A characterisation of spreads ovally-derived from Desarguesian spreads

We characterise all spreads that are obtainable from Desarguesian spreads by replacing a partial spread consisting of translation ovals; the corresponding ovally-derived planes are generalised André planes, of order 2 ^N , although not all generalised André planes are ovallyderived from Desarguesian planes. Our analysis allows us to obtain a complete classification of all nearfield planes that are ovally-derived from Desarguesian planes. It turns out that whether or not a nearfield plane is ovally-derived from a Desarguesian plane depends solely on the parametersq andr, where GF (q) is the kern, andr is the dimension of the plane. Our results also imply that a Hall plane of even orderq ² can be ovally-derived from a Desarguesian spread if and only ifq is a square.     

Nonstandard models that are definable in models of Peano Arithmetic

In this paper, we investigate definable models of Peano Arithmetic PA in a model of PA. For any definable model N without parameters in a model M, we show that N is isomorphic to M if M is elementary extension of the standard model and N is elementarily equivalent to M. On the other hand, we show that there is a model M and a definable model N with parameters in M such that N is elementarily equivalent to M but N is not isomorphic to M. We also show that there is a model M and a definable model N with parameters in M such that N is elementarily equivalent to M, and N is isomorphic to M, but N is not definably isomorphic to M. And also, we give a generalization of Tennenbaum's theorem. At the end, we give a new method to construct a definable model by a refinement of Kotlarski's method. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On dual of square free modules

A module M is said to be square free if whenever its submodule is isomorphic to N² = N⊕N for some module N, then N = 0. Dually, a module M is said to be d-square free (dual square free) if whenever its factor module is isomorphic to N² for some module N, then N = 0. In this paper, we give some fundamental properties of d-square free modules and study rings whose d-square free modules are closed under submodules or essential extensions.     

Lattices isomorphic with their free powers

For any N3 there exists a lattice L isomorphic with NL (the free product of its own N copies) but not isomorphic with kL for any k–2,...,N-1.     

A Note on Solitary Subgroups of Finite Groups

We say that a subgroup H of a finite group G is solitary (respectively, normal solitary) when it is a subgroup (respectively, normal subgroup) of G such that no other subgroup (respectively, normal subgroup) of G is isomorphic to H. A normal subgroup N of a group G is said to be quotient solitary when no other normal subgroup K of G gives a quotient isomorphic to G/N. We show some new results about lattice properties of these subgroups and their relation with classes of groups and present examples showing a negative answer to some questions about these subgroups.     

Von Neumann Algebras Associated with the Group SL2（R）

Let M\mathcal{M} and N\mathcal{N} be the von Neumann algebras induced by the rational action of the group SL ₂(ℝ) and its subgroup P on the upper half plane \mathbbH\mathbb{H}. We have shown that N\mathcal{N} is spatial isomorphic to the group von Neumann algebra L_P\mathcal{L}_P and characterized M\mathcal{M} and its commutant M￠\mathcal{M}' and gotten a generalization of the Mautner’s lemma. It is also shown that the Berezin operator commutates with the Laplacian operator.     

Shear planes

A shear plane is a 2n-dimensional stable plane admitting a quasi-perspective collineation group which is a vector group of the same dimension 2n and fixes no point. We show that all of these planes can be derived from a special kind of partial spreads by a construction analogous to the construction of (punctured) dual translation planes from compact spreads. Finally we give a criterion (and examples) for shear planes which are not isomorphic to an open subplane of a topological projective plane.     

Solutions to the XXX type Bethe ansatz equations and flag varieties

We consider a version of the A _N Bethe equation of XXX type and introduce a reporduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from a given one is called a population. We show that a population is isomorphic to the sl _N +1 flag variety and that the populations are in one-to-one correspondence with intersection points of suitable Schubert cycles in a Grassmanian variety. We also obtain similar results for the root systems B _N and C _N . Populations of B _N and C _N type are isomorphic to the flag varieties of C _N and B _N types respectively.     

N-free orders and minimal interval extensions

We investigate problems related to the set of minimal interval extensions of N-free orders. This leads us to a correspondence between this set for an arbitrary order and a certain set of its maximal N-free reductions. We also get a 1-1-correspondence between the set of linear extensions of an arbitrary order and the set of minimal interval extensions of the linegraph of that order. This has an algorithmic consequence, namely the problem of counting minimal interval extensions of an N-free order is #P-complete. Finally a characterization of all N-free orders with isomorphic root graph is given in terms of their lattice of maximal antichains; the lattices are isomorphic iff the root graphs agree.This work was supported by the PROCOPE Program. The first author is supported by the DFG.     

