We shall discuss in this paper some problems in non-linear prediction theory. An Ornstein-Uhlenbeck process {U(t)} is taken to be a basic process, and we shall deal with stochastic processes X(t) that are transformed by functions f satisfying certain condition. Actually, observed processes are expressed in the form X(t) = f(U(t)). Our main problem is to obtain the best non-linear predictor X̂(t, τ) for X(t + τ), τ > 0, assuming that X(s), s ≤t, are observed. The predictor is therefore a non-linear functional of the values X(s), s ≤ t.

In the following, for any group G and any g ∈ G, Ad[g] stands for the inner automorphism defined by Ad[g] (x) = gxg−1 for any x ∈ G.

The famous Picard theorem states that a holomorphic mapping f: C → P1(C) omitting distinct three points must be constant. Borel [1] showed that a non-degenerate holomorphic curve can miss at most n + 1 hyperplanes in Pn(C) in general position, thus extending Picard’s theorem (n = 1). Recently, Fujimoto [3], Green [4] and [5] obtained many Picard type theorems using Borel’s methods for holomorphic mappings.

In a previous article, we proved that Shimura curves have no points rational over number fields under a certain assumption. In this article, we give another criterion of the nonexistence of rational points on Shimura curves and obtain new counterexamples to the Hasse principle for Shimura curves. We also prove that such counterexamples obtained from the above results are accounted for by the Manin obstruction.

Let be a transitive group of degree n, and let be the stabilizer of a symbol in ®. Then we owe to J.S. Frame the following remarkable relations between the lengths ni of the orbits of and the degrees fi of the absolutely irreducible components of the permutation matrix representation * of :

(A) If the irreducible constituents of * are all different, then the rational number

is an integer, where k is the number of the orbits of

We are concerned with representation of positive definite quadratic forms by a positive definite quadratic form. Let us consider the following assertion

In this paper, we perform a detailed spectral study of the liberation process associated with two symmetries of arbitrary ranks: $(R,S)\mapsto (R,U_{t}SU_{t}^{\ast })_{t\geqslant 0}$, where $(U_{t})_{t\geqslant 0}$ is a free unitary Brownian motion freely independent from $\{R,S\}$. Our main tool is free stochastic calculus which allows to derive a partial differential equation (PDE) for the Herglotz transform of the unitary process defined by $Y_{t}:=RU_{t}SU_{t}^{\ast }$. It turns out that this is exactly the PDE governing the flow of an analytic function transform of the spectral measure of the operator $X_{t}:=PU_{t}QU_{t}^{\ast }P$ where $P,Q$ are the orthogonal projections associated to $R,S$. Next, we relate the two spectral measures of $RU_{t}SU_{t}^{\ast }$ and of $PU_{t}QU_{t}^{\ast }P$ via their moment sequences and use this relationship to develop a theory of subordination for the boundary values of the Herglotz transform. In particular, we explicitly compute the subordinate function and extend its inverse continuously to the unit circle. As an application, we prove the identity $i^{\ast }(\mathbb{C}P+\mathbb{C}(I-P);\mathbb{C}Q+\mathbb{C}(I-Q))=-\unicode[STIX]{x1D712}_{\text{orb}}(P,Q)$.

Let ⌈ be a subgroup of the modular group PSL(2,Z) then ⌈ acts on the upper half plane H = {zЄC : Im z > 0} and we can form the Riemann surface M = H/⌈, see [3]. The complex line bundles on a Riemann surface M form a group H1(M, *), see [4], and whenever we raise a line bundle to a power it will be in this group. Let κ denote the canonical bundle on M then a modular form of weight ν is a section of the bundle . A modular n-jet is then a section of Jn the n-th jet bundle, see [7]. We can reformulate these ideas in the following terms. A modular form can be viewed as a function Φ: H → C and a modular n-jet as a vector valued function Φ: H → Cn+1 both of which satisfy a transformation law under the elements of Γ.

