Zur Kohomologietheorie rigid analytischer Räume

Let X be a rigid analytic space and assume that X admits a formal covering inducing on X the structure of a formal scheme. Then, dependent on such a formal structure one associates to X the reduction which is a scheme of locally finite type over the residue field corresponding to the ground field in question. In this paper the cohomology groups of X are compared with the cohomology groups of and a dimension formula is proved. As a consequence it is shown that X is affinoid is affine and that an analytic map :XY which is compatible with the formal structures on X and Y is finite if and only if is finite.     

QUADRATIC ESTIMATORS OF QUADRATIC FUNCTIONS WITH PARAMETERS IN NORMAL LINEAR MODELS

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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.     

Note on irregularities of distribution

Letf be a periodic function on with period 1, piecewise continuously differentiable, satisfying . For an arbitrary sequence = ( _i ) in [0,1) put and . If then _n (f,) >c· logn holds for some positive constantc (depending onf only) and almost alln. In a certain sense the converse is also true: there is a class of functionsf with such that _n (f,) =o (logn).Support has been received from Netherlands Organization for the Advancement of Pure Research (Z. W. O.).     

An open mapping theorem for measures

LetX andY be Hausdorff spaces and denote byM (X) andM (Y) the corresponding spaces of finite and non-negative Borel measures, endowed with the weak topology. A Borel map :XY induces the map :M (X)M (Y). We give necessary and sufficient conditions for to be open. In case of being a surjection between Suslin spaces, is open if and only if is.     

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").     

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.     

Singular perturbations in foliations

Summary We define a constraint system , [0,₀), which is a kind of family of vector fields on a manifold. This is a generalized version of the family of the equations , [0,₀),x ^m ,y ⁿ . Finally, we prove a singular perturbation theorem for the system , [0,₀).Dedicated to Professor Kenichi Shiraiwa on his 60th birthday     

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.     

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.     

Sequential gradient-restoration algorithm for optimal control problems

This paper considers the problem of minimizing a functionalI which depends on the statex(t), the controlu(t), and the parameter . Here,I is a scalar,x ann-vector,u anm-vector, and ap-vector. At the initial point, the state is prescribed. At the final point, the statex and the parameter are required to satisfyq scalar relations. Along the interval of integration, the state, the control, and the parameter are required to satisfyn scalar differential equations. Asequential algorithm composed of the alternate succession of gradient phases and restoration phases is presented. This sequential algorithm is contructed in such a way that the differential equations and boundary conditions are satisfied at the end of each iteration, that is, at the end of a complete gradient-restoration phase; hence, the value of the functional at the end of one iteration is comparable with the value of the functional at the end of any other iteration.In thegradient phase, nominal functionsx(t),u(t), satisfying all the differential equations and boundary conditions are assumed. Variations x(t), u(t), leading to varied functions (t),(t), are determined so that the value of the functional is decreased. These variations are obtained by minimizing the first-order change of the functional subject to the linearized differential equations, the linearized boundary conditions, and a quadratic constraint on the variations of the control and the parameter.Since the constraints are satisfied only to first order during the gradient phase, the functions (t),(t), may violate the differential equations and/or the boundary conditions. This being the case, a restoration phase is needed prior to starting the next gradient phase. In thisrestoration phase, the functions (t),(t), are assumed to be the nominal functions. Variations (t), (t), leading to varied functions (t),û(t), consistent with all the differential equations and boundary conditions are determined. These variations are obtained by requiring the least-square change of the control and the parameter subject to the linearized differential equations and the linearized boundary conditions. Of course, the restoration phase must be performed iteratively until the cumulative error in the differential equations and boundary conditions becomes smaller than some preselected value.If the gradient stepsize is , an order-of-magnitude analysis shows that the gradient corrections are x=O(), u=O(), =O(), while the restoration corrections are . Hence, for sufficiently small, the restoration phase preserves the descent property of the gradient phase: the functionalI decreases between any two successive iterations.Methods to determine the gradient stepsize in an optimal fashion are discussed. Examples are presented for both the fixed-final-time case and the free-final-time case. The numerical results show the rapid convergence characteristics of the sequential gradient-restoration algorithm.The portions of this paper dealing with the fixed-final-time case were presented by the senior author at the 2nd Hawaii International Conference on System Sciences, Honolulu, Hawaii, 1969. The portions of this paper dealing with the free-final-time case were presented by the senior author at the 20th International Astronautical Congress, Mar del Plata, Argentina, 1969. This research, supported by the NASA-Manned Spacecraft Center, Grant No. NGR-44-006-089, Supplement No. 1, is a condensation of the investigations presented in Refs. 1–5. The authors are indebted to Professor H. Y. Huang for helpful discussions.     

Spectral Properties of Solutions of the Burgers Equation with Small Dissipation

We study the asymptotic behavior as of the Sobolev norm of the solution to the Cauchy problem for the one-dimensional quasilinear Burgers type equation (It is assumed that the problem is , the boundary conditions are periodic, and .) We show that the locally time-averaged Sobolev norms satisfy the estimate . The estimates obtained as a consequence for the Fourier coefficients justify Kolmogorov's spectral theory of turbulence for the case of the Burgers equation.     

