Symmetry, Integrability and Geometry: Methods and Applications, 2007
We survey some recent results concerning Stanilov-Tsankov-Videv theory, conformal Osserman geomet... more We survey some recent results concerning Stanilov-Tsankov-Videv theory, conformal Osserman geometry, and Walker geometry which relate algebraic properties of the curvature operator to the underlying geometry of the manifold.
We agree with the interpretation of W. Schlosser, that the Nebra Sky Disc is a reminder of a meth... more We agree with the interpretation of W. Schlosser, that the Nebra Sky Disc is a reminder of a method of determining a start date (and possibly also an end date) of the farming year. We extend this interpretation. We think that we found the constellation Taurus on the Disc, which forms by addition of three stars from the constellation Gemini the pattern of a plough of Bronze Age. Moreover we found a line on the disc consisting of the stars epsilon Gem, theta Aur, beta Aur and alpha Aur, which we called the Auriga line. We think that the Nebra people used the Auriga line to determine the day (which we call beta day) on which the Pleiades are vertically below beta Aur at dusk in February. We found a second representation of the Auriga line on the Disc where the distance ratios between the stars are very precisely equal to the distance ratios in the sky, and where the Pleiades are vertically below beta Aur. This proves that the Nebra people must have measured the distances, and that our hypothesis is correct.
The beta day could have been used to harmonize a lunisolar calendar with the solar year. However, the most likely possibility seems to us that a good sowing date could be determined by setting the sowing on the second round lunar phase after the beta day. (By round lunar phases we mean the full moon and the new moon.) Such a sowing date makes it possible to start a week count based on the lunar phases with the sowing in order to determine other agricultural dates.
The astronomical knowledge for this procedure can be gained by astronomical observations only. No mathematical calculations and import of knowledge from a Mediterranean culture are necessary.
One can determine a point on the ecliptic (beta point) such that the beta day occurs when the sun is at the beta point. The beta point moves only slowly relative to the vernal equinox. However, since the Bronze Age, the beta point and vernal equinox have swapped their order on the ecliptic.
Conformally invariant gravitational field equations on the one hand and fourth order field equati... more Conformally invariant gravitational field equations on the one hand and fourth order field equations on the other were discussed in the early history of general relativity (Weyl Einstein, Bach et al.) and have recently gained some new interest (Deser, P. Gunther, Treder, et al.). The equations Bab = 0 or Ba,q = xT=b, where Ba,g denotes the Bach tensor and Tap a suitable energy-momentum tensor, possess both the mentioned properties. We construct exact solutions ds' = g,pdxcLdx@ of the Bach equations: (2,2)-decomposable, centrally symmetric and pp-wave solutions. The gravitational field g,b is coupled by Bus = xT,p to an electromagnetic field Fa,g = -FM obeying the Maxwell equations or to a neutrino field VA obeying the Weyl equations respectively. Among interesting new metrics ds' there appear some physically well-known ones, such as the De Sitter universe, the Weyl-Trefftz metric, and the plane-fronted gravitational waves with parallel rays (ppwaves) known from Einstein's theory. The solutions are built up by means of special techniques: A separation method for (2,2)-decomposable solutions, simplification of centrally symmetric metrics by a suitable conformal transformation, and complex function methods for pp-wave solutions. ' Notations and conventions are explained later in this introduction. 'I This is a very special an&z. (10) with a (2,2)-decomposable metric has many other solutions (cf. Ml.
We show that the symmetry classes of torsion-free covariant derivatives ∇T of r-times covariant t... more We show that the symmetry classes of torsion-free covariant derivatives ∇T of r-times covariant tensor fields T can be characterized by Littlewood-Richardson products σ[1] where σ is a representation of the symmetric group S r which is connected with the symmetry class of and all primitive idempotents belonging to that sum can be calculated from a generating idempotent e of the symmetry class of T by means of the irreducible characters or of a discrete Fourier transform of S r+1 . We apply these facts to derivatives ∇S, ∇A of symmetric or alternating tensor fields. The symmetry classes of the differences ∇S -sym(∇S) and ∇A-alt(∇A) = ∇A-dA are characterized by Young frames (r, 1) ⊢ r + 1 and (2, 1 r-1 ) ⊢ r + 1, respectively. However, while the symmetry class of ∇A -alt(∇A) can be generated by Young symmetrizers of (2, 1 r-1 ), no Young symmetrizer of (r, 1) generates the symmetry class of ∇S -sym(∇S). Furthermore we show in the case r = 2 that ∇S -sym(∇S) and ∇A -alt(∇A) can be applied in generator formulas of algebraic covariant derivative curvature tensors. For certain symbolic calculations we used the Mathematica packages Ricci and PERMS. 1 A first proof that y * t ′ (S ⊗ S ′ ) and y * t ′ (S ′ ⊗ S) are generators for R ′ was given by P. B. Gilkey [16, p.236].
