A Second Descent Problem for Quadratic Forms

Let F be a field of characteristic different from 2. We discuss a new descent problem for quadratic forms, complementing the one studied by Kahn and Laghribi. More precisely, we conjecture that for any quadratic form q over F and any Im(W(F) W(F(q))), there exists a quadratic form W(F) such that dim 2 dim and _F(q), where F(q) is the function field of the projective quadric defined by q = 0. We prove this conjecture for dim 3 and any q, and get partial results for dim {4, 5,6}. We also give other related results.     

Free Algebras over Biregularizations of Varieties

Let : F N be a type of algebras, where F is a set of fundamental operation symbols and N is the set of nonnegative integers. An identity of type is called biregular if the sets of variables in and are identical and the sets of fundamental operation symbols in and are identical. If K is a variety of type , we denote by K_b the variety of type defined by all biregular identities from Id(K). K_b will be called the biregularization of K. In this paper we give a representation of free algebras over K_b by means of free algebras over K.     

Kennzeichnungen von Minimalschraubflächen Mittels Vertikalisophoten und Horizontalschattengrenzen

In Euclidean space E³, let be a (regular C-) minimal surface without planar points having locally (without loss of generality) the spherical representation n(u,v)=(cos v/cosh u, sin v/cosh u, tanh u), (u,v)G². The corresponding (isothermal) parametrization : x(u,v), (u,v)G can be expressed using agenerating Function (u,v) which satisfies _{uu + vv} – 2_utanh u + =0; the v-curves (coordinate curves u=u₀) in , along each of which the angle between the normal n(u,v) of and the x₃-axis is constant, are thevertical- isophotes of , the u-curves (v=v₀) being their orthogonal trajectories (theorems 1, 2). Considering u-curves and/or v-curves of having additional geometric properties (curves of constant/steepest slope, curves of constant Gaussian curvature, asymptotic curves, lines of curvature or geodesies of ) we prove many newgeometric characterizations of theright helicoid, thecatenoid andScherk's second surface (theorems 3–7). All of these surfaces areminimal hélicoidal surfaces.     

Einschließungsaussagen für gewöhnliche Differentialoperatoren

Summary For differential operatorsM of second order (as defined in (1.1)) we describe a method to prove Range-Domain implications—Mu–u and an algorithm to construct these functions , , , . This method has been especially developed for application to non-inverse-positive differential operators. For example, for non-negativea ₂ and for given functions = we require =C ₀[0, 1] C ₂([0, 1]–T) whereT is some finite set), (M) (t)(t), (t[0, 1]–T) and certain additional conditions for eachtT. Such Range-Domain implications can be used to obtain a numerical error estimation for the solution of a boundary value problemMu=r; further, we use them to guarantee the existence of a solution of nonlinear boundary value problems between the bounds- and .     

On Tabor's problem concerning a certain quasi-ordering of iterative roots of functions

Summary Using the Isaacs-Zimmermann's theory of iterative roots of functions, we prove a theorem concerning the problemP 250 posed by J. Tabor:Letf: E E be a given mapping. Denote byF the set of all iterative roots off. InF we define the following relation: if and only if is an iterative root of. The relation is obviously reflexive and transitive. The question is: Is it also antisymmetric? If we consider iterative roots of a monotonic function the answer is yes. But in general the question is open.Here we prove that there exists a three-element decomposition { _i ;i = 1, 2, 3} of the setE ^E with blocks _i of the same cardinality 2^cardE such that the functions from ₁ do not possess any proper iterative root, the quasi-ordering is not antisymmetric onF(f) for anyf ₂, and is an ordering onF(f) for anyf ₃. Iff is a strictly increasing continuous self-bijection ofE, then the relation is an ordering onF(f) ifff is different from the identity mapping of the setE.     

Some extensions of the principles of idealization transfer and choice in the relative internal set theory

The results established in this paper are in connection with the Relative Internal Set Theory (R.I.S.T.). The main result is the general principle of choice: Let be a level and let (x, y) be anexternalbounded formula of the language of R.I.S.T.. Suppose that to each elementx, dominated by , corresponds an elementy _x such that (x, y _x ) holds, then there exists a function of choice such that, which is a very general principle of choice, for everyx dominated by , (x, (x)) holds. More than that, we establish that if all the elementsy _x are uniformly dominated by a level then we can prescribe that the function of choice is also dominated by .     

On Vector Valued Measure Spaces of Bounded Φ-variation containing copies of ℓ∞

Given a Young function , we study the existence of copies of c ₀ and in cabv (,X) and in cabsv (,X), the countably additive, -continuous, and X-valued measure spaces of bounded -variation and bounded -semivariation, respectively.     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

Generalized ornstein-uhlenbeck process having a characteristic operator with polynomial coefficients

Summary Let be a weighted Schwartz's space of rapidly decreasing functions, the dual space and (t) a perturbed diffusion operator with polynomial coefficients from into itself. It is proven that (t) generates the Kolmogorov evolution operator from into itself via stochastic method. As applications, we construct a unique solution of a Langevin's equation on : whereW(t) is a Brownian motion and ^*(t) is the adjoint of (t) and show a central limit theorem for interacting multiplicative diffusions.     

Über die Minimalflächen des isotropen Raumes,welche zugleich Affinminimalflächen sind

One determines all the minimal surfaces of the isotropic space, which simultaneously are affinminimal surfaces. A characteristic property of those surfaces is that the isotropic spherical imagines of the asymptotic lines of form two orthogonal pencils of circles. There are three types of such surfaces : first the well known right helicoid _I , second an interesting transcendental surface _II , and third the isotropic analogy _III of the minimal surface ofEnneper. The surfaces permit cinematic generations. Especially _II and _III can be generated byClifford screws in a certain indefinite quasielliptic space.In the isotropic space conjugate to the surfaces are isotropic minimal surfaces ^* with plane lines of curvature. There are also three types of such surfaces: _I ^* is a logarithmic surface of revolution, _II ^* is an interesting transcendental surface, and _III ^* is again the isotropic minimal surface ofEnnerper.     

