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.

Let S = K[x1,x2,…,xn] be a polynomial ring over a field K. Let Δ be a simplicial complex whose vertex set is contained in {1, 2,…,n}. For an integer k ≥ 0, we investigate the k-Buchsbaum property of residue class rings S/I(t); and S/It for the Stanley-Reisner ideal I = IΔ. We characterize the k-Buchsbaumness of such rings in terms of the simplicial complex Δ and the power t. We also give a characterization in the case where I is the edge ideal of a simple graph.

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.

Let G be a Lie group, and let M be a smooth proper G-manifold. Let M/G denote the orbit space, and let π : M → M/G be the natural map. It is known that M/G is homeomorphic to a polyhedron. In the present paper we show that there exist a piecewise linear (PL) manifold P, a polyhedron L, and homeomorphisms τ : P → M and σ : M/G → L such that σ o π o τ is PL. This is an application of the theory of subanalytic sets and subanalytic maps of Shiota. If M and the G-action are, moreover, subanalytic, then we can choose τ and σ subanalytic and P and L unique up to PL homeomorphisms.

The complement of an arrangement of hyperplanes is a good example of the mixed Hodge structure on the fundamental group of an algebraic variety. We compute its isomorphic class using iterated integrals in the fiber type case and then get the combinatorial and projective invariant.

We introduce a framework for a nonlinear potential theory without a kernel on a reflexive, strictly convex and smooth Banach space of functions. Nonlinear potentials are defined as images of nonnegative continuous linear functionals on that space under the duality mapping. We study potentials and reduced functions by using a variant of the Gauss-Frostman quadratic functional. The framework allows a development of other main concepts of nonlinear potential theory such as capacities, equilibrium potentials and measures of finite energy.

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.

It is proved that the pseudodifferential operators σt(X, D) belong to the Schatten p-class Cp, 0 < p ≤ 2, if the symbol σ(x,ω) is in certain modulation spaces on

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)

In der Arbeit [2] hat L. Rédei den Begriff des einstufig nichtkommutativen Ringes eingefiihrt, und die alien möglichen Typen der einstufig nichtkommutativen endlichen Ringe aufgezàhlt. Sie bestehen aus drei Grundtypen und ihren homomorphen Bildern. In der oben genannten Arbeit hat Rédei einstufig nichtkommutativen unendlichen Ringes ganz kurz erwähnt, und nur darauf hingewiesen, dass man nicht einmal weiss, ob solche Ringe überhaupt existieren. In der vorliegenden Arbeit handelt es sich um die Nichtexistenz der einstufig nichtkommutativen unendlichen Ringe. Obwohl die Nichtexistenz der allge-meinen unendlichen Ringe vom genannten Charakter hoch glaubwürdig scheint, kann ich doch zur Zeit die Aufgabe in ihrer vollen Allgemeinheit nicht losen, sondem ich kann nur zeigen, dass ein einstufig nichtkommutativer Ring mit einer zusätzlichen Bedingung notwendigerweise endlich wird. Im folgenden soil nämlich der untenbenannte Satz bewiesen werden.