In this paper, an inequality of Simons type for $p$-Yang-Mills fields is established over compact Riemannian manifolds, and then, the energy gaps are obtained.

We consider a group presented as $G(p,q) = \langle x, y|x^p = y^q\rangle$, with integers $p$ and $q$ satisfying $2 \leq p \leq q$. The group is an amalgamated free product of two infinite cyclic groups and is geometrically realized as the fundamental group of a Seifert fiber space over the 2-dimensional disk with two cone points whose associated cone angles are $\frac{2\pi}{p}$ and $\frac{2\pi}{q}$. We present a formula for the spherical growth series of the group $G(p,q)$ with respect to the generating set $\{x,y,x^{-1}, y^{-1}\}$, from which a rational function expression for the spherical growth series of $G(p,q)$ is derived concretely, once $p$ and $q$ are given. In fact, an elementary computer program constructed from the formula yields an explicit form of a single rational fraction expression for the spherical growth series of $G(p,q)$. Such expressions for several pairs $(p,q)$ appear in this paper. In 1999, C. P. Gill already provided a similar formula for the same group. The formula given here takes a different form from his formula, because the method we used here is independent of that introduced by him.

By examining reciprocal relations of basic well-poised, quadratic and cubic series, we establish $q$-analogues of three infinite series for $1/\pi^2$ due to Guillera (2003) and $\lambda$-parameter extensions of three infinite series for $1/\pi$ due to Ramanujan (1914). Several further infinite series identities of Ramanujan-type are also derived as consequences.

Let $f$ be an entire function with the form $f(z)=P(e^z)/e^z$, where $P$ is a polynomial with $\deg(P)\geq2$ and $P(0)\neq 0$. We prove that the area of the complement of the fast escaping set (hence the Fatou set) of $f$ in a horizontal strip of width $2\pi$ is finite. In particular, the corresponding result can be applied to the sine family $\alpha\sin(z+\beta)$, where $\alpha\neq 0$ and $\beta\in\mathbf{C}$.

It is known that the Bers map, induced by Schwarzian derivative differential operator, is a holomorphic split submersion in Weil-Petersson Teichmüller space. We prove that the higher Bers maps which induced by some higher Schwarzian differential operators in Weil-Petersson Teichmüller space are holomorphic and its differentials at the origin are bounded and surjective.

We formulate a generalization of Vojta's conjecture in terms of log pairs and variants of multiplier ideals.

The aim of this paper is to study the derived category with respect to Gorenstein AC-projective modules. We characterize the bounded Gorenstein AC derived category and obtain some triangle equivalences. We also establish a right recollement related with Gorenstein AC derived category.

Let $S$ be a closed connected oriented surface of genus $g \gt 0$. We study a Poisson subalgebra $W_1(g)$ of $C^{\infty}(\mathrm{Hom}(\pi_1(S), \mathrm{GL}(1, \mathbf{R}))/\mathrm{GL}(1, \mathbf{R}))$, the smooth functions on the moduli space of flat $\mathrm{GL}(1, \mathbf{R})$-bundles over $S$. There is a surjective Lie algebra homomorphism from the Goldman Lie algebra onto $W_1(g)$. We classify all cobrackets on $W_1(g)$ up to coboundary, that is, we compute $H^1(W_1(g), W_1(g) \wedge W_1(g)) \cong \mathrm{Hom}(\mathbf{Z}^{2g}, \mathbf{R})$. As a result, there is no cohomology class corresponding to the Turaev cobracket on $W_1(g)$.

We consider Łojasiewicz inequalities for a non-degenerate holomorphic function with an isolated singularity at the origin. We give an explicit estimation of the Łojasiewicz exponent in a slightly weaker form than the assertion in Fukui [10]. We also introduce Łojasiewicz inequality for strongly non-degenerate mixed functions and generalize this estimation for mixed functions.

We introduce a Zariskian analogue of the theory of Huber's adic spaces.