Stability of infinite-dimensional sampled-data systems

Suppose that a static-state feedback stabilizes a continuous-time linear infinite-dimensional control system. We consider the following question: if we construct a sampled-data controller by applying an idealized sample-and-hold process to a continuous-time stabilizing feedback, will this sampled-data controller stabilize the system for all sufficiently small sampling times? Here the state space and the control space are Hilbert spaces, the system is of the form , where is the generator of a strongly continuous semigroup on , and the continuous time feedback is . The answer to the above question is known to be ``yes' if and are finite-dimensional spaces. In the infinite-dimensional case, if is not compact, then it is easy to find counterexamples. Therefore, we restrict attention to compact feedback. We show that the answer to the above question is ``yes', if is a bounded operator from into . Moreover, if is unbounded, we show that the answer ``yes' remains correct, provided that the semigroup generated by is analytic. We use the theory developed for static-state feedback to obtain analogous results for dynamic-output feedback control.

     

Spines and topology of thin Riemannian manifolds

Consider Riemannian manifolds for which the sectional curvature of and second fundamental form of the boundary are bounded above by one in absolute value. Previously we proved that if has sufficiently small inradius (i.e. all points are sufficiently close to the boundary), then the cut locus of exhibits canonical branching behavior of arbitrarily low branching number. In particular, if is thin in the sense that its inradius is less than a certain universal constant (known to lie between and ), then collapses to a triply branched simple polyhedral spine.

We use a graphical representation of the stratification structure of such a collapse, and relate numerical invariants of the graph to topological invariants of when is simply connected. In particular, the number of connected strata of the cut locus is a topological invariant. When is -dimensional and compact, has complexity in the sense of Matveev, and is a connected sum of copies of the real projective space , copies chosen from the lens spaces , and handles chosen from or , with 3-balls removed, where . Moreover, we construct a thin metric for every graph, and hence for every homeomorphism type on the list.

     

A classification and examples of rank one chain domains

A chain order of a skew field is a subring of so that implies Such a ring has rank one if , the Jacobson radical of is its only nonzero completely prime ideal. We show that a rank one chain order of is either invariant, in which case corresponds to a real-valued valuation of or is nearly simple, in which case and are the only ideals of or is exceptional in which case contains a prime ideal that is not completely prime. We use the group of divisorial of with the subgroup of principal to characterize these cases. The exceptional case subdivides further into infinitely many cases depending on the index of in Using the covering group of and the result that the group ring is embeddable into a skew field for a skew field, examples of rank one chain orders are constructed for each possible exceptional case.

     

Criteria for large deviations

We give the general variational form of

for any bounded above Borel measurable function on a topological space , where is a net of Borel probability measures on , and a net in converging to . When is normal, we obtain a criterion in order to have a limit in the above expression for all continuous bounded, and deduce new criteria of a large deviation principle with not necessarily tight rate function; this allows us to remove the tightness hypothesis in various classical theorems.

     

Stationary sets for the wave equation in crystallographic domains

Let be a crystallographic group in generated by reflections and let be the fundamental domain of We characterize stationary sets for the wave equation in when the initial data is supported in the interior of The stationary sets are the sets of time-invariant zeros of nontrivial solutions that are identically zero at .

We show that, for these initial data, the -dimensional part of the stationary sets consists of hyperplanes that are mirrors of a crystallographic group , This part comes from a corresponding odd symmetry of the initial data.

In physical language, the result is that if the initial source is localized strictly inside of the crystalline , then unmovable interference hypersurfaces can only be faces of a crystalline substructure of the original one.

     

West's problem on equivariant hyperspaces and Banach-Mazur compacta

Let be a compact Lie group, a metric -space, and the hyperspace of all nonempty compact subsets of endowed with the Hausdorff metric topology and with the induced action of . We prove that the following three assertions are equivalent: (a) is locally continuum-connected (resp., connected and locally continuum-connected); (b) is a -ANR (resp., a -AR); (c) is an ANR (resp., an AR). This is applied to show that is an ANR (resp., an AR) for each compact (resp., connected) Lie group . If is a finite group, then is a Hilbert cube whenever is a nondegenerate Peano continuum. Let be the hyperspace of all centrally symmetric, compact, convex bodies , , for which the ordinary Euclidean unit ball is the ellipsoid of minimal volume containing , and let be the complement of the unique -fixed point in . We prove that: (1) for each closed subgroup , is a Hilbert cube manifold; (2) for each closed subgroup acting non-transitively on , the -orbit space and the -fixed point set are Hilbert cubes. As an application we establish new topological models for tha Banach-Mazur compacta and prove that and have the same -homotopy type.

