Showing papers in "Nagoya Mathematical Journal in 2000"

An analogue of Pitman's 2M-X theorem for exponential Wiener functionals, Part I : A time-inversion approach

Abstract: Let $\{B_t^{(\mu)},t\geqq0\}$ be a one-dimensional Brownian motion with constant drift $\mu\in{\bf R}$ starting from $0$. In this paper we show that $$ Z_t^{(\mu)} = \exp(-B_t^{(\mu)}) \int_0^t \exp(2B_s^{(\mu)}) ds $$ gives rise to a diffusion process and we explain how this result may be considered as an extension of the celebrated Pitman's $2M-X$ theorem. We also derive the infinitesimal generator and some properties of the diffusion process $\{Z_t^{(\mu)},t\geqq0\}$ and, in particular, its relation to the generalized Bessel processes.

Special polynomials and the Hirota Bilinear relations of the second and the fourth Painlevé equations

Abstract: We give a purely algebraic proof that the rational functions $P_n(t),\, Q_n(t)$ inductively defined by the recurrence relation (1), (2) respectively, are polynomials. The proof reveals the Hirota bilinear relations satisfied by the $\tau$-functions.

The joint universality and the functional independence for Lerch zeta-functions

Abstract: The joint universality theorem for Lerch zeta-functions L(λl, αl, s) (1 ≤ l ≤ n) is proved, in the case when λls are rational numbers and αls are transcendental numbers. The case n = 1 was known before ([12]); the rationality of λls is used to establish the theorem for the “joint” case n ≥ 2. As a corollary, the joint functional independence for those functions is shown.

A note on weighted Bergman spaces and the Cesaro operator

Abstract: Let $B$ denote the unit ball in $\mathbb{C}^n$, and $dV(z)$ normalized Lebesgue measure on $B$. For $\alpha > -1$, define $dV^\alpha (z)=(1-|z|^2)^\alpha dV(z)$. Let ${\cal H}(B)$ denote the space of holomorhic functions on $B$, and for $0 < p <\infty$, let ${\mathcal{A}}^p(dV_\alpha)$ denote $L^p(dV_\alpha)\cap {\cal H}(B)$. In this note we characterize ${\mathcal{A}}^p(dV_\alpha)$ as those functions in ${\cal H}(B)$ whose images under the action of a certain set of differential operators lie in $L^p(dV_\alpha)$. This is valid for $1 \le p <\infty$. We also show that the Ces\`aro operator is bounded on ${\mathcal{A}}^p(dV_\alpha)$ for $0

Remarks on the Yablonskii-Vorob'ev polynomials

Abstract: We study the Yablonskii-Vorob'ev polynomial associated with the second Painleve equation. To study other special polynomials (Okamoto polynomials, Umemura polynomials)associated with the Painleve equations, our purely algebraic approach is useful.

Kähler identity on Levi flat manifolds and application to the embedding

Abstract: It is shown that any compact Levi flat manifold admitting a positive line bundle is embeddable into $\mathbb{P}^{n}$ by a CR mapping with an arbitrarily high, though finite, order of regularity.

On the domain and range of the maximal operator

Abstract: 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.

Affine structure on Weil bundles

Abstract: . For every r-th order Weil functor T(A), we introduce the underliyng k-th order Weil functors T(Ak), k=1,...,r-1. We deduce that T(A)M -> T(Ar-1)M is an affine bundle for every manifold M. Generalizing the classical concept of contakt element by C. Ehresmann, we define the bundle of contact elements of type A on M and we describe some affine properties of this bundle.

On the Auslander-Reiten quiver of an infinitesimal group

Abstract: Let ${\cal G}$ be an infinitesimal group scheme, defined over an algebraically closed field of characteristic $p$. We employ rank varieties of ${\cal G}$-modules to study the stable Auslander-Reiten quiver of the distribution algebra of ${\cal G}$. As in case of finite groups, the tree classes of the AR-components are finite or infinite Dynkin diagrams, or Euclidean diagrams. We classify the components of finite and Euclidean type in case ${\cal G}$ is supersolvable or a Frobenius kernel of a smooth, reductive group.

