Über ebene affine Inzidenzgruppen

For a translation plane P with respect to f we consider the group of collineations generated by all elations fixing f and a point F of f. All subgroups or are determined which operate regularly on the points of the affine plane P. Group-theoretic and operating properties of the groups are stated especially for the finite and the Desarguesian cases. In the latter case the companion NL-near modules are constructed. Finally we characterize the groups within PGL(3, K) with commutative field K of finite characteristic.     

Über fastgeordnete affine Ebenen

An example of an affine plane is constructed, which cannot be ordered, but can be embedded in an ordered projective plane. There is no such an example, if the affine plane is a translation plane with kernel GF(2).

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Herrn Rafael Artzy zum 75. Geburtstag gewidmet     

Degenerate homogeneous affine surfaces in ℝ3

We study degenerate homogeneous affine surfaces in ³. It is proved that such a surface is either an open part of a plane, a cylinder on an ellipse, parabola or hyperbola or of the surface given byxz – 1/2y ²=0.     

Piecewise affine functions and polyhedral sets ∗

In this paper we present a number of characterizations of piecewise affine and piecewise linear functions defined on finite dimesional normed vector spaces. In particular we prove that a real-valued function is piecewise affine [resp. piecewise linear] if both its epigraph and its hypograph are (nonconvex) polyhedral sets[resp..Polyhedral cones]. Also,We show that the collection of all piecewise affine[resp.piecewise linear] functions. Furthermore, we prove that a function is piecewise affine[resp.piecewise linear] if it can be represented as a difference of two convex [resp.,sublinear] polyhedral fucntions.     

Tubular algebras and affine Kac-Moody algebras

The purpose of this paper is to construct quotient algebras L(A)1C/I(A) of complex degenerate composition Lie algebras L(A)1C by some ideals, where L(A)1C is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)1C/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)1C generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)1C generated by simple A-modules.     

On affine Sawada–Kotera equation

It is shown that motion of plane curves in affine geometry induces naturally the Sawada–Kotera hierarchy. The affine Sawada–Kotera equation is obtained in view of the equivalence of equations for the curvature and graph of plane curves when the curvature satisfies the Sawada–Kotera equation. The affine Sawada–Kotera equation can be viewed as an affine version of the WKI equation since they have similarity properties, such as they have loop-solitons, they are solved by the AKNS-scheme and are obtained by choosing the normal velocity to be the derivative of the curvature with respect to the arc-length. Its symmetry reductions to ordinary differential equations corresponding to an one-dimensional optimal system of its Lie symmetry algebras are discussed.     

Gearhart–Koshy acceleration for affine subspaces

The method of cyclic projections finds nearest points in the intersection of finitely many affine subspaces. To accelerate convergence, Gearhart & Koshy proposed a modification which, in each iteration, performs an exact line search based on minimising the distance to the solution. When the subspaces are linear, the procedure can be made explicit using feasibility of the zero vector. This work studies an alternative approach which does not rely on this fact, thus providing an efficient implementation in the affine setting.     

Weierstraß-type representation of affine spheres

Affine spheres are discussed in the context of loop groups. We show that every affine sphere can be obtained by solving two ordinary differential equations followed by an application of a generalized Birkhoff Decomposition Theorem (which we proof in the Appendix). A geometric interpretation of the coefficients of the ODE is given. Finally the method is applied to construct all ruled surfaces. Most of this work was done while the second named author was visiting the University of Kansas.     

Completely splittable representations of affine Hecke-Clifford algebras

We classify and construct irreducible completely splittable representations of affine and finite Hecke-Clifford algebras over an algebraically closed field of characteristic not equal to 2.     

The Erdős-Ko-Rado theorem for finite affine spaces

In [W.N. Hsieh, Intersection theorems for finite vector spaces, Discrete Math. 12 (1975) 1–16], Hsieh obtained the Erd?s-Ko-Rado theorem for finite vector spaces. This paper generalizes Hsieh’s result and obtains the Erd?s-Ko-Rado theorem for finite affine spaces.     

An affine interpretation of Bäcklund transformations

The paper is devoted to an affine interpretation of Bäcklundmaps (Bäcklund transformations are a particular case of Bäcklund maps) for second order differential equations with unknown function of two arguments. Note that up to now there are no papers where Bäcklund transformations are interpreted as transformations of surfaces in a space other than Euclidean space. In this paper, we restrict our considerations to the case of so-called Bäcklund maps of class 1. The solutions of a differential equation are represented as surfaces of an affine space with induced connection determining a representation of zero curvature. We show that, in the case when a second order partial differential equation admits a Bäcklund map of class 1, for each solution of the equation there is a congruence of straight lines in an affine space formed by the tangents to the affine image of the solution. This congruence is an affine analog of a parabolic congruence in Euclidean space. The Bäcklund map can be interpreted as a transformation of surfaces of an affine space under which the affine image of a solution of the differential equation is mapped into a particular boundary surface of the congruence.     