     

Twisted sums with spaces

If is a separable Banach space, we consider the existence of non-trivial twisted sums , where or For the case we show that there exists a twisted sum whose quotient map is strictly singular if and only if contains no copy of . If we prove an analogue of a theorem of Johnson and Zippin (for ) by showing that all such twisted sums are trivial if is the dual of a space with summable Szlenk index (e.g., could be Tsirelson's space); a converse is established under the assumption that has an unconditional finite-dimensional decomposition. We also give conditions for the existence of a twisted sum with with strictly singular quotient map.

     

Integration of multivalued operators and cyclic submonotonicity

We introduce a notion of cyclic submonotonicity for multivalued operators from a Banach space to its dual. We show that if the Clarke subdifferential of a locally Lipschitz function is strictly submonotone on an open subset of , then it is also maximal cyclically submonotone on , and, conversely, that every maximal cyclically submonotone operator on is the Clarke subdifferential of a locally Lipschitz function, which is unique up to a constant if is connected. In finite dimensions these functions are exactly the lower C functions considered by Spingarn and Rockafellar.

     

A path-transformation for random walks and the Robinson-Schensted correspondence

The author and Marc Yor recently introduced a path-transformation with the property that, for belonging to a certain class of random walks on , the transformed walk has the same law as the original walk conditioned never to exit the Weyl chamber . In this paper, we show that is closely related to the Robinson-Schensted algorithm, and use this connection to give a new proof of the above representation theorem. The new proof is valid for a larger class of random walks and yields additional information about the joint law of and . The corresponding results for the Brownian model are recovered by Donsker's theorem. These are connected with Hermitian Brownian motion and the Gaussian Unitary Ensemble of random matrix theory. The connection we make between the path-transformation and the Robinson-Schensted algorithm also provides a new formula and interpretation for the latter. This can be used to study properties of the Robinson-Schensted algorithm and, moreover, extends easily to a continuous setting.

     

On extendability of group actions on compact Riemann surfaces

The question of whether a given group which acts faithfully on a compact Riemann surface of genus is the full group of automorphisms of (or some other such surface of the same genus) is considered. Conditions are derived for the extendability of the action of the group in terms of a concrete partial presentation for associated with the relevant branching data, using Singerman's list of signatures of Fuchsian groups that are not finitely maximal. By way of illustration, the results are applied to the special case where is a non-cyclic abelian group.

     

Poset block equivalence of integral matrices

Given square matrices and with a poset-indexed block structure (for which an block is zero unless ), when are there invertible matrices and with this required-zero-block structure such that ? We give complete invariants for the existence of such an equivalence for matrices over a principal ideal domain . As one application, when is a field we classify such matrices up to similarity by matrices respecting the block structure. We also give complete invariants for equivalence under the additional requirement that the diagonal blocks of and have determinant . The invariants involve an associated diagram (the ``-web') of -module homomorphisms. The study is motivated by applications to symbolic dynamics and -algebras.

     

Connections with prescribed first Pontrjagin form

Let be a principal bundle over a manifold of dimension . If , then we prove that every differential 4-form representing the first Pontrjagin class of is the Pontrjagin form of some connection on .     

On the canonical rings of covers of surfaces of minimal degree

In one of the main results of this paper, we find the degrees of the generators of the canonical ring of a regular algebraic surface of general type defined over a field of characteristic , under the hypothesis that the canonical divisor of determines a morphism from to a surface of minimal degree . As a corollary of our results and results of Ciliberto and Green, we obtain a necessary and sufficient condition for the canonical ring of to be generated in degree less than or equal to . We construct new examples of surfaces satisfying the hypothesis of our theorem and prove results which show that many a priori plausible examples cannot exist. Our methods are to exploit the -algebra structure on . These methods have other applications, including those on Calabi-Yau threefolds. We prove new results on homogeneous rings associated to a polarized Calabi-Yau threefold and also prove some existence theorems for Calabi-Yau covers of threefolds of minimal degree. These have consequences towards constructing new examples of Calabi-Yau threefolds.

     

Finiteness theorems for positive definite -regular quadratic forms

An integral quadratic form of variables is said to be -regular if globally represents all quadratic forms of variables that are represented by the genus of . For any , it is shown that up to equivalence, there are only finitely many primitive positive definite integral quadratic forms of variables that are -regular. We also investigate similar finiteness results for almost -regular and spinor -regular quadratic forms. It is shown that for any , there are only finitely many equivalence classes of primitive positive definite spinor or almost -regular quadratic forms of variables. These generalize the finiteness result for 2-regular quaternary quadratic forms proved by Earnest (1994).

