Ideal Theorem Research Articles

HITCHIN'S CONNECTION AND DIFFERENTIAL OPERATORS WITH VALUES IN THE DETERMINANT BUNDLE

For a family of smooth curves, we have the associated family of moduli spaces of stable bundles with fixed determinant on the curves. There exists a so-called theta line bundle on the family of moduli spaces. When the Kodaira-Spencer map of the family of curves is an isomorphism, we prove in this paper an identification theorem between sheaves of differential operators on the theta line bundle and higher direct images of vector bundles on curves. As an application, the so-called Hitchin connection on the direct image of (powers of) the theta line bundle is derived naturally from the identification theorem. A logarithmic extension to certain singular stable curves is also presented in this paper.

A GENERIC IDENTIFICATION THEOREM FOR GROUPS OF FINITE MORLEY RANK

The paper contains a final identification theorem for the ‘generic’ K*-groups of finite Morley rank.

A signalizer functor theorem for groups of finite Morley rank

There is a longstanding conjecture, due to Gregory Cherlin and Boris Zilber, that all simple groups of finite Morley rank are simple algebraic groups. Towards this end, the development of the theory of groups of finite Morley rank has achieved a good theory of Sylow 2-subgroups. It is now common practice to divide the Cherlin–Zilber conjecture into different cases depending on the nature of the connected component of the Sylow 2-subgroup, known as the Sylow◦ 2-subgroup. We shall be working with groups whose Sylow◦ 2-subgroup is divisible, or odd type groups. To date, the main theorem in the area of odd type groups is Borovik’s trichotomy theorem. The “trichotomy” here is a case division of the minimal counterexamples within odd type. More technically, Borovik’s result represents a major success at transferring signalizer functors and their applications from finite group theory to the finite Morley rank setting. The major difference between the two settings is the absence of a solvable signalizer functor theorem. This forced Borovik to work only with nilpotent signalizer functors, and the trichotomy theorem ends up depending on the assumption of tameness to assure that the necessary signalizer functors are nilpotent. The present paper shows that one may obtain a connected nilpotent signalizer functor from any sufficiently non-trivial solvable signalizer functor. This result plugs seamlessly into Borovik’s work to eliminate the assumption of tameness from his trichotomy theorem. In the meantime, a new approach to the trichotomy theorem has been developed by Borovik [7], based on the “generic identification theorem” of Berkman and Borovik [5]. Borovik uses his original signalizer functor arguments, and incorporates the result of the present paper. The paper is organized as follows. The first section will develop a limited characteristic zero notion of unipotence to complement the usual p-unipotence theory. The section on centralizers and generation which follows will establish some background needed in the rest of the paper. In Section 4 we prove our main result on signalizer functors, and in Section 5 we discuss some applications. With Borovik’s kind permission, we include a proof of the nilpotent signalizer functor theorem as an appendix. The results of Section 3 are based in part on a section of an unpublished version of [3].

DUAL GEOMETRY OF THE WIGNER–YANASE–DYSON INFORMATION CONTENT

We develop a non-parametric information geometry on finite-dimensional matrix manifolds by using the Fréchet differentiation. Taking the simplest prototype Riemannian metric form [Formula: see text], [Formula: see text] with the Fréchet derivative D on a pair of smooth functions g(ρ) and g*(ρ) of the density matrix ρ, we prove the WYD identification theorem: this metric is identified with the normalized Wigner–Yanase–Dyson skew information (the WYD metric), if and only if the metric form satisfies the monotonicity under every stochastic mapping T on ρ and A: [Formula: see text]. On this basis, we establish (a) a fine structure of the partial order in the set ℱ of all monotone metrics such that ℱ power ∪ℱ WYD forms a linearly ordered subset of ℱ with the same mini-max bound, where ℱ power (the power-mean metrics) interpolates the Bures and the WYD metrics (b) a characterization of the quasi-entropy S (ρ, σ)= Tr F (Δσ, ρ)ρ induced by the metric-characterizing function f WYD (x) of the WYD metric (c) an affine connection on the above metric which is torsionless to guarantee the quantum version of the ±α-connection, provided α∈[-3, 3].