A remark on algebraic surfaces with polyhedral Mori cone

Abstract: We denote by FPMC the class of all non-singular projective algebraic surfaces X over ℂ with finite polyhedral Mori cone NE(X) ⊂ NS(X) ⊗ ℝ. If ρ(X) = rk NS(X) ≥ 3, then the set Exc(X) of all exceptional curves on X ∈ FPMC is finite and generates NE(X). Let δE(X) be the maximum of (-C 2) and pE(X) the maximum of pa(C) respectively for all C ∈ Exc(X). For fixed ρ ≥ 3, δE and pE we denote by FPMCρ,δE,pE the class of all algebraic surfaces X ∈ FPMC such that ρ(X) = ρ, δE(X) = δE and pE(X) = pE . We prove that the class FPMCρ,δE,pE is bounded in the following sense: for any X ∈ FPMCρ,δE,pE there exist an ample effective divisor h and a very ample divisor h′ such that h 2 ≤ N(ρ, δE) and h′2 ≤ N′(ρ, δE, pE) where the constants N(ρ, δE) and N′(ρ, δE, pE) depend only on ρ, δE and ρ, δE, pE respectively. One can consider Theory of surfaces X ∈ FPMC as Algebraic Geometry analog of the Theory of arithmetic reflection groups in hyperbolic spaces.

Nonexistence of real analytic Levi flat hypersurfaces in {${\bf P}\sp 2$}

Abstract: A real hypersurface $M$ in a complex manifold $X$ is said to be Levi flat if it separates $X$ locally into two Stein pieces. It is proved that there exist no real analytic Levi flat hypersurfaces in ${\Bbb P}^2$.

Hooke's law in statistical manifolds and divergences

Abstract: The concept of the canonical divergence is defined for dually flat statistical manifolds in terms of the Legendre transform between dual affine coordinates In this article, we introduce a new two point function defined for any triple $(g, abla, abla^{*})$ of a Riemannian metric $g$ and two affine connections $ abla$ and $ abla^{*}$ We show that this interprets the canonical divergence without refering to the existence of special coordinates (dual affine coordinates) but in terms of only classical mechanics concerning $ abla$- and $ abla^{*}$-geodesics We also discuss the properties of the two point function and show that this shares some important properties with the canonical divergence defined on dually flat statistical manifolds

Conformally flat hypersurfaces in Euclidean 4-space

Abstract: We study generic and conformally flat hypersurfaces in Euclidean four-space. What kind of conformally flat three manifolds are really immersed generically and conformally in Euclidean space as hypersurfaces? According to the theorem due to Cartan [1], there exists an orthogonal curvature-line coordinate system at each point of such hypersurfaces. This fact is the first step of our study. We classify such hypersurfaces in terms of the first fundamental form. In this paper, we consider hypersurfaces with the first fundamental forms of certain specific types. Then, we give a precise representation of the first and the second fundamental forms of such hypersurfaces, and give exact shapes in Euclidean space of them.

Distribution of units of real quadratic number fields

Abstract: Let k be a real quadratic field and k, E the ring of integers and the group of units in k. Denoting by E() the subgroup represented by E of ( k/)× for a prime ideal , we show that prime ideals for which the order of E() is theoretically maximal have a positive density under the Generalized Riemann Hypothesis.

On the Dirichlet problem of prescribed mean curvature equations without $H$-convexity condition

Abstract: The Dirichlet problem of prescribed mean curvature equations is well posed, if the boundery is H-convex In this article we eliminate the H-convexity condition from a portion $\Gamma$ of the boundary and prove the existence theorem, where the boundary condition is satisfied on $\Gamma$ in the weak sense

Twistor theory of manifolds with Grassmannian structures

Abstract: As a generalization of the conformal structure of type $(2, 2)$, we study Grassmannian structures of type $(n, m)$ for $n, m \geq 2$ We develop their twistor theory by considerin the complete integrability of the associated null distributions The integrability corresponds to global solutions of the geometric structuresA Grassmannian structure of type $(n, m)$ on a manifold $M$ is, by definition, an isomorphism from the tangent bundle $TM$ of $M$ to the tensor product $V \otimes W$ of two vector bundles $V$ and $W$ with rank $n$ and $m$ over $M$ respectively Because of the tensor product structure, we have two null plane bundles with fibres $P^{m-1}(\mathbb{R})$ and $P^{n-1}(\mathbb{R})$ over $M$ The tautological distribution is defined on each two bundles by a connection We relate the integrability condition to the half flatness of the Grassmannian structures Tanaka's normal Cartan connections are fully used and the Spencer cohomology groups of graded Lie algebras play a fundamental roleBesides the integrability conditions corr[e]sponding to the twistor theory, the lifting theorems and the reduction theorems are derived We also study twistor diagrams under Weyl connections