One can characterize Siegel cusp forms among Siegel modular forms by growth properties of their Fourier coefficients. We give a new proof, which works also for more general types of modular forms. Our main tool is to study the behavior of a modular form for $Z=X+iY$ when $Y\longrightarrow 0$.

We consider the continuous valuation of logic, where the certainty of a statement is measured with a number in the closed unit interval I =[0, 1].

The idea originates in continuous logics, which have been investigated from various standpoints, in [1] ~ [4] and [8] ~ [10], for example. More comprehensive information can be seen in [4] and others.

For a character of the absolute Galois group of a complete discrete valuation field, we define a lifting of the refined Swan conductor, using higher dimensional class field theory.

The many unanswerable questions (1) which arise in the study of finite groups have lead to a review of fundamental ideas, e.g. the Theorem of Burnside (3, p. 299; 2, 6) that if λ be any faithful irreducible representation of G over a field K, then every irreducible representation of G over K is contained in some tensor power of λ.

Let $S$ be a smooth minimal complex surface of general type with $p_{g}=0$ and $K^{2}=7$. We prove that any involution on $S$ is in the center of the automorphism group of $S$. As an application, we show that the automorphism group of an Inoue surface with $K^{2}=7$ is isomorphic to $\mathbb{Z}_{2}^{2}$ or $\mathbb{Z}_{2}\times \mathbb{Z}_{4}$. We construct a $2$-dimensional family of Inoue surfaces with automorphism groups isomorphic to $\mathbb{Z}_{2}\times \mathbb{Z}_{4}$.

Let A be a real analytic set, M be a compact real analytic manifold and f : A × M → R be a real analytic function. Then we have a family of real analytic functions fa, a ∈ A, on M defined by fa(X) = f(a, x).

In this paper we discuss the discrete spectrum of the Schrödinger operator HNZ(b), defined as below, for an atomic system in a magnetic field. Let where xj is a point in R3 (1 ≥ j ≥ N), and ∇j be the gradient in R3 with respect to xj (1 ≥ j ≥ N). Then we consider the following operator:

(1.1)

defined on , where 3 being real-valued and

(1.2)

For a smooth variety proper over a curve having a fibre with isolated ordinary quadratic singularities, it is well-known that we have the vanishing cycles associated to the singularities in the étale cohomology of the geometric generic fibre. The base-change by a double cover of the base curve ramified at the image of the singular fibre has singularities corresponding to the singularities in the fibre. In this note, we show that in the even relative-dimensional case, there exist elements of the bivariant Chow group of the base-change with supports in the singularities and hence their images in the bivariant Chow group with supports in the special fibre and that the usual cohomological vanishing cycles are obtained as their images by a natural map, a kind of “cycle map” so that the elements in the bivariant Chow groups can be regarded as the vanishing cycles. The bivariant Chow group with supports in the special fibre has a ring structure and the natural map is a ring homomorphism to the cohomology ring of the geometric generic fibre. Also discussed is the relation of the bivariant Chow group with supports in the special fibre to the specialization map of Chow groups.

We prove that every symplectic filling of the link of a rational surface singularity with reduced fundamental cycle admits a rational compactification, possibly after a modification of the filling in a collar neighbourhood of the link.

Let Γ be a discrete cocompact subgroup of SL2(ℂ). We conjecture that the quotient manifold X = SL2(ℂ) / Γ contains infinitely many non-isogenous elliptic curves and prove this is indeed the case if Schanuel’s conjecture holds. We also prove it in the special case where Γ ∩ SL2(∝) is cocompact in SL2(ℝ).

Furthermore, we deduce some consequences for the geodesic length spectra of real hyperbolic 2- and 3-folds.

Bounds for the maximum degree of a minimal Gröbner basis of simplicial toric ideals with respect to the reverse lexicographic order are given. These bounds are close to the bound stated in Eisenbud-Goto’s Conjecture on the Castelnuovo-Mumford regularity.