The Martin boundary and completion of Markov chains

Summary In this paper we treat a time-symmetrical Martin boundary theory for continuous parameter Markov chains. This is done by reversing the time sense of a Markov chainX _t in such a way as to obtain a dual Markov chain , and considering the two chains together. Various relations between the Martin exit boundaries and of these processes are studied. The exit boundary of , is in a sense an entrance boundary forX _t and vice versa. After a natural identification of certain points in and one can topologizeI in such a way thatboth X _t and have standard modifications in this space which are right continuous, have left limits, and are strongly Markov.Research supported in part at Stanford University, Stanford, California under AFOSR 0049.     

About existence and uniqueness of the restricted total least squares estimation

Summary Total Least Squares (TLS) is an estimation method for the solutiona of the linear system when both data sets and are subject to error. The TLS-method minimizes the functional with weighting parameter . In this paper the TLS-functional is analyzed by the technique of Lagrangian multipliers. The main part of the work deals with the case when the estimatea is restricted by an inequality of the formD a–b0, D a diagonal matrix. It is shown that there exists a unique estimatea if the weighting parameter is chosen sufficiently large.     

Completion of an additive measure to a σ-additive measure by an extension of the space

For an algebra of subsets of a set X there is constructed a set and an algebra of its subsets so that the mapping is a one-to-one correspondence between and and for each additive measure on the measure on defined by the equation is countably additive.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 71–76, January, 1968.The author wishes to express his deep appreciation to S. V. Fomin, under whose guidance this paper was written.     

Path processes and historical superprocesses

Summary A superprocessX over a Markov process can be obtained by a passage to the limit from a branching particle system for which describes the motion of individual particles.The historical process for is the process whose state at timet is the path of over time interval [0,t]. The superprocess over the historical superprocess over —reflects not only the particle distribution at any fixed time but also the structure of family trees. The principal property of a historical process is that is a function of for alls<t. Every process with this property is calleda path process. We develop a theory of superprocesses over path processes whose core is the integration with respect to measure-functionals. By applying this theory to historical superprocesses we construct the first hitting distributions and prove a special Markov property for superprocesses.Partially supported by National Science Foundation Grant DMS-8802667     

Numerical stability of the cyclic Richardson iteration

Summary The success of the cyclic Richardson iteration depends on the proper ordering of the acceleration parameters. We give a rigorous error analysis to show that, with the proper ordering, the relative error in the iterative method, when properly terminated, is not larger than the error incurred in stable direct methods such as Cholesky factorization. For both the computed approximation tou=L ^–1f satisfies cond (L)u2^–t and this bound is attainable. We also show that the residual norm is bounded by L cond . This bound is attainable for a small cycle lengthN. Our analysis suggests that for a larger cycle lengthN the residuals are bounded by . We construct a theoretical example in which this bound is attainable. However we observed in all numerical tests that ultimately the residual norms were of order . We explain why in practice even the factor is never encountered. Therefore the residual stopping criterion for the Richardson iteration appears to be very reliable and the method itself appears to be stable.The author gratefully acknowledges partial support from ONR Contract N00014-85-K-0180On leave from the University of Krakow, Poland, during the spring semester 1989     

Computing on-line the lattice of maximal antichains of posets

We consider the on-line computation of the lattice of maximal antichains of a finite poset . This on-line computation satisfies what we call the linear extension hypothesis: the new incoming vertex is always maximal in the current subposet of . In addition to its theoretical interest, this abstraction of the lattice of antichains of a poset has structural properties which give it interesting practical behavior. In particular, the lattice of maximal antichains may be useful for testing distributed computations, for which purpose the lattice of antichains is already widely used. Our on-line algorithm has a run time complexity of , where |P| is the number of elements of the poset, , |MA(P)| is the number of maximal antichains of and (P) is the width of . This is more efficient than the best off-line algorithms known so far.     

Spectral pictures of Aluthge transforms of operators

In this paper we continue our study, begun in [12], of the relationships between an arbitrary operatorT on Hilbert space and its Aluthge transform . In particular, we show that in most cases the spectral picture ofT coincides with that of , and we obtain some interesting connections betweenT and as a consequence.     

Un théorème de Vitali-Hahn-Saks pour les semimartingales

Summary Let (, , P) be a complete probability space; let _t0 be an increasing right-continuous family of -complete sub--fields of ; let be a sequence of semimartingales. Assume that for all positive t and for all bounded predictable processes H, the r.v.'s converge in probability to a limit J(t, H) when n tends to infinity. Then there exists a semimartingale X such that, for all t and H, J(t, H)= .     

Isometrische Immersionen der Kodimension 2 von Raumformen

Let M, resp. , denote Riemannian manifolds of dimensions m>4, resp. =m+2, and of constant sectional curvatures C, resp. , with 0 and is a standard space form, then the foliation L is a (globally) trivial fibre bundle with fibre S^m–1.     