Integers free of small prime factors in arithmetic progressions

Abstract: For real x ≥ y ≥ 2 and positive integers a, q, let Φ(x, y; a, q) denote the number of positive integers ≤ x, free of prime factors ≤ y and satisfying n ≡ a (mod q). By the fundamental lemma of sieve, it follows that for (a,q) = 1, Φ(x,y;a,q) = φ(q)-1, Φ(x, y){1 + O(exp(-u(log u- log2 3u- 2))) + (u = log x log y) holds uniformly in a wider ranges of x, y and q. Let χ be any character to the modulus q, and L(s, χ) be the corresponding L-function. Let be a (‘exceptional’) real character to the modulus q for which L(s, ) have a (‘exceptional’) real zero satisfying > 1 - c0/log q. In the paper, we prove that in a slightly short range of q the above first error term can be replaced by where ρ(u) is Dickman function, and ρ′(u) = dρ(u)/du. The result is an analogue of the prime number theorem for arithmetic progressions. From the result can deduce that the above first error term can be omitted, if suppose that 1 < q < (log q)A .

The determination of caloric morphisms on Euclidean domains

Abstract: Let D be a domain in ℝ m+1 and E be a domain in ℝ n+1. A pair of a smooth mapping f : D → E and a smooth positive function ϕ on D is called a caloric morphism if ϕ ˙ u o f is a solution of the heat equation in D whenever u is a solution of the heat equation in E. We give the characterization of caloric morphisms, and then give the determination of caloric morphisms. In the case of m n, under some assumption on f, every caloric morphism is obtained by composing a projection with a direct sum of caloric morphisms of ℝ n+1.

Malgrange's vanishing theorem in 1-concave CR manifolds

Abstract: We prove a vanishing theorem for the -cohomology in top degree on 1-concave CR generic manifolds.

A remark on the theorem of Ohsawa-Takegoshi

Abstract: If D ⊂ ℂ n is a pseudoconvex domain and X ⊂ D a closed analytic subset, the famous theorem B of Cartan-Serre asserts, that the restriction operator r : (D) → (X) mapping each function F to its restriction F|X is surjective. A very important question of modern complex analysis is to ask what happens to this result if certain growth conditions for the holomorphic functions on D and on X are added.

On separable A^1-forms

Abstract: We show that for any field k, separable A 1-forms over commutative k-algebras are trivial.

On the transformation group of the second Painlevé equation

Abstract: We show that for the second Painleve equation $y'' = 2y^{3}+ty+\alpha$, the Backlund transformation group $G$, which is isomorphic to the extended affine Weyl group of type $\hat{A}_{1}$, operates regularly on the natural projectification ${\mathcal X}(c)/\mathbb{C}(c, t)$ of the space of initial conditions, where $c = \alpha-1/2$ ${\cal X}(c)/\mathbb{C}(c, t)$ has a natural model ${\mathcal X}[c]/\mathbb{C}(t)[c]$ The group $G$ does not operate, however, regularly on $\mathcal {X}[c]/\mathbb{C}(t)[c]$ To have a family of projective surfaces over $\mathbb{C}(t)[c]$ on which $G$ operates regularly, we have to blow up the model ${\mathcal X}[c]$ along the projective lines corresponding to the Riccati type solutions

A unitary representation of the basical central extension of a loop group

Abstract: We construct a measure over the string bundle associated to the loop space of a Riemannian manifold. We deduce a representation of a finite energy Kac-Moody group analoguous to the energy representation.

Values of zeta functions and class number 1 criterion for the simplest cubic fields

Abstract: Let K be the simplest cubic field defined by the irreducible polynomial where m is a nonnegative rational integer such that m 2 + 3 m + 9 is square-free. We estimate the value of the Dedekind zeta function ζ K (s ) at s = −1 and get class number 1 criterion for the simplest cubic fields.

