For shifts with weak specification, we show that the set of points for which the Birkhoff averages of a continuous function diverge is residual. This includes topologically transitive topological Markov chains, sofic shifts and more generally shifts with specification. In addition, we show that the set of points for which the Birkhoff averages of a continuous function have a prescribed set of accumulation points is also residual. The proof consists of bridging together strings of sufficiently large length corresponding to a dense set of limits of Birkhoff averages. Finally, we consider intersections of finitely many irregular sets and show that they are again residual. As an application, we show that the set of points for which the Lyapunov exponents on a conformal repeller are not limits is residual.

We prove that the Ricci operator on a contact Riemannian 3-manifold $M$ is invariant along the Reeb flow if and only if $M$ is Sasakian or locally isometric toSU(2) (or SO(3)), SL(2, $\boldsymbol{R}$) (or $O(1,2)$), the group $E(2)$ of rigid motions of Euclidean 2-plane with a contact left invariant Riemannian metric.

In this paper, we introduce the 4-th multiple residue symbol $[p_1,p_2,p_3,p_4]$ for certain four prime numbers $p_i$'s, which extends the Legendre symbol $\big(\frac{p_1}{p_2}\big)$ and the Rédei triple symbol $[p_1,p_2,p_3]$ in a natural manner. For this we construct concretely a certain nilpotent extension $K$ over $\boldsymbol{Q}$ of degree 64, where ramified prime numbers are $p_1, p_2$ and $p_3$, such that the symbol $[p_1,p_2,p_3,p_4]$ describes the decomposition law of $p_4$ in the extension $K/\boldsymbol{Q}$. We then establish the relation of our symbol $[p_1,p_2,p_3,p_4]$ and the 4-th arithmetic Milnor invariant $\mu_2(1234)$ (an arithmetic analogue of the 4-th order linking number) by showing $[p_1,p_2,p_3,p_4] = (-1)^{\mu_2(1234)}$.

Let $\{X_{t}\}_{t \geq 0}$ be the $\alpha$-stable-like or relativistic $\alpha$-stable-like process on $\boldsymbol{R}^{d}$ generated by a certain symmetric jump-type regular Dirichlet form $(\mathcal{E, F})$. It is known in [5, 7] that the transition probability density $p(t, x, y)$ of $\{X_{t}\}_{t \geq 0}$ admits the two-sided estimates. Let $\mu$ be a positive smooth Radon measure in a certain class and consider the perturbed form $\mathcal{E}^{\mu}(u, u) = \mathcal{E}(u, u) - (u, u)_\mu$. Denote by $p^{\mu}(t, x, y)$ the fundamental solution associated with $\mathcal{E}^{\mu}$. In this paper, we establish a necessary and sufficient condition on $\mu$ for $p^{\mu}(t, x, y)$ having the same two-sided estimates as $p(t, x, y)$ up to positive constants.

Let $v, \omega_1, \omega_2$ be weights and let $1<p_1, p_2<\infty$. Suppose that $1/p=1/p_1+1/p_2$ and the couple of weights $(\omega_1, \omega_2)$ satisfies the reverse Hölder's condition. For the multisublinear maximal operator $\mathfrak{M}$ on martingale spaces, we characterize the weights for which $\mathfrak{M}$ is bounded from $L^{p_1}(\omega_1)\times L^{p_2}(\omega_2)$ to $L^{p,\infty}(v)\hbox{ or }L^p(v)$. If $v=\omega_2^{p/p_2}\omega_2^{p/p_2}$, we partially give the bilinear version of one-weight theory.

We provide a construction of limit linear series for families of curves and justify dimension bounds. We show how to extend the sections of a vector bundle to twists by line bundles and to elementary transformations.

Let $G$ be a linear connected complex reductive Lie group. The purpose of this paper is to construct a $G$-equivariant symplectomorphism in terms of local coordinates from a holomorphic twisted cotangent bundle of the generalized flag variety of $G$ onto the semisimple coadjoint orbit of $G$. As an application, one can obtain an explicit embedding of a noncompact real coadjoint orbit into the twisted cotangent bundle.

In this paper we consider a nonlinear parametric Dirichlet problem driven by a nonhomogeneous differential operator (special cases are the $p$-Laplacian and the $(p,q)$-differential operator) and with a reaction which has the combined effects of concave ($(p-1)$-sublinear) and convex ($(p-1)$-superlinear) terms. We do not employ the usual in such cases AR-condition. Using variational methods based on critical point theory, together with truncation and comparison techniques and Morse theory (critical groups), we show that for all small $\lambda > 0$ ($\lambda$ is a parameter), the problem has at least five nontrivial smooth solutions (two positive, two negative and the fifth nodal). We also prove two auxiliary results of independent interest. The first is a strong comparison principle and the second relates Sobolev and Hölder local minimizers for $C^1$ functionals.