An identification theorem for groups of finite Morley rank and even type

The paper contains a construction of a definable BN-pair in a simple group of finite Morley rank and even type with a sufficiently good system of 2-local parabolic subgroups. This provides ‘the final identification theorem’ for simple groups of finite Morley rank and even type.

Noncritical holomorphic functions on Stein manifolds

We prove that every Stein manifold X of dimension n admits [(n+1)/2] holomorphic functions with pointwise independent differentials, and this number is maximal for every n. In particular, X admits a holomorphic function without critical points; this extends a result of Gunning and Narasimhan from 1967 who constructed such functions on open Riemann surfaces. Furthermore, every surjective complex vector bundle map from the tangent bundle TX onto the trivial bundle of rank q < n=dim X is homotopic to the differential of a holomorphic submersion of X to C^q. It follows that every complex subbundle E in the tangent bundle TX with trivial quotient bundle TX/E is homotopic to the tangent bundle of a holomorphic foliation of X. If X is parallelizable, it admits a submersion to C^{n-1} and nonsingular holomorphic foliations of any dimension; the question whether such X also admits a submersion (=immersion) in C^n remains open. Our proof involves a blend of techniques (holomorphic automorphisms of Euclidean spaces, solvability of the di-bar equation with uniform estimates, Thom's jet transversality theorem, Gromov's convex integration method). A result of possible independent interest is a lemma on compositional splitting of biholomorphic mappings close to the identity (Theorem 4.1).

Aspherical pro--groups

The notion of an aspherical pro--group is introduced. It is proved that if a group is aspherical, where is a free pro--group, then the relation -module satisfies an assertion of the type of Lyndon's identity theorem. The finite subgroups and the centre of are described. The structure of an aspherical pro--group with a soluble normal subgroup is studied. In particular, if , then contains a subgroup of finite index of the form where is a free pro--group.

A generalization to the q-convex case of a theorem of Fornæss and Narasimhan

Fornaess and Narasimhan proved (in [8, Thm. 5.3.1]) that, for any complex space X, the identity WPSH(X) = PSH(X) holds, where WPSH(X) denotes the weakly plurisubharmonic functions on X and PSH(X) denotes, as usual, the plurisubharmonic functions on X. When X has no singularities, this identity is clear. For the singular case, however, the inclusion WPSH(X) ⊆ PSH(X) is no longer trivial; one must find locally a plurisubharmonic extension to the ambient space of an embedding of X. In this paper we give another proof for this identity (Theorem 3.3). It is shorter and easier and has the advantage that it can be generalized to q-plurisubharmonic functions (Theorem 4.16). However it has the disadvantage that it works only for continuous functions. The q-plurisubharmonic functions were introduced by Hunt and Murray in [10] (see also [9]), but we will call here q-plurisubharmonic what they call (q − 1)-plurisubharmonic. We also obtain a generalization of a theorem of Siu [16]; namely, we show (Lemma 4.18) that every q-complete subspace with corners of a complex spaceX admits a neighborhood in X that is q-complete with corners. This will be needed in the proof of our main result. The results and proofs of this paper have been announced in [13]. This paper is part of the author’s doctoral thesis written in Wuppertal. I thank Prof. M. Colţoiu and Prof. K. Diederich for many helpful discussions during the whole time of preparing my thesis. I thank the Department of Mathematics of the University of Wuppertal for providing me a nice working atmosphere.

A PWL circuit approach to the definition of a ε-approximation model of scalar static hysteresis

A piecewise-linear (PWL) circuit model of hysteresis phenomena is presented. The proposed model involves the use of parallel-connected simple PWL circuits. The PWL characteristic of the model allows a direct determination of the condition on a rate-independent hysteresis behavior. The basic features of the model are discussed through proper demonstrations. In particular, it is shown that the parameters of the PWL circuit model can be set by using an identification theorem that ensures an arbitrary precision (/spl epsiv/-approximation). The theorem results from the generalization of analogous theorems for single-valued functions to many-valued functions. Finally, both numerical comparisons with other models and experimental fittings are presented to verify the robustness of the /spl epsiv/-approximation theorem to finite-precision arithmetic.