General Topology — the Monadic Case, Examples, Applications

The paper deals with monadic as well as monadic-free topological notions. For defining these monadic-free notions the notion of basic triple is introduced. A lot of monadic-free topological notions are presented, for instance that of -convergence structure, -hull operator and -uniform structure. By means of a generalized metric, e.g. a probabilistic metric, and the general notion of -zero approach introduced in this paper, a -uniform structure is generated. In case of a fuzzy metric the related -uniform structure defines in a canonic way a fuzzy topology which is used for developing a fuzzy analysis and fuzzy calculus.     

Inequalities Between the Radii of Some Spheres That Are Connected with a Convex Surface in Space of Constant Curvature

With a convex surface in space of constant curvature, we associate four numbers (,M,), where is the radius of a largerst sphere freely rolling over the interior side of , is the inradius of , M is the outradius of , and is the radius of a sphere over whose interior may roll freely. Exact inequalities connecting these four numbers are found.     

Hauptregelflächen einer Verallgemeinerten Regelfläche mit Zentralregelfläche

We consider generalized ruled surfaces in euclidean n-space ⁿ with k-dimensional generators and central ruled surface of dimension k–m+1 (O < m < k). Every orthogonal trajectoryy of the generators of defines a principal ruled surface _y with generators totally orthogonal to the generators of . In each generator of _y there exists an ellipsoid — called the indicatrix of the distribution parameters — which is defined by the distribution parameters of the tangent spaces to or _y. Formulars will be given for the distribution parameters of and _y .

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Herrn Prof. Dr. H.R. Müller zum 70. Geburtstag     

Über die mittelfläche einer isotropen geradenkongruenz

Let be the middle surface of an isotropic rectilinear congruence of class C³ in the real Euclidean space E³. When the spherical image of is parametrized by special isothermal coordinates (u,v) G ², can be described by a generating harmonic function A(u,v). Using such a C-representation of , the basic properties of regularity and curvature of are discussed. Moreover, the cases that be a minimal (regular) surface ₁, or a plane surface ₂ are solved explicitly. In connection with the latter results (which are already well-known from Ribaucour) several new characterizations for being a regular surface ₁ resp. ₂ are given: they are based on special properties (like: being asymptotic lines resp. lines of curvature of ) of those curves c (-Spurlinien) in the tangents of which form in each point Xc a minimal angle with the straight line of passing through X.

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Herrn Prof. Dr. Karl Strubecker zum 80. Geburtstag gewidmet     

When do Toeplitz and Hankel operators commute?

We completely classify all Toeplitz and Hankel operators which commute; namely, we prove that that a non-trivial Hankel operator and a non-trivial Toeplitz operator commute if and only if the Hankel operator has symbolz, where is the symbol of the Toeplitz operator, and is an affine function of the characteristic function of certain anti-symmetric sets of the unit circle.     

Group Classification of Nonlinear Schrödinger Equations

We propose an approach to problems of group classification. By using this approach, we perform a complete group classification of nonlinear Schrödinger equations of the form i _t + + F(, *) = 0.     

Anticommutative Algebras Satisfying Standard Identities of Degree Four

We define an anticommutative -algebra A(D,a) whose multiplication generalizes the concept of a Jacobi bracket in the form (4). It is proved that A(D,a) is a J-algebra and that it satisfies a standard identity of degree four. A subclass of algebras A(D,a) over which is connected with some class of 3-Lie algebras is distinguished. We establish a criterion of being simple for factor algebras of non-Lie algebras in , given a 1-dimensional annihilator, and then use it to construct examples of simple infinite-dimensional (of dimension p³-1) non-Lie J-algebras over a field satisfying standard identities of degree 4, if the characteristic p of is zero (for p > 2). Also, the criterion of algebras belonging to is given.     

Compact groups on compact projective planes

LetP=(P, L) be a compact projective plane with 0P< and let be a compact connected subgroup of Aut(P). If dim dimE – dimP, whereE is the elliptic motion group of the corresponding classical plane, then E or is isomorphic to a point stabilizerE ₀ inE, cf. [31]. Here we consider the case E ₀. It is shown that the action of on the point spaceP is equivalent to the classical action ofE ₀. For dimP {8, 16} the planeP is uniquely determined by a 2-dimensional subplane with SO₂ Aut().Für H. Reiner Salzmann zum 65. Geburtstag     

Tuning the mesh of a mixed method for the stream function Vorticity formulation of the Navier-Stokes equations

Summary In this article we study a new mixed method for the Stokes and Navier-Stokes equations. The method uses two meshes, one very fine for and a coarser one for . Error estimates show that boundary layers do not require to refine the mesh for the stream function as much as for the vorticity when the Reynolds number is large. We prove estimates and study implementation problems.     

A Selection Theorem for Strongly Regular Multivalued Mappings

We prove the following generalization of a theorem of Ferry concerning selections of strongly regular multivalued maps onto the class of paracompact spaces: Let : X (Z, ) be a map of a paracompact space X into a metric space (Z, ) whose values (x) are complete subspaces of Z and absolute extensors (AE), for every x X. Suppose that can be represented as = , where : X Y is a continuous singlevalued map of X onto some finite-dimensional paracompact space Y and : Y (Z, ) is a strongly regular map. Then for every closed subset A X and every selection r : A Z of the map |A : A Z, there exists an extension : X Z of r such that is a selection of the map . We also prove a local version of this theorem.