Topologische Divisionsalgebren ohne zugehörige topologische affine Ebene

We prove that many (non-associative) topological division algebrasD of dimensionn ∈ N over the centreK do not yield topological affine or projective planes (of Lenz-Barlotti type V) in contrast to the results of SKORNJAKOV [20], SALZMANN [18] and [19], GRUNDHÖFER [7], HARTMANN [11] and RINK [17] concerning projective planes coordinatized by compact or special topological ternary fields. In particular, this holds for every non-trivial and non-archimedian valuation topology ofK distinct from the order topology ifK is a real-closed field, and if the division algebraD =K ⁿ carries the product topology.     

On the bigℳ in the affine scaling algorithm

When we apply the affine scaling algorithm to a linear program, we usually construct an artificial linear program having an interior feasible solution from which the algorithm starts. The artificial linear program involves a positive number called the big. Theoretically, there exists an ^* such that the original problem to be solved is equivalent to the artificial linear program if > ^*. Practically, however, such an ^* is unknown and a safe estimate of is often too large. This paper proposes a method of updating to a suitable value during the iteration of the affine scaling algorithm. As becomes large, the method gives information on infeasibility of the original problem or its dual.This paper is dedicated to Phil Wolfe on the occasion of his 65th birthday.Supported by Grant-in-Aids for Co-Operative Research (03832017) of the Japan Ministry of Education, Science and Culture.     

A forward–backward SDE approach to affine models

We consider factor models for interest rates and asset prices where the risk- neutral dynamics of the factors process is modelled by an affine diffusion. We characterize the factors process and bond price in terms of forward–backward stochastic differential equations (FBSDEs), prove an existence and uniqueness theorem which gives the solution explicitly, and characterize the bond price as an exponential affine function of the factors in a new way. Our approach unifies the results, based on stochastic flows, of Elliott and van der Hoek (Finance Stoch 5:511–525, 2001) with the approach, based on the Feynman-Kac formula, of Duffie and Kan (Math Finance 6(4):379–406, 1996), and addresses a mistake in the approach of Elliott and van der Hoek (Finance Stoch 5:511–525, 2001). We extend our results on the bond price to consider the futures and forward price of a risky asset or commodity.      

Algebraic volume density property of affine algebraic manifolds

We introduce the notion of algebraic volume density property for affine algebraic manifolds and prove some important basic facts about it, in particular that it implies the volume density property. The main results of the paper are producing two big classes of examples of Stein manifolds with volume density property. One class consists of certain affine modifications of ℂ ⁿ equipped with a canonical volume form, the other is the class of all Linear Algebraic Groups equipped with the left invariant volume form.     

Cuspidal systems for affine Khovanov–Lauda–Rouquier algebras

A cuspidal system for an affine Khovanov–Lauda–Rouquier algebra $R_\alpha $ yields a theory of standard modules. This allows us to classify the irreducible modules over $R_\alpha $ up to the so-called imaginary modules. We describe minuscule imaginary modules, laying the groundwork for future study of imaginary Schur–Weyl duality. We introduce colored imaginary tensor spaces and reduce a classification of imaginary modules to one color. We study the characters of cuspidal modules. We show that under the Khovanov–Lauda–Rouquier categorification, cuspidal modules correspond to dual root vectors.     

An extension of Wilf’s conjecture to affine semigroups

Let \(\mathcal {C}\subset \mathbb {Q}^p_+\) be a rational cone. An affine semigroup \(S\subset \mathcal {C}\) is a \(\mathcal {C}\)-semigroup whenever \((\mathcal {C}\setminus S)\cap \mathbb {N}^p\) has only a finite number of elements. In this work, we study the tree of \(\mathcal {C}\)-semigroups, give a method to generate it and study the \(\mathcal {C}\)-semigroups with minimal embedding dimension. We extend Wilf’s conjecture for numerical semigroups to \(\mathcal {C}\)-semigroups and give some families of \(\mathcal {C}\)-semigroups fulfilling the extended conjecture. Other conjectures formulated for numerical semigroups are also studied for \(\mathcal {C}\)-semigroups.     

Über geschlossene affine Zwangläufe in der Ebene

We consider integral coverings y:{1,2,..,} of an affine plane which occur when is moved under a continuous periodic affine motion(t):. One can distinguish normal points × , i.e. is constant in a certain neighborhood of x, and singular points. If (x) is the number of times x passes through its orbit (t)x all normal points x have (x)=1, and the set of all singular points consists of a number of isolated points and lines. If (x) is the tangent rotation number of the orbit of x all singular points lie on the moving pole curve.