\infty $ automorphisms $\{F_t\}_{t\in \mathbb R }$ of the principal $G$ -bundle $E_G$ with the following property: Let $\nabla ^t$ be the connection on $E_G$ corresponding to the Hermitian structure and the new holomorphic structure on $E_G$ constructed using $F_t$ from the original holomorphic structure. As $t\rightarrow -\infty $ , the connection $\nabla ^t$ converges in $C^\infty $ Fréchet topology to the connection on $E_G$ given by the Hermitian–Einstein connection on the polystable principal bundle associated to $E_G$ . In particular, as $t\rightarrow -\infty $ , the curvature of $\nabla ^t$ converges in $C^\infty $ Fréchet topology to the curvature of the connection on $E_G$ given by the Hermitian–Einstein connection on the polystable principal bundle associated to $E_G$ . The family $\{F_t\}_{t\in \mathbb R }$ is constructed by generalizing the method of [6]. Given a holomorphic vector bundle $E$ on $X$ , in [6] a $1$ -parameter family of $C^\infty $ automorphisms of $E$ is constructed such that as $t\rightarrow -\infty $ , the curvature converges, in $C^0$ topology, to the curvature of the Hermitian–Einstein connection of the associated graded bundle.     

Character varieties of abelian groups

We prove that for every reductive group $G$ with a maximal torus ${\mathbb {T}}$ and the Weyl group $W,\, {\mathbb {T}}^N/W$ is the normalization of the irreducible component, $X_G^0({\mathbb {Z}}^N)$ , of the $G$ -character variety $X_G({\mathbb {Z}}^N)$ of ${\mathbb {Z}}^N$ containing the trivial representation. We also prove that $X_G^0({\mathbb {Z}}^N)={\mathbb {T}}^N/W$ for all classical groups. Additionally, we prove that even though there are no irreducible representations in $X_G^0({\mathbb {Z}}^N)$ for non-abelian $G$ , the tangent spaces to $X_G^0({\mathbb {Z}}^N)$ coincide with $H^1({\mathbb {Z}}^N, Ad\, \rho )$ . Consequently, $X_G^0({\mathbb {Z}}^2)$ , has the “Goldman” symplectic form for which the combinatorial formulas for Goldman bracket hold.     

Appell Bases for Monogenic Functions of Three Variables

Monogenic (or hyperholomorphic) functions are well known in general Clifford algebras but have been little studied in the particular case ${\mathbb{R}^{3} \rightarrow \mathbb{R}^{3}}$ R 3 → R 3 . We describe for this case the collection of all Appell systems: bases for the finite-dimensional spaces of monogenic homogeneous polynomials which respect the operator ${D = \partial_{0} - \vec{\partial}}$ D = ? 0 ? ? → . We prove that no purely algebraic recursive formula (in a specific sense) exists for these Appell systems, in contrast to the existence of known constructions for ${\mathbb{R}^{3} \rightarrow \mathbb{R}^{4}}$ R 3 → R 4 and ${\mathbb{R}^{4} \rightarrow \mathbb{R}^{4}}$ R 4 → R 4 . However, we give a simple recursive procedure for constructing Appell bases for ${\mathbb{R}^{3} \rightarrow \mathbb{R}^{3}}$ R 3 → R 3 which uses the operation of integration of polynomials.     

On the signed star domination number of regular multigraphs

Let $G$ be a graph with the vertex set $V(G)$ and the edge set $E(G)$ . A function $f: E(G)\longrightarrow \{-1, 1\}$ is said to be a signed star dominating function of $G$ if $\sum _{e \in E_G(v)}f (e)\ge 1 $ , for every $v \in V(G)$ , where $E_G(v) = \{uv\in E(G)\,|\,u \in V (G)\}$ . The minimum values of $\sum _{e \in E_G(v)}f (e)$ , taken over all signed star dominating functions $f$ on $G$ , is called the signed star domination number of $G$ and denoted by $\gamma _{SS}(G)$ . In this paper we determine the signed star domination number of regular multigraphs.     

Convergence of the eigenvalues of the p-Laplace operator as p goes to 1

Let $(\lambda ^k_p)_k$ be the usual sequence of min-max eigenvalues for the $p$ -Laplace operator with $p\in (1,\infty )$ and let $(\lambda ^k_1)_k$ be the corresponding sequence of eigenvalues of the 1-Laplace operator. For bounded $\Omega \subseteq \mathbb{R }^n$ with Lipschitz boundary the convergence $\lambda ^k_p\rightarrow \lambda ^k_1$ as $p\rightarrow 1$ is shown for all $k\in \mathbb{N }$ . The proof uses an approximation of $BV(\Omega )$ -functions by $C_0^\infty (\Omega )$ -functions in the sense of strict convergence on $\mathbb{R }^n$ .     