On the Hellinger type distances for filtered experiments

Summary We study the Hellinger type distances on a filtered space. Herep2 is an arbitrary number andP _T and are two probability measures stopped at a random timeT. We give lower and upper bounds for in predictable terms.     

Positive elements in the algebra of the quantum moment problem

Summary Let denote the extended Weyl algebra, , the Weyl algebra. It is well known that every element of of the formA=B _k ^* B _k is positive. We prove that the converse implication also holds: Every positive elementA in has a quadratic sum factorization for some finite set of elements (B _k ) in . The corresponding result is not true for the subalgebra . We identify states on which do not extend to states on . It follows from a result of Powers (and Arveson) that such states on cannot be completely positive. Our theorem is based on a certain regularity property for the representations which are generated by states on , and this property is not in general shared by representations generated by states defined only on the subalgebra .Work supported in part by the NSF     

A Comparison on the Commutative Neutrix Convolution of Distributions and the Exchange Formula

Let , be ultradistributions in and let and where is a sequence in which converges to the Dirac-delta function . Then the neutrix product is defined on the space of ultradistributions as the neutrix limit of the sequence provided the limit exist in the sense that

Topological and bornological characterisations of ideals in von Neumann algebras: I

Suppose is a von Neumann algebra on a Hilbert space and is any ideal in . We determine a topology on , for which the members of that are to norm continuous are exactly those in ; and a bornology on such that the elements of which map the unit ball to an element of , equivalently those members of that are norm to bounded, are exactly those in . This is achieved via analogues of the notions of injectivity and surjectivity in the theory of operator ideals on Banach spaces.     

Birth delay of a Markov process and the stopping distributions for regular processes

Summary We consider Markov processes with a fixed transition functionp(r, x; t, B) and with random birth times. We show that a process can be obtained from (X _t ,P) by birth delay if and only if for allt andB. As an application, we give a new version and a new proof of the results of Rost [R] and Fitzsimmons [F2] on stopping distributions of Markov processes. The key Lemma 1.1 replaces the filling scheme used by the previous authors.Birth delay was considered from a different prospective in [F1].Partially supported by the National Science Foundation Grant DMS-8802667     

Completions of orthomodular lattices II

If is a variety of orthomodular lattices generated by a set of orthomodular lattices having a finite uniform upper bound om the length of their chains, then the MacNeille completion of every algebra in again belongs to .The author gratefully acknowledges the support of NSERC.     

On some properties of the metric subalgebras ofℓ ∞

The objects studied are the subalgebras of which contain c_o. These are isometrically isomorphic to the algebras C( ) where is a compactification of a discrete denumerable set N . It is shown: 1) If is metric then there is a projection of norm 1, P: C( ) C( ) with kernel c_o defined by PF = f o where is a retraction of onto = – N . 2) If is metric, then the group of homeomorphisms of is isomorphic to a complete group of permutations of the natural numbers . 3) The group of homeomorphisms of a compact metric space is the homomorphic image of a complete group of permutations of ("complete" means "no outer automorphisms, trivial center").     

Bemerkungen zur Deformationstheorie nichtrationaler Singularitäten

Let be a 1-convex holomorphic mapping between complex spaces resp.S, and let be the blowingdown factorization of over S. We prove in part 1 of the present note: The fiber ^–1(s₀) over a point s₀S is the Remmert quotient of if and only if every holomorphic function on (defined in a neighborhood of the exceptional subvariety of that fiber) can be extended holomorphically to . This is true, for instance, in the case: flat, S reduced at s₀ and dim , =const for all sS. In part 2, we use this result to obtain the following: For any Riemann surface R with genus g2 there exists a 2-dimensional normal complex analytic singularity X such that the minimal resolution of X contains R as exceptional subvariety, and has a deformation over the unit disc S={|s|<1} which can not be blown down to a deformation of X.     

On the convergence of powers of interval matrices (2)

Summary Let be a real irreduciblen×n interval matrix. Then a necessary and sufficient condition is given for the sequence of the powers of an interval matrix to converge to a matrix which is not the null matrix. In addition a criterion for is proved to decide whether the limit matrix satisfies the condition of symmetry .     

A Helly-type theorem for simple polygons

Let be a family of simple polygons in the plane. If every three (not necessarily distinct) members of have a simply connected union and every two members of have a nonempty intersection, then {P:P in } . Applying the result to a finite family of orthogonally convex polygons, the set {C:C in } will be another orthogonally convex polygon, and, in certain circumstances, the dimension of this intersection can be determined.Supported in part by NSF grant DMS-9207019.     

On a theorem of Seidel and Walsh

Given a sequence ( _n ) _n in with there are functions such that , is a dense subset of , and the set of functions with this property is residual in . We will show that in and some related Banach spaceX there are functionsf with is dense in , and we will give a sufficient condition when the set of such functions is residual inX.