A structure theorem for weak buildings of spherical type

Following earlier work of Tits [8], this paper deals with the structure of buildings which are not necessarily thick; that is, possessing panels (faces of codimension 1) which are contained in two chambers, only. To every building , there is canonically associated a thick building whose Weyl group W( ) can be considered as a reflection subgroup of the Weyl group W() of . One can reconstruct from together with the embedding W( ) W(). Conversely, if is any thick building and W any reflection group containing W( ) as a reflection subgroup, there exists a weak building with Weyl group W and associated thick building .     

Überschallströmung eines Gases mit festen Partikeln um einen Keil

Zusammenfassung Kleine feste Partikeln, die in einem Gas enthalten sind, können schnellen Änderungen der Geschwindigkeit und der Temperatur des Gases nicht unmittelbar folgen. Hier wird der Einfluss dieser Relaxationseffekte auf zweidimensionale, stationäre Überschallströmungen um einen ebenen, symmetrischen, schlanken Keil mit Hilfe von Laplace-Transformationen untersucht. Dabei ergibt sich die Gestalt des gefrorenen Verdichtungsstosses in der physikalischen Ebene analytisch. Für kleinen Massenanteil der Partikeln lassen sich ausserdem analytische Beziehungen für den Verlauf der Strömmungsgrössen an der Keiloberfläche angeben.

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Summary Small solid particles contained in a gas cannot follow rapid changes of velocity and temperature of the gas immediately. The influence of these relaxation effects on two-dimensional steady, supersonic flow around a plane, symmetric, slender wedge is studied. The problem is treated by Laplace transformation. In the physical plane, the shape of the frozen shock is given analytically. Also, for small mass fraction of the particles, analytical solutions are obtained for all flow variables on the surface of the wedge.

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Bezeichnungen a=ã/ã _A Schallgeschwindigkeit im Gas - isobare spezifische Wärme des Gases - spezifische Wärme des Partikelmaterials - isochore spezifische Wärme des Gases - innere Energie des Gases, bezogen auf die Masseneinheit - M Mach-Zahl - N=(M ²–1)^–1/2 Abkürzung - Druck - Pr Prandtl-Zahl des Gases - s unabhängige Variable, die bei der Laplace-Transformation der Variablen im Bildraum entspricht - Entropie des Gases, bezogen auf die Masseneinheit - Temperatur - x-Komponente der Geschwindigkeit - y-Komponente der Geschwindigkeit - Koordinate in Anströmrichtung - Abkürzungen - Koordinate senkrecht zur Anströmrichtung - =tg Entwicklungsparameter - Verhältnis zwischen isobarer und isochorer spezifischer Wärme des Gases - Verhältnis der spezifischen Wärme des Partikelmaterials zur isobaren spezifischen Wärme des Gases - Winkel des gefrorenen Verdichtungsstosses gegen die Anströmrichtung - = _P / _G Verhältnis der Partikeldichte zur Gasdichte - Winkel der Gasteilchenbahnen gegen diex-Achse - _K halber Öffnungswinkel des Keils - Steigung der linksläufigen Machschen Linien - Dichte des Gases - Dichte der Partikeln, bezogen auf die Volumeneinheit der Mischung - Bezugszeit - Relaxationszeit für den Geschwindigkeitsausgleich durch Reibung zwischen Gas und Partikeln - Relaxationszeit für den Temperaturausgleich durch Wärmeleitung zwischen Gas und Partikeln - charakteristische Koordinate - Funktion, die den Ort des gefrorenen Verdichtungsstosses in charakteristischen Koordinaten angibt - charakteristische Koordinate - IndexA Bezugszustand - IndexE Gleichgewichtszustand am Ende der Relaxationszone - IndexF Zustand hinter dem gefrorenen Verdichtungsstoss - IndexG Gas - IndexK Keiloberfläche - IndexP Partikel - ^~(Tilde) dimensionsbehaftete Grössen Herrn Professor Dr.-Ing. Karl Klotter zur Vollendung seines 70. Lebensjahres gewidmet     

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.     

A Version of the Ruh--Vilms Theorem for Surfaces of Constant Mean Curvature in S 3

We study a version of the Gauss map for a surface immersed in and prove an analog of the Ruh--Vilms theorem which states that this map is harmonic iff has a constant mean curvature. As a corollary, we conclude that an embedded flat torus with constant mean curvature is a spherical Delonay surface.     

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 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.     

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

Zur Stetigkeit der mengenwertigen metrischen Projektion in endlichdimensionalen Räumen

Let X be a real normed linear space, X^* its dual, V a linear subspace of X and S(V) the unit sphere in the orthogonal space In this note we prove in the case of finite-dimensional X the following sufficient condition for the continuity of the set-valued metric projection in terms of the mapping and : If the restriction of T to S(V) is lower semi-continuous then P_V is lower semi-continuous.