Lower semicontinuity for functionals with (p, q)-growth in Carnot groups

Let ?? be a bounded open subset of ${\mathbb{G}}$ , where ${\mathbb{G}}$ is a Carnot group, and let ${u: \Omega \rightarrow \mathbb{R}^d}$ be a vector valued function. We prove a lower semicontinuity result in the weak topology of the horizontal Sobolev space ${W^{1,p}_X(\Omega,\mathbb{R}^d)}$ , with p?>?1, of the integral functional of the calculus of variations of the type $$F(u)=\int\limits_\Omega f(Xu)\,dx$$ where f is a X-quasiconvex function satisfying a non-standard growth conditions and Xu is the horizontal gradient of u.     

Existence and Multiplicity Results for Some Elliptic Systems in Unbounded Cylinders

We study the following nonlinear elliptic system of Lane–Emden type $$\left\{\begin{array}{ll} -\Delta u = {\rm sgn}(v) |v| ^{p-1} \qquad \qquad \qquad \; {\rm in} \; \Omega , \\ -\Delta v = - \lambda {\rm sgn} (u)|u| \frac{1}{p-1} + f(x, u)\; \; {\rm in}\; \Omega , \\ u = v = 0 \qquad \qquad \qquad \quad \quad \;\;\;\;\; {\rm on}\; \partial \Omega , \end{array}\right.$$ where ${\lambda \in \mathbb{R}}$ . If ${\lambda \geq 0}$ and ${\Omega}$ is an unbounded cylinder, i.e., ${\Omega = \tilde \Omega \times \mathbb{R}^{N-m} \subset \mathbb{R}^{N}}$ , ${N - m \geq 2, m \geq 1}$ , existence and multiplicity results are proved by means of the Principle of Symmetric Criticality and some compact imbeddings in partially spherically symmetric spaces. We are able to state existence and multiplicity results also if ${\lambda \in \mathbb{R}}$ and ${\Omega}$ is a bounded domain in ${\mathbb{R}^{N}, N \geq 3}$ . In particular, a good finite dimensional decomposition of the Banach space in which we work is given.     

Compactification de Chabauty de l’espace des sous-groupes de Cartan de {\text{ SL}}_n(\mathbb{R })

Let $G$ be a real semisimple Lie group with finite center, with a finite number of connected components and without compact factor. We are interested in the homogeneous space of Cartan subgroups of $G$ , which can be also seen as the space of maximal flats of the symmetric space of $G$ . We define its Chabauty compactification as the closure in the space of closed subgroups of $G$ , endowed with the Chabauty topology. We show that when the real rank of $G$ is 1, or when $G={\text{ SL}}_3(\mathbb{R })$ or ${\text{ SL}}_4(\mathbb{R })$ , this compactification is the set of all closed connected abelian subgroups of dimension the real rank of $G$ , with real spectrum. And in the case of ${\text{ SL}}_3(\mathbb{R })$ , we study its topology more closely and we show that it is simply connected.     

Finite groups with \mathbb{P }-subnormal subgroups

A subgroup $H$ of a group $G$ is called $\mathbb{P }$ -subnormal in $G$ whenever either $H=G$ or there is a chain of subgroups $H=H_0\subset H_1\subset \cdots \subset H_n=G$ such that $|H_i:H_{i-1}|$  is a prime for all $i$ . In this paper we study groups with $\mathbb{P }$ -subnormal 2-maximal subgroups, and groups with $\mathbb{P }$ -subnormal primary cyclic subgroups.     

Global symplectic coordinates on gradient Kähler–Ricci solitons

A classical result of McDuff [14] asserts that a simply connected complete Kähler manifold $(M,g,\omega )$ with non positive sectional curvature admits global symplectic coordinates through a symplectomorphism $\Psi \ : M \rightarrow \mathbb{R }^{2n}$ (where $n$ is the complex dimension of $M$ ), satisfying the following property (proved by E. Ciriza in [4]): the image $\Psi (T)$ of any complex totally geodesic submanifold $T\subset M$ through the point $p$ such that $\Psi (p)=0$ , is a complex linear subspace of $\mathbb C ^n\simeq \mathbb{R }^{2n}$ . The aim of this paper is to exhibit, for all positive integers $n$ , examples of $n$ -dimensional complete Kähler manifolds with non-negative sectional curvature globally symplectomorphic to $\mathbb{R }^{2n}$ through a symplectomorphism satisfying Ciriza’s property.     

Finite-time blowup for a complex Ginzburg–Landau equation with linear driving

In this paper, we consider the complex Ginzburg–Landau equation ${u_t = e^{i\theta} [\Delta u + |u|^\alpha u] + \gamma u}$ on ${\mathbb{R}^N}$ , where ${\alpha > 0,\,\gamma \in \mathbb{R}}$ and ${-\pi /2 < \theta < \pi /2}$ . By convexity arguments, we prove that, under certain conditions on ${\alpha,\theta,\gamma}$ , a class of solutions with negative initial energy blows up in finite time.