We demonstrate the use of several tools from Algebraic Combinatorics such as Young tableaux, symm... more We demonstrate the use of several tools from Algebraic Combinatorics such as Young tableaux, symmetry operators, the Littlewood-Richardson rule and discrete Fourier transforms of symmetric groups in investigations of algebraic curvature tensors. In we constructed and investigated generators of algebraic curvature tensors and algebraic covariant derivative curvature tensors. These investigations followed the example of the paper [14] by S.A. Fulling, R.C. King, B.G.Wybourne and C.J. Cummins and applied tools from Algebraic Combinatorics such as Young tableaux, symmetry operators (in particular Young symmetrizers), the Littlewood-Richardson rule, but also discrete Fourier transforms of symmetric groups. The present paper is a short summary of ] in which we want to demonstrate the use of these methods.
For a positive definite fundamental tensor all known examples of Osserman algebraic curvature ten... more For a positive definite fundamental tensor all known examples of Osserman algebraic curvature tensors have a typical structure. They can be produced from a metric tensor and a finite set of skew-symmetric matrices which fulfil Clifford commutation relations. We show by means of Young symmetrizers and a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins that every algebraic curvature tensor has a structure which is very similar to that of the above Osserman curvature tensors. We verify our results by means of the Littlewood-Richardson rule and plethysms. For certain symbolic calculations we used the Mathematica packages MathTensor, Ricci and PERMS.
We consider generators of algebraic curvature tensors R which can be constructed by a Young symme... more We consider generators of algebraic curvature tensors R which can be constructed by a Young symmetrization of product tensors U ⊗ w or w ⊗ U , where U and w are covariant tensors of order 3 and 1. We assume that U belongs to a class of the infinite set S of irreducible symmetry classes characterized by the partition (2 1). We show that the set S contains exactly one symmetry class S 0 ∈ S whose elements U ∈ S 0 can not play the role of generators of tensors R. The tensors U of all other symmetry classes from S \ {S 0 } can be used as generators for tensors R. Using Computer Algebra we search for such generators whose coordinate representations are polynomials with a minimal number of summands. For a generic choice of the symmetry class of U we obtain lengths of 8 summands. In special cases these numbers can be reduced to the minimum 4. If this minimum occurs then U admits an index commutation symmetry. Furthermore minimal lengths are possible if U is formed from torsion-free covariant derivatives of alternating 2-tensor fields. We apply ideals and idempotents of group rings C[S r ] of symmetric groups S r , Young symmetrizers, discrete Fourier transforms and Littlewood-Richardson products. For symbolic calculations we used the Mathematica packages Ricci and PERMS.
We consider generators of algebraic covariant derivative curvature tensors R ′ which can be const... more We consider generators of algebraic covariant derivative curvature tensors R ′ which can be constructed by a Young symmetrization of product tensors W ⊗ U or U ⊗ W , where W and U are covariant tensors of order 2 and 3. W is a symmetric or alternating tensor whereas U belongs to a class of the infinite set S of irreducible symmetry classes characterized by the partition (2 1). Using Computer Algebra we search for such generators whose coordinate representations are polynomials with a minimal number of summands. For a generic choice of the symmetry class of U we obtain lengths of 16 or 20 summands if W is symmetric or skew-symmetric, respectively. In special cases these numbers can be reduced to the minima 12 or 10. If these minima occur then U admits an index commutation symmetry. Furthermore minimal lengths are possible if U is formed from torsion-free covariant derivatives of symmetric or alternating 2-tensor fields. Foundation of our investigations is a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins about a Young symmetrizer that generates the symmetry class of algebraic covariant derivative curvature tensors. Furthermore we apply ideals and idempotents in group rings C[S r ] and discrete Fourier transforms for symmetric groups S r . For symbolic calculations we used the Mathematica packages Ricci and PERMS.