A group theoretic characterization of finite derivable nets

In this article, it is shown that a finite net of orderq ² and degreeq + 1 is a derivable net if and only if the net admits a collineation group isomorphic toPSL(4,q)_N (whereN is a line of the associated 3-dimensional projective space upon which the abstract group acts).     

The uniqueness of Hering's translation plane of order 27

It is shown that the following conjecture of Kallaher and Ostrom [2] is correct: Hering's translation plane of order 27 is the only translation plane of odd dimension over its kernel which has a collineation group isomorphic to SL(2, w) with w prime to 5 and to the characteristic, and having no affine perspectivity.     

Disegni unitari nei piani di Hughes

Let H(q ²) be the Hughes plane on the regular nearfield of order q ₂ whose center is a field of order q. We construct in H(q ²) a unital of parameter q.

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Lavoro eseguito nell'ambito di attività di ricerca finanziate dal M.P.I. (fondi 60% e 40%).     

A projective plane of order 16

A projective plane of order 16 is constructed. It is a translation plane and appears to be new. The representation of the collineation group on the axis of the plane has a normal subgroup isomorphic to L₃ (2) with factor group isomorphic to S₃. The orbits of this representation have lengths 14 and 3. If two points in the latter orbit are chosen to define a sharply doubly transitive set of permutations, the permutations from the multiplicative loop generate a group isomorphic to A₇. The plane is of Lenz-Barlotti class IVa.1.     

Geometrical implications of the existence of very smooth bump functions in Banach spaces

We show that ifX is a Banach space and if there is a non-zero real-valuedC ^∞-smooth function onX with bounded support, then eitherX contains an isomorphic copy ofc ₀(N), or there is an integerk greater than or equal to 1 such thatX is of exact cotype 2k and, in this case,X contains an isomorphic copy ofl ^2k(N). We also show that ifX is a Banach space such that there is onX a non-zero real-valuedC ⁴-smooth function with bounded support and ifX is of cotypeq forq<4, thenX is isomorphic to a Hilbert space.     

Conditions ensuring starter induced 1‐factorizations are not isomorphic

In general, it is difficult to determine whether two starter induced 1‐factorizations of K_2n are isomorphic. However, when one of the 1‐factorizations has a unique starter group to within conjugacy, we show that two starter induced 1‐factorizations on K_2n are isomorphic, if and only if, the corresponding starters are isomorphic. Two starters are isomorphic if there is a group isomorphism between the respective starter groups which takes one starter to the other. It is relatively easy to check whether starters are isomorphic in many examples. The difficulty comes in showing there is a unique starter group to within conjugacy. A number of sufficient conditions for this are given; one such condition is the assumption that the 1‐factorization be irreducible, a condition which applies to all almost perfect 1‐factorizations. Also, generalized Mullin Nemeth starters are introduced and discussed in this context. © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 124–143, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10023     

Extended genus of torsion-free finitely generated nilpotent groups of class two

The extended genus of a nilpotent group N is the set of isomorphism classes of nilpotent groups M, not necessarily finitely generated, such that the p-localizations M _p , N _p are isomorphic for all primes p. In this article, for any torsion-free finitely generated nilpotent group N of nilpotency class 2, the extended genus of N is analyzed by assigning to each of its members a sequence of triads of matrices with rational entries, generalizing the sequential representation which has been exploited elsewhere in the case when N is abelian. This approach allows, among other things, to obtain examples of groups in the ordinary (Mislin) genus of N     

A Pigeonhole Property for Relational Structures

We study those relational structures S with the property (P) that each partition of S contains a block isomorphic to S. We show that the Fraïsse limits of parametric classes K. have property (P); over a binary language, every countable structure in K satisfying (P) along with a condition on 1-extensions must be isomorphic to this limit.     

Designs arising from Buekenhout‐Metz unitals

An affine 2–(q³,q², q + 1) design is constructed from a Buekenhout‐Metz unital of the affine plane AG(2,q²), with q > 2. It is also shown that such a design is isomorphic to the point‐plane design of the affine space AG(3,q). © 2003 Wiley Periodicals, Inc. J Combin Designs 11: 79–88, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/jcd.10010