     

Explicit Lower bounds for residues at of Dedekind zeta functions and relative class numbers of CM-fields

Let be a given set of positive rational primes. Assume that the value of the Dedekind zeta function of a number field is less than or equal to zero at some real point in the range . We give explicit lower bounds on the residue at of this Dedekind zeta function which depend on , the absolute value of the discriminant of and the behavior in of the rational primes . Now, let be a real abelian number field and let be any real zero of the zeta function of . We give an upper bound on the residue at of which depends on , and the behavior in of the rational primes . By combining these two results, we obtain lower bounds for the relative class numbers of some normal CM-fields which depend on the behavior in of the rational primes . We will then show that these new lower bounds for relative class numbers are of paramount importance for solving, for example, the exponent-two class group problem for the non-normal quartic CM-fields. Finally, we will prove Brauer-Siegel-like results about the asymptotic behavior of relative class numbers of CM-fields.

     

The periodic Euler-Bernoulli equation

We continue the study of the Floquet (spectral) theory of the beam equation, namely the fourth-order eigenvalue problem

where the functions and are periodic and strictly positive. This equation models the transverse vibrations of a thin straight (periodic) beam whose physical characteristics are described by and . Here we develop a theory analogous to the theory of the Hill operator .

We first review some facts and notions from our previous works, including the concept of the pseudospectrum, or -spectrum.

Our new analysis begins with a detailed study of the zeros of the function , for any given ``quasimomentum' , where is the Floquet-Bloch variety of the beam equation (the Hill quantity corresponding to is , where is the discriminant and the period of ). We show that the multiplicity of any zero of can be one or two and (for some ) if and only if is also a zero of another entire function , independent of . Furthermore, we show that has exactly one zero in each gap of the spectrum and two zeros (counting multiplicities) in each -gap. If is a double zero of , it may happen that there is only one Floquet solution with quasimomentum ; thus, there are exceptional cases where the algebraic and geometric multiplicities do not agree.

Next we show that if is an open -gap of the pseudospectrum (i.e., ), then the Floquet matrix has a specific Jordan anomaly at and .

We then introduce a multipoint (Dirichlet-type) eigenvalue problem which is the analogue of the Dirichlet problem for the Hill equation. We denote by the eigenvalues of this multipoint problem and show that is also characterized as the set of values of for which there is a proper Floquet solution such that .

We also show (Theorem 7) that each gap of the -spectrum contains exactly one and each -gap of the pseudospectrum contains exactly two 's, counting multiplicities. Here when we say ``gap' or ``-gap' we also include the endpoints (so that when two consecutive bands or -bands touch, the in-between collapsed gap, or -gap, is a point). We believe that can be used to formulate the associated inverse spectral problem.

As an application of Theorem 7, we show that if is a collapsed (``closed') -gap, then the Floquet matrix is diagonalizable.

Some of the above results were conjectured in our previous works. However, our conjecture that if all the -gaps are closed, then the beam operator is the square of a second-order (Hill-type) operator, is still open.

     

Examples for the mod motivic cohomology of classifying spaces

Let be the classifying space of a compact Lie group . Some examples of computations of the motivic cohomology are given, by comparing with , and .

     

Contractive projections and operator spaces

Parallel to the study of finite-dimensional Banach spaces, there is a growing interest in the corresponding local theory of operator spaces. We define a family of Hilbertian operator spaces , , generalizing the row and column Hilbert spaces and , and we show that an atomic subspace that is the range of a contractive projection on is isometrically completely contractive to an -sum of the and Cartan factors of types 1 to 4. In particular, for finite-dimensional , this answers a question posed by Oikhberg and Rosenthal. Explicit in the proof is a classification up to complete isometry of atomic w-closed -triples without an infinite-dimensional rank 1 w-closed ideal.

     

On the Iwasawa -invariants of real abelian fields

For a prime number and a number field , let denote the projective limit of the -parts of the ideal class groups of the intermediate fields of the cyclotomic -extension over . It is conjectured that is finite if is totally real. When is an odd prime and is a real abelian field, we give a criterion for the conjecture, which is a generalization of results of Ichimura and Sumida. Furthermore, in a special case where divides the degree of , we also obtain a rather simple criterion.