Dynamics of a self-gravitating lightlike matter shell: A gauge-invariant Lagrangian and Hamiltonian description

A complete Lagrangian and Hamiltonian description of the theory of self-gravitating light-like matter shells is given in terms of gauge-independent geometric quantities. For this purpose the notion of an extrinsic curvature for a null-like hypersurface is discussed and the corresponding Gauss-Codazzi equations are proved. These equations imply Bianchi identities for spacetimes with null-like, singular curvature. Energy-momentum tensor-density of a light-like matter shell is unambiguously defined in terms of an invariant matter Lagrangian density. Noether identity and Belinfante-Rosenfeld theorem for such a tensor-density are proved. Finally, the Hamiltonian dynamics of the interacting system: ``gravity + matter'' is derived from the total Lagrangian, the latter being an invariant scalar density.

Uniqueness theorems for classical four-vector fields in Euclidean and Minkowski spaces

Euclidean and Minkowski four-space uniqueness theorems are derived which yield a new perspective of classical four-vector fields. The Euclidean four-space uniqueness theorem is based on a Euclidean four-vector identity which is analogous to an identity used in Helmholtz’s theorem on the uniqueness of three-vector fields. A Minkowski space identity and uniqueness theorem can be formulated from first principles and the space components of this identity turn out to reduce to the three-vector Helmholtz’s identity in a static Newtonian limit. A further result is a uniqueness theorem for scalar fields based on an identity which is proved to be a static Newtonian limit of the zeroth or scalar component of the Minkowski space extension of the Helmholtz identity. Last, the three-vector Helmholtz identity and uniqueness theorem and their four-space extensions to Minkowski space are generalized to mass damped fields.

Some Aspects of General Relativistic Vortex Dynamics and Ertel's Identity

According to Erters identity and Ertel's vorticity theorem some aspects on general relativistic vortex dynamics and relationships are discussed in detail.

Infinite imprimitive homogeneous 3-edge-colored complete graphs

Introduction. We will refer to a complete graph with a 3-edge coloring as a 3-graph in the present paper, for lack of a better name; equivalently, a 3-graph is a structure in a language consisting of three nontrivial symmetric 2-types. A 3-graph is said to be homogeneous if every isomorphism between finite induced 3-subgraphs extends to an automorphism of the whole 3-graph. The finite homogeneous 3-graphs are known [10, 6], and in fact there is a decent theory of finite homogeneous structures for relational languages [5, 9] which provides a very rough classification in general. The present paper classifies the infinite imprimitive homogeneous 3-graphs, as part of an ongoing project to explore the ramifications of a combinatorial method due primarily to Lachlan. There is no systematic theory of infinite homogeneous structures, even for binary languages. It is not clear whether the infinite homogeneous 3-graphs are classifiable. The classification of the infinite homogeneous directed graphs is known, and the case of 3-graphs appears harder but in some ways similar. We expect that an attempt to apply Lachlan's method to the classification of all infinite homogeneous 3-graphs will either lead to a considerable development of that method, or to some completely unfamiliar phenomena. The case treated here is one of the obvious special cases that needs to be dealt with first. The classification of imprimitive homogeneous directed graphs was dealt with in a similar spirit in [1]. In ? 1 we will present a catalog of the examples found with a brief discussion of their properties. With one exception, the five families listed are analogs of imprimitive directed graphs; the exceptional family is the class of product 3-graphs. Homogeneous product structures exist quite generally but are not encountered in the context of directed graphs, as the type structure is too limited. We will see that the first three families in our catalog are rather special in character, while the last two families are of more or less generic type. The bulk of the analysis is directed toward identification theorems for the structures lying in these last two families. In particular the fifth and last family has only two members, each of which needs its own identification theorem. As a result, about a third of our analysis will be devoted to these two structures. The paper is organized as follows. In ? 1 we describe our five families of infinite imprimitive homogeneous 3-graphs, which we refer to as: composite 3-graphs; product 3-graphs; double covers; restricted generic; and generic type. The first four families contain infinitely many examples, while the last family consists of just two

Volterra system identification and Kramer's sampling theorem

Nonlinear systems increase signal bandwidth, but Volterra systems may be identified uniquely even when the output is sampled at the Nyquist rate of the input. We present a more transparent and general proof. Using the theory of sampling expansions for general integral transforms, we permit the sampling to be nonuniform.

