Magnetic structure of Pr6Fe13Ge studied by M?ssbauer effect and neutron diffraction

Magnetic structure of Pr₆Fe₁₃Ge at room temperature has been investigated by magnetic measurement, M?ssbauer effect and neutron diffraction. Magnetic atoms are closely packed up and down the mirror planes atz = 0 and 1/2 and separated by the non-magnetic atoms located on the planes atz= 1/4 and 3/4, which constructs an M/NM/M sandwich structure (M: magnetic, NM: non-magnetic). The intralayer magnetic moments couple ferromagnetically and are out ofab plane at a small angle so as to form a component alongc-axis. The interlayer coupling is antiferromagnetic for theab component while ferromagnetic for thec component.     

Magnetic structure of Pr6Fe13Ge studied by Mössbauer effect and neutron diffraction

Magnetic structure of Pr₆Fe₁₃Ge at room temperature has been investigated by magnetic measurement, Mössbauer effect and neutron diffraction. Magnetic atoms are closely packed up and down the mirror planes atz = 0 and 1/2 and separated by the non-magnetic atoms located on the planes atz= 1/4 and 3/4, which constructs an M/NM/M sandwich structure (M: magnetic, NM: non-magnetic). The intralayer magnetic moments couple ferromagnetically and are out ofab plane at a small angle so as to form a component alongc-axis. The interlayer coupling is antiferromagnetic for theab component while ferromagnetic for thec component.     

Effective interactions in the periodic Anderson model in the regime of mixed valency with strong correlations

For the periodic Anderson model in the strong correlation regime, we construct the effective Hamiltonian H _eff up to terms of the fourth order in the parameter V/U, where V is the hybridization interaction intensity and U is the intra-atom Coulomb repulsion strength. This Hamiltonian contains interactions inducing both magnetic ordering and Cooper instability under conditions of a mixed valency of rare-earth ions. Based on numerical calculations, we obtain information about the dependences of the effective interaction parameters on the distance between crystal lattice sites. We demonstrate that realizing exchange interactions corresponds to a strongly frustrated system of localized spin moments and facilitates the suppression of the antiferromagnetic order parameter with a possible transition to the state of a quantum spin liquid. It is essential that among the terms in H _eff inducing the transition to the superconductivity phase, there are terms resulting in the d-type symmetry of the superconductivity order parameter; such a symmetry is realized in many heavy-fermion compounds. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 2, pp. 235–249, November, 2008.     

Smooth Hughes planes are classical

We prove that the only compact projective Hughes planes which are smooth projective planes are the classical planes over the complex numbers \Bbb C \Bbb C , the quaternions \Bbb H \Bbb H , and the Caley numbers \Bbb O \Bbb O . As a by-product this shows that an 8-dimensional smooth projective plane which admits a collineation group of dimension d 3 17d \geq 17 is isomorphic to the quaternion projective plane P ₂\Bbb H {\cal P _2\Bbb H }. For topological compact projective planes this is true if d 3 19d \geq 19, and this bound is sharp.     

Spreads,ovoids andS 5

We consider translation planes of orderq ² (whereq andq ² - 1 are coprime to 30) such thatS ₅ acts on the line at infinity. It turns out that the Klein correspondence is in particular useful for the investigation of these planes. Representations of the planes, automorphisms and examples of low order are studied in detail. In view of a problem of Ostrom (Math. Z. 156 (1977), 59–71), series of translation planes are constructed with the following property: the translation complement is nonsolvable and has an order coprime to the characteristic of the plane.     

The inverse Abel transform for an exceptional bounded symmetric domain

By using the first shift operator of thebc ₂- root system, the elementary spherical functions on the exceptional bounded symmetric domainE ₆/spin(10) ×T are obtained, and the inverse Abel transform for this space is derived. Finally an expression of the heat kernel of this domain is given. Project partially supported by the National Natural Science Foundation of China.     

Wave interactions for a nonlinear degenerate wave equations

This paper is concerned with the interactions of the elementary waves for the nonlinear degenerate wave equations. By analyzing the expressions of the elementary waves and the relative locations of the left state U _l and the right state U _r in the phase plane (u, v) we deal with the interactions of the elementary waves, especially the overtaking of shock wave and rarefaction wave from the same family.     

k-Plane Clustering

A finite new algorithm is proposed for clustering m given points in n-dimensional real space into k clusters by generating k planes that constitute a local solution to the nonconvex problem of minimizing the sum of squares of the 2-norm distances between each point and a nearest plane. The key to the algorithm lies in a formulation that generates a plane in n-dimensional space that minimizes the sum of the squares of the 2-norm distances to each of m₁ given points in the space. The plane is generated by an eigenvector corresponding to a smallest eigenvalue of an n × n simple matrix derived from the m₁ points. The algorithm was tested on the publicly available Wisconsin Breast Prognosis Cancer database to generate well separated patient survival curves. In contrast, the k-mean algorithm did not generate such well-separated survival curves.     

Compact projective planes with homogeneous ovals

Closed ovals exist only in 2-or 4-dimensional compact projective planes. We show that a plane of dimension 2 or 4 contains ahomogeneous closed oval iff the automorphism group contains SO₂ or SO₃, respectively. In 4-dimensional planes, existence of a homogeneous oval and existence of a homogeneous Baer subplane are equivalent. We determine the possible full automorphism groups of planes containing homogeneous ovals except for two possibilities in dimension 4 whose existence remains uncertain.     