We show that the space of algebraic covariant derivative curvature tensors R ′ is generated by Yo... more We show that the space of algebraic covariant derivative curvature tensors R ′ is generated by Young symmetrized product tensors T ⊗ T or T ⊗ T , where T and T are covariant tensors of order 2 and 3 whose symmetry classes are irreducible and characterized by the following pairs of partitions: {(2), (3)}, {(2), (2 1)} or {(1 2 ), (2 1)}. Each of the partitions (2), ( ) and ( 2 ) describes exactly one symmetry class, whereas the partition (2 1) characterizes an infinite set S of irreducible symmetry classes. This set S contains exactly one symmetry class S 0 ∈ S whose elements T ∈ S 0 can not play the role of generators of tensors R ′ . The tensors T of all other symmetry classes from S \ {S 0 } can be used as generators for tensors R ′ . Foundation of our investigations is a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins about a Young symmetrizer that generates the symmetry class of algebraic covariant derivative curvature tensors. Furthermore we apply ideals and idempotents in group rings C[S r ], the Littlewood-Richardson rule and discrete Fourier transforms for symmetric groups S r . For certain symbolic calculations we used the Mathematica packages Ricci and PERMS.
Séminaire Lotharingien de Combinatoire [electronic only], 2001
Symmetry properties of r-times covariant tensors T can be described by certain linear subspaces W... more Symmetry properties of r-times covariant tensors T can be described by certain linear subspaces W of the group ring K[S r ] of a symmetric group S r . If for a class of tensors T such a W is known, the elements of the orthogonal subspace W ⊥ of W within the dual space K[S r ] * of K[S r ] yield linear identities needed for a treatment of the term combination problem for the coordinates of the T . We give the structure of these W for every situation which appears in symbolic tensor calculations by computer. Characterizing idempotents of such W can be determined by means of an ideal decomposition algorithm which works in every semisimple ring up to an isomorphism. Furthermore, we use tools such as the Littlewood-Richardson rule, plethysms and discrete Fourier transforms for S r to increase the efficience of calculations. All described methods were implemented in a Mathematica package called PERMS.
Singularity-free static centrally symmetric solutions of some fourth order gravitational field equations
Astronomische Nachrichten, 1983
Die von TREDER vorgeschlagenen Gravitationsfeldgleichungen 4. Ordnung mit einer Linearkombination... more Die von TREDER vorgeschlagenen Gravitationsfeldgleichungen 4. Ordnung mit einer Linearkombination des Bachtensors und des Einsteintensors auf der linken Seite besitzen statische zentralsymmetrische Lösungen, welche in einer Umgebung des Symmetriezentrums analytisch und nicht flach sind.
We consider the problern to determine normal forms of the Coordinates of covariant tensors T E Tr... more We consider the problern to determine normal forms of the Coordinates of covariant tensors T E Tr V of order r over a finite-dimensional!K-vector space, lK = IR, C. A connection between such tensors and the group ring IK[Sr] can be established by assigning a group ring element n := L:pESr T(vp(l), . . .
We consider the problern to determine normal forms of the Coordinates of covariant tensors T E Tr... more We consider the problern to determine normal forms of the Coordinates of covariant tensors T E Tr V of order r over a finite-dimensional!K-vector space, lK = IR, C. A connection between such tensors and the group ring IK[Sr] can be established by assigning a group ring element n := L:pESr T(vp(l), . . .
Abstract. For a positive definite fundamental tensor all known examples of Osserman algebraic cur... more Abstract. For a positive definite fundamental tensor all known examples of Osserman algebraic curvature tensors have a typical structure. They can be produced from a metric tensor and a finite set of skew-symmetric matrices which fulfil Clifford commutation relations. We show by means of Young symmetrizers
We consider generators of algebraic covariant derivative curvature tensors R ′ which can be const... more We consider generators of algebraic covariant derivative curvature tensors R ′ which can be constructed by a Young symmetrization of product tensors W ⊗ U or U ⊗ W , where W and U are covariant tensors of order 2 and 3. W is a symmetric or alternating tensor whereas U belongs to a class of the infinite set S of irreducible symmetry classes characterized by the partition (2 1). Using Computer Algebra we search for such generators whose coordinate representations are polynomials with a minimal number of summands. For a generic choice of the symmetry class of U we obtain lengths of 16 or 20 summands if W is symmetric or skew-symmetric, respectively. In special cases these numbers can be reduced to the minima 12 or 10. If these minima occur then U admits an index commutation symmetry. Furthermore minimal lengths are possible if U is formed from torsion-free covariant derivatives of symmetric or alternating 2-tensor fields. Foundation of our investigations is a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins about a Young symmetrizer that generates the symmetry class of algebraic covariant derivative curvature tensors. Furthermore we apply ideals and idempotents in group rings C[S r ] and discrete Fourier transforms for symmetric groups S r . For symbolic calculations we used the Mathematica packages Ricci and PERMS.