Multipliers on vector spaces of holomorphic functions

Abstract: Let G be a domain in the complex plane containing zero and H(G ) be the set of all holomorphic functions on G . In this paper the algebra M(H(G) ) of all coefficient multipliers with respect to the Hadamard product is studied. Central for the investigation is the domain introduced by Arakelyan which is by definition the union of all sets with w ∈ G c . The main result is the description of all isomorphisms between these multipliers algebras. As a consequence one obtains: If two multiplier algebras M(H(G 1 )) and M(H(G 2 ) ) are isomorphic then is equal to Two algebras H(G1 ) and H(G 2 ) are isomorphic with respect to the Hadamard product if and only if G 1 is equal to G 2 . Further the following uniqueness theorem is proved: If G 1 is a domain containing 0 and if M(H(G) ) is isomorphic to H(G1 ) then G 1 is equal to .

On the Steinitz module and capitulation of ideals

Abstract: Let L be a finite extension of a number field K with ring of integers O L and O K respectively. One can consider O L as a projective module over O K . The highest exterior power of O L as an O K module gives an element of the class group of O K , called the Steinitz module. These considerations work also for algebraic curves where we prove that for a finite unramified cover Y of an algebraic curve X, the Steinitz module as an element of the Picard group of X is the sum of the line bundles on X which become trivial when pulled back to Y. We give some examples to show that this kind of result is not true for number fields. We also make some remarks on the capitulation problem for both number field and function fields. (An ideal in O K is said to capitulate in L if its extension to O L is a principal ideal.)

Two-weighted inequalities for the derivatives of holomorphic functions and Carleson measures on the unit ball

Abstract: We characterize those positive measure µ ’s on the higher dimensional unit ball such that “two-weighted inequalities” hold for holomorphic functions and their derivatives. Characterizations are given in terms of the Carleson measure conditions. The results of this paper also distinguish between the fractional and the tangential derivatives.

Hecke structure of spaces of half-integral weight cusp forms

Abstract: We investigate the connection between integral weight and half-integral weight modular forms. Building on results of Ueda [14], we obtain structure theorems for spaces of half-integral weight cusp forms Sk/2 (4N,χ) where k and N are odd nonnegative integers with k ≥ 3, and χ is an even quadratic Dirichlet character modulo 4N. We give complete results in the case where N is a power of a single prime, and partial results in the more general case. Using these structure results, we give a classical reformulation of the representation-theoretic conditions given by Flicker [5] and Waldspurger [17] in results regarding the Shimura correspondence. Our version characterizes, in classical terms, the largest possible image of the Shimura lift given our restrictions on N and χ, by giving conditions under which a newform has an equivalent cusp form in Sk/2(4N, χ). We give examples (computed using tables of Cremona [4]) of newforms which have no equivalent half-integral weight cusp forms for any such N and χ. In addition, we compare our structure results to Ueda’s [14] decompositions of the Kohnen subspace, illustrating more precisely how the Kohnen subspace sits inside the full space of cusp forms.

Tangent loci and certain linear sections of adjoint varieties

Abstract: An adjoint variety X(g )associated to a complex simple Lie algebra is by definition a projective variety in ℙ*(g) obtained as the projectivization of the (unique) non-zero, minimal nilpotent orbit in g. We first describe the tangent loci of X(g ) in terms of triples. Secondly for a graded decomposition of contact type we show that the intersection of X(g ) and the linear subspace ℙ*(g1 ) in ℙ*(g) coincides with the cubic Veronese variety associated to g .

On the deduction of the class field theory from the general reciprocity of power residues

Abstract: We denote by (A) Artin’s reciprocity law for a general abelian extension of a finite degree over an algebraic number field of a finite degree, and denote two special cases of (A) as follows: by (AC) the assertion (A) where K/F is a cyclotomic extension; by (AK) the assertion (A) where K/F is a Kummer extension. We will show that (A) is derived from (AC) and (AK) only by routine, elementarily algebraic arguments provided that n = (K : F) is odd. If n is even, then some more advanced tools like Proposition 2 are necessary. This proposition is a consequence of Hasse’s norm theorem for a quadratic extension of an algebraic number field, but weaker than the latter.