Large Almost Simple Groups Acting on 16-Dimensional Compact Projective Planes

 This paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E₆(−26). A 16-dimensional, compact projective plane ? admitting an automorphism group of dimension 41 or more is clasical, [23] 87.5 and 87.7. For the special case of a semisimple group Δ acting on ? the same result can be obtained if dim , see [22]. Our aim is to lower this bound. We show: if Δ is semisimple and dim , then ? is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin₉ (ℝ, r), r∈{0, 1}. The proof consists of two parts. In [16] it has been shown that Δ is in fact almost simple or isomorphic to SL₃?ċSpin₃ℝ. In the underlying paper we can therefore restrict our considerations to the case that Δ is almost simple, and the corresponding planes are classified. Received 10 February 1997; in final form 19 December 1997     

Large Almost Simple Groups Acting on 16-Dimensional Compact Projective Planes

 This paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E₆(−26). A 16-dimensional, compact projective plane ? admitting an automorphism group of dimension 41 or more is clasical, [23] 87.5 and 87.7. For the special case of a semisimple group Δ acting on ? the same result can be obtained if dim , see [22]. Our aim is to lower this bound. We show: if Δ is semisimple and dim , then ? is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin₉ (ℝ, r), r∈{0, 1}. The proof consists of two parts. In [16] it has been shown that Δ is in fact almost simple or isomorphic to SL₃?ċSpin₃ℝ. In the underlying paper we can therefore restrict our considerations to the case that Δ is almost simple, and the corresponding planes are classified.     

Numerical Burniat and Irregular Surfaces

In this paper, we construct three numerical Burniat surfaces as desingularizations of double planes of degree >10. Two are surfaces having the bigenus P ₂=4 and the third is a surface having the bigenus P ₂=5. In addition, another surface of general type is constructed as a desingularization of a double plane of degree 12 having the birational invariants: q=p _g =1, P ₂=4. One of the numerical Burniat surfaces with P ₂=4 is obtained as a desingularization of a double plane of degree 22 with an irreducible branch locus, so it is a good candidate for having torsion zero. Moreover, its bicanonical transformation seems to be birational.     

Projective planes over rings of stable rank 2

This paper gives an axiomatic characterization of projective planes over rings of stable rank 2. These rings, which are known from algebraic K-theory, beautifully reflect simple geometric properties. The basic relations in the plane are incidence and the neighbor relation. The axioms consist of a number of axioms expressing elementary relations between points and lines such as, e. g., the existence of a unique line joining any two non-neighboring points, and a couple of axioms ensuring the existence of transvections and dilatations.     

Static spin susceptibility in the <Emphasis Type="Italic">t</Emphasis>-<Emphasis Type="Italic">J</Emphasis> model

Based on the equation-of-motion method for the relaxation function expressed in terms of Hubbard operators, we evaluate the static spin susceptibility and correlation functions in the two-dimensional t-J model. We show that at zero temperature, a long-range antiferromagnetic order exists in the lattice for the hole concentration δ < δ_c ≃ 0.018. The calculated dependences of the spin susceptibility and correlation length on the temperature and hole concentration agree qualitatively with the experiment. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 538–550, September, 2007.     

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.     

Das kinematische Bild einer Regelfläche des einfach isotropen Raumes

TheClifford parallelism of the three-dimensional isotropic space J ₃ ⁽¹⁾ induces thekinematic mapping of an element of surface in J ₃ ⁽¹⁾ onto a pair of points in a fixed plane ₀ . By identifying a regular surface with the manifold of its osculating elements of surface thekinematic image of is defined and equivalent to an area preserving transformation of the plane ₀ . In this paper we will examine the connections between the isotropic invariants of a skew ruled surface and its kinematic image. Since the striction line y of exactly consists of the singular points of the kinematic images of the osculating planes of y are considered in addition. In this way we obtain a correspondence between the theory of ruled surfaces and the elementary geometry of two pairs of plane curves with a common middle curve.     

The Mean Perimeter of Some Random Plane Convex Sets Generated by a Brownian Motion

If C ₁ is the convex hull of the curve of a standard Brownian motion in the complex plane watched from 0 to 1, we consider the convex hulls of C ₁ and several rotations of it and compute the mean of the length of their perimeter by elementary calculations. This can be seen geometrically as a study of the exit time by a Brownian motion from certain polytopes having the unit circle as an inscribed one.     

On the transposed Rao’s flag-transitive plane of order 49

Rao’s flag-transitive plane π of order 49 and π ^t , the plane obtained by transposing matrices of a representative set of π, has been studied. It is shown that π ^t is flag-transitive, π ^t is not isomorphic to π, and π ^t is obtained from π by replacement of a net of degree 25. Further, (1) the flag-transitive planes associated with 1-spread sets S _2b and S _2a in the classified list of translation planes of order 49 enumerated by Mathon et al, are respectively isomorphic to π and π ^t (2) The flag-transitive planes associated with the 1-spread sets of 0an* in the classified list of translation planes of order 49 enumerated by Charnes et al are isomorphic to π and π ^t in some order.     

A Semifield Plane of Odd Order Admitting an Autotopism Subgroup Isomorphic to <Emphasis Type="Italic">A</Emphasis><Subscript>5</Subscript>

We develop an approach to constructing and classifying semifield projective planes with the use of a spread set. The famous conjecture is discussed on the solvability of the full collineation group of a finite semifield nondesarguesian plane. We construct a matrix representation of a spread set of a semifield plane of odd order admitting an autotopism subgroup isomorphic to the alternating group A₅ and find a series of semifield planes of odd order not admitting A₅.     

Cartesian groups not belonging to topological projective planes

A new class of topological Cartesian groups is introduced by bending the lines of the affine coordinate plane over an ordered skewfield countably infinitely often. These Cartesian groups belong to topological affine planes with continuous parallelism, but they need not yield a topological projective plane if the topology is not equal to the order topology of the skewfield, in contrast to the results of HARTMANN [6] for generalized Moulton planes. Dedicated to Professor Dr. Gerhard Grimeisen on his 70 ^th birthday