Let Γ be a continuous pseudo group acting on a manifold M. Denote by (M, M’, ρ) a fibered manifold preserved by transformations in Γ. Then any f in Γ locally induces a local transformation f′ of M′. Denote by Γ/ρ the set of all such f′. Then it might seem natural to expect that Γ/ρ is a continuous pseudo group acting on M′. However the matter is not so simple. For instance, take f′ and g′ in Γ/ρ such that the composition f′°g′ can be defined. Then they can be lifted to local transformations f and g belonging to Γ.

Let Λ be the path algebra of a Dynkin quiver. A recent result of Bo Chen asserts that Hom(X; Y/X) = 0 for any Gabriel-Roiter inclusion X ⊆ Y. The aim of the present note is to give an interpretation of this result in terms of Hall polynomials, and to extend it in this way to representation-directed split algebras. We further show its relevance when dealing with arbitrary representation-finite split algebras.

In this paper, coupled systems

of Korteweg-de Vries type are considered, where u = u(x, t), v = v(x, t) are real-valued functions and where x, t∈R. Here, subscripts connote partial differentiation and

are quadratic polynomials in the variables u and v. Attention is given to the pure initial-value problem in which u(x, t) and v(x, t) are both specified at t = 0, namely,

for x ∈ ℝ. Under suitable conditions on P and Q, global well-posedness of this problem is established for initial data in the L2-based Sobolev spaces Hs(ℝ) × Hs(ℝ) for any s > ‒3/4.

In the pioneering work by Dimitrov–Haiden–Katzarkov–Kontsevich, they introduced various categorical analogies from the classical theory of dynamical systems. In particular, they defined the entropy of an endofunctor on a triangulated category with a split generator. In the connection between the categorical theory and the classical theory, a stability condition on a triangulated category plays the role of a measured foliation so that one can measure the “volume” of objects, called the mass, via the stability condition. The aim of this paper is to establish fundamental properties of the growth rate of mass of objects under the mapping by the endofunctor and to clarify the relationship between it and the entropy. We also show that they coincide under a certain condition.

Let P be a principal fibre bundle over M with group G and with projection π : P → M. By definition of a principal fibre bundle, G acts on P on the right. We shall denote this transformation law by ρ

Let Sm(r) be an m-sphere of constant sectional curvature 1/r2 and M an n-dimensional compact minimal submanifold of Sm(r). If Sm(r) is imbedded in Em+1 by its first standard imbedding, then, by a well-known result of Takahashi [11], the Euclidean coordinate functions restricted to M are eigenfunctions of Δ on M with the same eigenvalue n/r2. Moreover, the center of mass of M in Em+1 coincides with the center of the hypersphere Sm(r) in Em+1. Thus, M is mass-symmetric in Sm(r) ⊂ Em+\ Consequently, we see that if one wants to study the spectral geometry of a submanifold of Sm(r), it is natural to immerse Sm(r) by its k-th standard immersion, in particular, by its second standard immersion.

We shall investigate a system of spin configurations S = {S(t, x); t ≥ 0, x ∊ ℤ} on a one-dimensional lattice ℤ changing randomly in time. The evolution law is described by an infinite-dimensional stochastic differential equation (SDE):

where {β(t, x); t > 0, x ∊ Z} is a family of independent standard Wiener processes and U′ is the derivative of a self-potential U: R → R.

A locally projective amalgam is formed by the stabilizer G(x) of a vertex x and the global stabilizer G{x, y} of an edge (containing x) in a group G, acting faithfully and locally finitely on a connected graph Γ of valency 2n - 1 so that (i) the action is 2-arc-transitive; (ii) the subconstituent G(x)Γ(x) is the linear group SLn(2) = Ln(2) in its natural doubly transitive action and (iii) [t, G{x, y}] < O2(G(x) n G{x, y}) for some t G G{x, y} \ G(x). D. Z. Djokovic and G. L. Miller [DM80], used the classical Tutte’s theorem [Tu47], to show that there are seven locally projective amalgams for n = 2. Here we use the most difficult and interesting case of Trofimov’s theorem [Tr01] to extend the classification to the case n > 3. We show that besides two infinite series of locally projective amalgams (embedded into the groups AGLn(2) and O2n+(2)) there are exactly twelve exceptional ones. Some of the exceptional amalgams are embedded into sporadic simple groups M22, M23, Co2, J4 and BM. For each of the exceptional amalgam n = 3, 4 or 5.