Frontiers in Artificial Intelligence and Applications, 2003
We show that the space of algebraic covariant derivative curvature tensors R ′ is generated by Yo... more We show that the space of algebraic covariant derivative curvature tensors R ′ is generated by Young symmetrized product tensors T ⊗ T or T ⊗ T , where T and T are covariant tensors of order 2 and 3 whose symmetry classes are irreducible and characterized by the following pairs of partitions: {(2), (3)}, {(2), (2 1)} or {(1 2 ), (2 1)}. Each of the partitions (2), ( ) and ( 2 ) describes exactly one symmetry class, whereas the partition (2 1) characterizes an infinite set S of irreducible symmetry classes. This set S contains exactly one symmetry class S 0 ∈ S whose elements T ∈ S 0 can not play the role of generators of tensors R ′ . The tensors T of all other symmetry classes from S \ {S 0 } can be used as generators for tensors R ′ . Foundation of our investigations is a theorem of S. A. Fulling, R. C. King, B. G. Wybourne and C. J. Cummins about a Young symmetrizer that generates the symmetry class of algebraic covariant derivative curvature tensors. Furthermore we apply ideals and idempotents in group rings C[S r ], the Littlewood-Richardson rule and discrete Fourier transforms for symmetric groups S r . For certain symbolic calculations we used the Mathematica packages Ricci and PERMS.
We demonstrate the use of several tools from Algebraic Combinatorics such as Young tableaux, symm... more We demonstrate the use of several tools from Algebraic Combinatorics such as Young tableaux, symmetry operators, the Littlewood-Richardson rule and discrete Fourier transforms of symmetric groups in investigations of algebraic curvature tensors. In we constructed and investigated generators of algebraic curvature tensors and algebraic covariant derivative curvature tensors. These investigations followed the example of the paper [14] by S.A. Fulling, R.C. King, B.G.Wybourne and C.J. Cummins and applied tools from Algebraic Combinatorics such as Young tableaux, symmetry operators (in particular Young symmetrizers), the Littlewood-Richardson rule, but also discrete Fourier transforms of symmetric groups. The present paper is a short summary of ] in which we want to demonstrate the use of these methods.
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Papers by Bernd Fiedler
found the constellation Taurus on the Disc, which forms by addition of
three stars from the constellation Gemini the pattern of a plough of
Bronze Age. Moreover we found a line on the disc consisting of the
stars epsilon Gem, theta Aur, beta Aur and alpha Aur, which we called the Auriga line. We think that the Nebra people used the Auriga line to determine the day (which we call beta day) on which the Pleiades are vertically below beta Aur at dusk in February. We found a second representation of the Auriga line on the Disc where the distance ratios between the stars are very precisely equal to the distance ratios in the sky, and where the Pleiades are vertically below beta Aur. This proves that the Nebra people must have measured the distances, and that our hypothesis is correct.
The beta day could have been used to harmonize a lunisolar calendar with the solar year. However, the most likely possibility seems to us that a good sowing date could be determined by setting the sowing on the second round lunar phase after the beta day. (By round lunar phases we mean the full moon and the new moon.) Such a sowing date makes it possible to start a week count based on the lunar phases with the sowing in order to determine other agricultural dates.
The astronomical
knowledge for this procedure can be gained by astronomical
observations only. No mathematical calculations and import of knowledge from a Mediterranean culture are necessary.
One can determine a point on the ecliptic (beta point) such that the beta day occurs when the sun is at the beta point. The beta point moves only slowly relative to the vernal equinox. However, since the Bronze Age, the beta point and vernal equinox have swapped their order on the ecliptic.