Dual boundary element analysis for cracked bars under torsion

A dual integral formulation for a cracked bar under torsion is derived, and a dual boundary element method is implemented. It is shown that as the thickness of the crack becomes thinner, the ill‐posedness for the linear algebraic matrix becomes more serious if the conventional BEM is used. Numerical experiments for solution instability due to ill‐posedness are shown. To deal with this difficulty, the hypersingular equation of the dual boundary integral formulation is employed to obtain an independent constraint equation for the boundary unknowns. For the sake of computational efficiency, the area integral for the torsion rigidity is transformed into two boundary integrals by using Green’s second identity and divergence theorem. Finally, the torsion rigidities for cracks with different lengths and orientations are solved by using the dual BEM, and the results compare well with the analytical solutions and FEM results.

Identity Theorems for Functions of Bounded Characteristic

Suppose that f(z) is a meromorphic function of bounded characteristic in the unit disk Δ:∣z∣<1. Then we shall say that f(z)∈N. It follows (for example from [3, Lemma 6.7, p. 174 and the following]) that f ( z ) = c ∏ 1 ( z ) ∏ 2 ( z ) e h 1 ( z ) - h 2 ( z ) where h1(z), h2(z) are holomorphic in Δ and have positive real part there, while Π1(z), Π2(z) are Blaschke products, that is, ∏ 1 ( z ) = z p ∏ j { z - a j 1 - a ¯ j z } . { - a j ¯ | a j | } where p is a positive integer or zero, 0<∣aj∣<1, c is a constant and ∑(1−|aj|)<∞. We note in particular that, if c≠0, so that f(z)≢0, | f ( z ) | ⩾ | c ∏ 1 ( z ) e - h 2 ( z ) | ⩾ | c ∏ 1 ( z ) | e - | h 2 ( 0 ) | ( 1 + | z | ) / ( 1 - | z | ) (1.1) so that f(z)=0 only at the points aj. Suppose now that zj is a sequence of distinct points in Δ such that |zj|→1 as j→∞ and ∑(1−|zj|)=∞. (1.2) If f(zj)=0 for each j and f∈N, then f(z)≡0. N. Danikas [1] has shown that the same conclusion obtains if f(zj)→0 sufficiently rapidly as j→∞. Let εj, λj be sequences of positive numbers such that ∑εj<∞ and λj →∞ as j→∞. Danikas then defines δ j = min { ε j , 1 2 inf k ≠ j | z k - z j | } and proves Theorem A.

On Scott Consequence Systems

The notion of Scott consequence system (briefly, S-system) was introduced by D. Vakarelov in [32] in an analogy to a similar notion given by D. Scott in [26]. In part one of the paper we study the category SSyst of all S-systems and all their morphisms. We show that the category DLat of all distributive lattices and all lattice homomorphisms is isomorphic to a reflective full subcategory of the category SSyst. Extending the representation theory of D. Vakarelov [32] for S-systems in P-systems, we develop an isomorphism theory for S-systems and for Tarski consequence systems. In part two of the paper we prove that the separation theorem for S-systems is equivalent in ZF to some other separation principles, including the separation theorem for filters and ideals in Boolean algebras and separation theorem for convex sets in convexity spaces.

A recursive family of differential polynomials generated by the sylvester identity and addition theorems for hyperelliptic Kleinian functions

A recursive family of differential polynomials generated by the sylvester identity and addition theorems for hyperelliptic Kleinian functions

The Helson–Reiter theorem for a class of nilpotent discrete groups

It is shown that the analogue of the Helson–Reiter theorem for ideals in group algebras of locally compact abelian groups holds for a class of groups including all divisible nilpotent discrete groups.

The Inverse Segal–Bargmann Transform for Compact Lie Groups

In this paper I give a new inversion formula (Theorem 1) for the generalized Segal–Bargmann transform introduced in B. C. Hall,J. Funct. Anal.122(1994), 103–151. The inversion formula may be viewed as a formula for theinverseheat operator for a compact Lie group. I then use this formula to give a new direct proof of the unitary of theK-invariant form of the Segal–Bargmann transform (Theorem 2). The proof of the inversion formula relies on an identity (Theorem 5) which relates the Laplacian for a compact Lie groupKto the Laplacian for the non-compact dual symmetric space KC/K.