This paper contains a new proof of the classification of prime order elements of $\Bir(\P^2)$ up to conjugation. The first results on this topic can be traced back to classic works by Bertini and Kantor, among others. The innovation introduced by this paper consists of explicit geometric constructions of these Cremona transformations and the parameterization of their conjugacy classes. The methods employed here are inspired to [4], and rely on the reduction of the problem to classifying prime order automorphisms of rational surfaces. This classification is completed by combining equivariant Mori theory to the analysis of the action on anticanonical rings, which leads to characterize the cases that occur by explicit equations (see [28] for a different approach). Analogous constructions in higher dimensions are also discussed.

Okamoto has obtained a sequence of $\tau$-functions for the $P_{VI}$ system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be re-expressed as multi-dimensional integrals, and these in turn can be identified with averages over the eigenvalue probability density function for the Jacobi unitary ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these averages, which depend on four continuous parameters and the discrete parameter $N$, can be characterised as the solution of the second order second degree equation satisfied by the Hamiltonian in the $P_{VI}$ theory. We show that the Hamiltonian also satisfies an equation related to the discrete $P_{V}$ equation, thus providing an alternative characterisation in terms of a difference equation. In the case of the cJUE, the spectrum singularity scaled limit is considered, and the evaluation of a certain four parameter average is given in terms of the general $P_{V}$ e transcendent in $\sigma$ form. Applications are given to the evaluation of the spacing distribution for the circular unitary ensemble (CUE) and its scaled counterpart, giving formulas more succinct than those known previously; to expressions for the hard edge gap probability in the scaled Laguerre orthogonal ensemble (LOE) (parameter $a$ a non-negative integer) and Laguerre symplectic ensemble (LSE) (parameter $a$ an even non-negative integer) as finite dimensional combinatorial integrals over the symplectic and orthogonal groups respectively; to the evaluation of the cumulative distribution function for the last passage time in certain models of directed percolation; to the $\tau$-function evaluation of the largest eigenvalue in the finite LOE and LSE with parameter $a = 0$; and to the characterisation of the diagonal-diagonal spin-spin correlation in the two-dimensional Ising model.

We consider the behavior of the total energy for the wave equation with the dissipative term. When the dissipative term works well uniformly in every direction, several authors obtain uniform decay estimates of the total energy. On the other hand, if the dissipative term is small enough uniformly in every direction, it is known that there exists a solution whose total energy does not decay. We examine the case that the dissipative term vanishes only in a neighborhood of a half-line. We introduce a uniform decay property, which is a natural generalization of the uniform decay estimates, and show that this property does not hold in our case. We prove this by constructing asymptotic solutions supported in the place where the dissipative term vanishes.

In this paper, we consider the Cauchy problem for some semilinear heat equations with time-dependent external forces. Both the external force and the initial data are assumed to be small in some Morrey spaces. We first prove the unique existence of a small time-global solution. We next show the stability of that solution by proving the time-global sovability of perturbation problems.

We study Toeplitz operators on the harmonic Bergman spaces on bounded smooth domains. Two classes of symbols are considered; one is the class of positive symbols and the other is the class of uniformly continuous symbols. For positive symbols, boundedness, compactness, and membership in the Schatten classes are characterized. For uniformly continuous symbols, the essential spectra are described.

We classify the periodic monodromies which are realized as the monodromies of hyperelliptic families.

Let $E$ be a principal $G$--bundle over a smooth projective curve over an algebraically closed field $k$, where $G$ is a reductive linear algebraic group over $k$. We construct a canonical reduction of $E$. The uniqueness of canonical reduction is proved under the assumption that the characteristic of $k$ is zero. Under a mild assumption on the characteristic, the uniqueness is also proved when the characteristic of $k$ is positive.