In 1947, Lehmer conjectured that the Ramanujan $\tau$-function $\tau (m)$ never vanishes for all positive integers $m$, where $\tau (m)$ are the Fourier coefficients of the cusp form $\Delta_24$ of weight $12$. Lehmer verified the conjecture in 1947 for $m < 214928639999$. In 1973, Serre verified up to $m < 10^15$, and in 1999, Jordan and Kelly for $m < 22689242781695999$.

The theory of spherical $t$-design, and in particular those which are the shells of Euclidean lattices, is closely related to the theory of modular forms, as first shown by Venkov in 1984. In particular, Ramanujan's $\tau$-function gives the coefficients of a weighted theta series of the $E_8$-lattice. It is shown, by Venkov, de la Harpe, and Pache, that $\tau (m) = 0$ is equivalent to the fact that the shell of norm $2m$ of the $E_8$-lattice is an 8-design. So, Lehmer's conjecture is reformulated in terms of spherical $t$-design.

Lehmer's conjecture is difficult to prove, and still remains open. In this paper, we consider toy models of Lehmer's conjecture. Namely, we show that the $m$-th Fourier coefficient of the weighted theta series of the $\mathbf{Z}^2$-lattice and the $A_2$-lattice does not vanish, when the shell of norm $m$ of those lattices is not the empty set. In other words, the spherical 5 (resp. 7)-design does not exist among the shells in the $\mathbf{Z}^2$-lattice (resp. $A_2$-lattice).

Our aim in this paper is to deal with boundedness of fractional integral operators on Morrey spaces L^(1,φ)(G) and the Sobolev embeddings for generalized Riesz potentials. Target spaces are Orlicz-Morrey, Orlicz-Campanato, and generalized Hölder spaces.

In this paper, we study the invariant subspace and reducing subspace of the weighted Bergman space over bidisk. The minimal reducing subspace of Toeplitz operator $$T_{z^N}=T_{z_1^N{z}_2^N}$$ is completely described, and Beurling-type theorem of some invariant subspace of the weighted Bergman space over bidisk is also obtained.

We establish a necessary and sufficient condition for spectral bounds of a non-local Feynman-Kac semigroup being L^p-independent. This result is an extension of that in [24] to more general symmetric Markov processes; in [24], we only treated a symmetric stable process on R^d. For example, we consider a symmetric stable process on the hyperbolic space, the jump process generated by the fractional power of the Laplace-Beltrami operator, and prove that by adding a non-local potential, the associated Feynman-Kac semigroup satisfies the L^p-independence.

Recently we discovered a new geometry on submanifolds in hyperbolic n-space which is called horospherical geometry. Unfortunately this geometry is not invariant under the hyperbolic motions (it is invariant under the canonical action of SO(n)), but it has quite interesting features. For example, the flatness in this geometry is a hyperbolic invariant and the total curvatures are topological invariants. In this paper, we investigate the horospherical flat surfaces (flat surfaces in the sense of horospherical geometry) in hyperbolic 3-space. Especially, we give a generic classification of singularities of such surfaces. As a consequence, we can say that such a class of surfaces has quite a rich geometric structure.

We show that a Hermitian algebraic curvature model satisfies the Gray identity if and only if it is geometrically realizable by a Hermitian manifold. Furthermore, such a curvature model can in fact be realized by a Hermitian manifold of constant scalar curvature and constant $\star$-scalar curvature which satisfies the Kaehler condition at the point in question.

We give a derivation of tagged particle processes from unlabeled interacting Brownian motions. We give a criteria of the non-explosion property of tagged particle processes. We prove the quasi-regularity of Dirichlet forms describing the environment seen from the tagged particle, which were used in previous papers to prove the invariance principle of tagged particles of interacting Brownian motions.

In this paper we study a certain class of Hermitian n-manifolds whose geodesic flows admit n first integrals of certain kind. It is a generalization of Kähler-Liouville manifold in [3] and called Hermite-Liouville manifold. We completely determine the local structure of Hermite-Liouville manifolds “of type (A)”, and construct global examples over the complex projective space.

We show that the space of all Lelek fans in a Cantor fan, equipped with the Hausdorff metric, is homeomorphic to the separable Hilbert space. This result is a special case of a general theorem we prove about spaces of upper semicontinuous functions on compact metric spaces that are strongly discontinuous.

In this paper we consider a certain order-like relation for self-adjoint operators on a Hilbert space. This relation is defined by using the Jensen inequality. We will show that under some assumptions this relation is antisymmetric.

Any 7-colorable 1-knot has a 7-colored diagram such that exactly four colors of seven are assigned to the arcs of the diagram.

We analyze the expansions in terms of the approximate roots of a Weierstrass polynomial f ∈ C {x}[y], defining a plane branch (C,0), in the light of the toric embedded resolution of the branch. This leads to the definition of a class of (non-equisingular) deformations of a plane branch (C,0) supported on certain monomials in the approximate roots of f, which are essential in the study of Harnack smoothings of real plane branches by Risler and the author. Our results provide also a geometrical approach to Abhyankar's irreducibility criterion for power series in two variables and also a criterion to determine if a family of plane curves is equisingular to a plane branch.

The scaling limits for d-dimensional random walks perturbed by an attractive force toward the origin are studied under the critical situation that the rate functional of the corresponding large deviation principle admits two minimizers. Our results extend those obtained by [2] from the mean-zero Gaussian to non-Gaussian setting under the absence of the wall.