Is every isometry, or more generally, every affine transformation of a Kählerian manifold a complex analytic transformation? The answer is certainly negative in the case of a complex Euclidean space. This question has been recently studied by Lichnerowicz [8] and Schouten-Yano [11] from the infinitesimal point of view; they have found some conditions in order that every infinitesimal motion of a Kählerian manifold preserve the complex structure. (As a matter of fact, [11] has dealt with the case of a pseudo-Kählerian manifold, which does not differ essentially from a Kählerian manifold as far as the question at hand is concerned.)

In the present paper, we generalize their results by a different approach. In order to explain our main idea, we shall first give a few definitions (1 and 2) and state our main results (3). The proofs are given in the subsequent sections.

We characterize the adjoint G2 orbits in the Lie algebra g of G2 as fibered spaces over S6 with fibers given by the complex Cartan hypersurfaces. This combines the isoparametric hypersurfaces of case (g,m) = (6,2) with case (3,2). The fibrations on two singular orbits turn out to be diffeomorphic to the twistor fibrations of S6 and G2/SO(4). From the symplectic point of view, we show that there exists a 2-parameter family of Lagrangian submanifolds on every orbit.

We construct a Glauber dynamics on {0, 1}ℛ, ℛ a discrete space, with infinite range flip rates, for which a fermion point process is reversible. We also discuss the ergodicity of the corresponding Markov process and the log-Sobolev inequality.

P. Lévy introduced a generalized notion of Brownian motion in his monograph “Processus stochastiques et mouvement brownien” by taking the time parameter space to be a general metric space. Let (M, d) be a metric space and let O be a fixed point of M called the origin. Following his definition, a Brownian motion parametrized with the metric space (M, d) is a Gaussian system ℬ = {B(m); m ∈ M} such that the difference B(m) − B(m′) is a random variable with mean zero and variance d(m, m′), and that B(O) = 0.

Recently, A. Morimoto [5] proved that every isometry of a compact Riemannian manifold of positive dimension has not the pseudo-orbit tracing property, and that if a homeomorphism of a compact metric space has the pseudo-orbit tracing property then Eφ— 0φ(see § 1 for definition). The purpose of this paper is to show that every distal homeomorphism of a compact connected metric space has not the pseudo-orbit tracing property.

In this article, we give a new construction of a Kähler-Einstein metric on a smooth projective variety with ample canonical bundle. As a consequence, for a dominant projective morphism f: X → S with connected fibers such that a general fiber has an ample canonical bundle, and for a positive integer m, we construct a canonical singular Hermitian metric hE,m on with semipositive curvature in the sense of Nakano.

We discuss the Monge problem for under-determined systems of ordinary differential equations with an arbitrary degree of freedom and give a sufficient condition, in terms of truncated multi-flag systems, for the Monge property to hold. This condition extends in a natural way the Cartan criterion valid for systems with one degree of freedom.

Consider a relatively compact region Ω of a Riemann surface R. The term Dirichlet integral over Ω, DΩ(·), is used for the variation whose Euler-Lagrange equation is Δu = 0 on Ω and the term energy integral over Ω, , is used for the variation with Euler-Lagrange equation

The Enriques-Fano classification ([E.F], [F]) of the maximal connected algebraic subgroups of the three variable Cremona group, despite of its group theoretic feature, seems to be the most significant result on the rational threefolds so far known. In this paper as in [MU] we interpret the Enriques-Fano classification from a geometric view point, namely the geometry of minimal rational threefolds. We explained in [MU] the link between the two objects; the maximal algebraic subgroups and the minimal rational threefolds. Let (G, X) be a maximal algebraic subgroup of three variable Cremona group. We denote by ℓ(G, X) the set of all the algebraic operations (G, Y) such that Y is non-singular and projective and such that (G, Y) is isomorphic to (G, X) as law chunks of algebraic operation: namely (G, Y) is birationally equivalent to (G, X). Then we define an order in ℓ(G, X): for (G, Z), (G, W) ∊ ℓ(G, X), (G, Z)>(G, W) if there exists an G-equivariant birational morphism of Z onto W.

We show that the Hilbert-Kunz multiplicities of the reductions to positive characteristics of an irreducible projective curve in characteristic 0 have a well-defined limit as the characteristic tends to infinity.

We give a detailed survey, known and new results on the domain and the range of the maximal operator. In particular we employ the grand Lp spaces and logarithmic Lebesgue spaces.

E. S. Barnes and I recently constructed a series of positive quadratic forms fN in N = 2n variables (n = 1, 2,…) with relative minima of order for large N. I continue this investigation by determining the minimal vectors of fN and showing that, for its group of automorphs is the Clifford group This suggests a generalization. Replacing by where p is an odd prime, I derive a new series of positive forms in N =(p−1)pn variables (§4). The relative minima are again of order (p fixed, N → ∞ ), the “best” forms being those for p = 3, 5. All forms are eutactic though only those for p = 3,5 are extreme.