131,003 research outputs found
Freudenthal Gauge Theory
No full textWe present a novel gauge field theory, based on the Freudenthal Triple System (FTS), a ternary algebra with mixed symmetry (not completely symmetric) structure constants. The theory, named Freudenthal Gauge Theory (FGT), is invariant under two (off-shell) symmetries: the gauge Lie algebra constructed from the FTS triple product and a novel global non-polynomial symmetry, the so-called Freudenthal duality. Interestingly, a broad class of FGT gauge algebras is provided by the Lie algebras "of type e7" which occur as conformal symmetries of Euclidean Jordan algebras of rank 3, and as U-duality algebras of the corresponding (super)gravity theories in D = 4. We prove a No-Go Theorem, stating the incompatibility of the invariance under Freudenthal duality and the coupling to space-time vector and/or spinor fields, thus forbidding non-trivial supersymmetric extensions of FGT. We also briefly discuss the relation between FTS and the triple systems occurring in BLG-type theories, in particular focusing on superconformal Chern-Simons-matter gauge theories in D = 3.We present a novel gauge field theory, based on the Freudenthal Triple System (FTS), a ternary algebra with mixed symmetry (not completely symmetric) structure constants. The theory, named Freudenthal Gauge Theory (FGT), is invariant under two (off-shell) symmetries: the gauge Lie algebra constructed from the FTS triple product and a novel global non-polynomial symmetry, the so-called Freudenthal duality.We present a novel gauge field theory, based on the Freudenthal Triple System (FTS), a ternary algebra with mixed symmetry (not completely symmetric) structure constants. The theory, named Freudenthal Gauge Theory (FGT), is invariant under two (off-shell) symmetries: the gauge Lie algebra constructed from the FTS triple product and a novel global non-polynomial symmetry, the so-called Freudenthal duality. Interestingly, a broad class of FGT gauge algebras is provided by the Lie algebras "of type e7" which occur as conformal symmetries of Euclidean Jordan algebras of rank 3, and as U-duality algebras of the corresponding (super)gravity theories in D = 4. We prove a No-Go Theorem, stating the incompatibility of the invariance under Freudenthal duality and the coupling to space-time vector and/or spinor fields, thus forbidding non-trivial supersymmetric extensions of FGT. We also briefly discuss the relation between FTS and the triple systems occurring in BLG-type theories, in particular focusing on superconformal Chern-Simons-matter gauge theories in D = 3
Learning geometry in self-made tutorials: the impact of producing mathematical videos on emotions, motivation and achievement in mathematical learning
No full textBarton D. Learning geometry in self-made tutorials: the impact of producing mathematical videos on emotions, motivation and achievement in mathematical learning. In: Jankvist UT, van den Heuvel-Panhuizen M, Veldhuis M, Utrecht University and ERME, eds. Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education. Utrecht, the Netherlands: Freudenthal Group & Freudenthal Institute; 2019: 1401-1402
Black holes admitting a Freudenthal dual
No full textThe quantized charges x of four-dimensional stringy black holes may be assigned to elements of an integral Freudenthal triple system whose automorphism group is the corresponding U duality and whose U-invariant quartic norm Δ(x) determines the lowest-order entropy. Here, we introduce a Freudenthal duality x→˜x, for which ˜˜x=−x. Although distinct from U duality, it nevertheless leaves Δ(x) invariant. However, the requirement that ˜x be an integer restricts us to the subset of black holes for which Δ(x) is necessarily a perfect square. The issue of higher-order corrections remains open as some, but not all, of the discrete U-duality invariants are Freudenthal invariant. Similarly, the quantized charges A of five-dimensional black holes and strings may be assigned to elements of an integral Jordan algebra, whose cubic norm N(A) determines the lowest-order entropy. We introduce an analogous Jordan dual A⋆, with N(A) necessarily a perfect cube, for which A⋆⋆=A and which leaves N(A) invariant. The two dualities are related by a 4D/5D lift
Black holes and general Freudenthal transformations
No full textWe study General Freudenthal Transformations (GFT) on black hole solutions in Einstein-Maxwell-Scalar (super)gravity theories with global symmetry of type E7. GFT can be considered as a 2-parameter, a, b ∈ ℝ, generalisation of Freudenthal duality: x→xF=ax+bx~ , where x is the vector of the electromagnetic charges, an element of a Freudenthal triple system (FTS), carried by a large black hole and x~ is its Freudenthal dual. These transformations leave the Bekenstein-Hawking entropy invariant up to a scalar factor given by a2 ± b2. For any x there exists a one parameter subset of GFT that leave the entropy invariant, a2 ± b2 = 1, defining the subgroup of Freudenthal rotations. The Freudenthal plane defined by spanℝ{x, x~ } is closed under GFT and is foliated by the orbits of the Freudenthal rotations. Having introduced the basic definitions and presented their properties in detail, we consider the relation of GFT to the global symmetries or U-dualities in the context of supergravity. We consider explicit examples in pure supergravity, axion-dilaton theories and N = 2, D = 4 supergravities obtained from D = 5 by dimensional reductions associated to (non-degenerate) reduced FTS’s descending from cubic Jordan Algebras
Simulating the referential properties of Dutch, German and English Root Infinitives in MOSAIC
No full textChildren learning many languages go through an Optional Infinitive stage in which they produce non-finite verb forms in contexts in which a finite verb form is required (e.g. ‘That go there’ instead of ‘That goes there’). MOSAIC (Model of Syntax Acquisition in Children) is a computational model of language learning that successfully simulates the developmental patterning of the Optional Infinitive (OI) phenomenon in English, Dutch, German and Spanish (Freudenthal, Pine, Aguado-Orea & Gobet, 2007). In the present study, MOSAIC is applied to the simulation of certain subtle but theoretically important phenomena in the cross-linguistic patterning of the OI phenomenon that are typically assumed to require a more complex formal analysis. MOSAIC is shown to successfully simulate 1) The Modal Reference Effect: the finding that Dutch and German children tend to use Root Infinitives in modal contexts, 2) The Eventivity constraint: the finding that Dutch and German Root Infinitives refer predominantly to actions rather than static situations, and 3) The absence or reduced size of these effects in English. These results provide strong support for input-driven explanations of the Modal Reference Effect as well as MOSAIC’s mechanism for producing Root Infinitives, and the wider claim that it is possible to explain key aspects of children’s early multi-word speech in terms of the interaction between a resource-limited distributional learning mechanism and the surface properties of the language to which children are exposed
Freudenthal Duality and Generalized Special Geometry
No full textFreudenthal duality, introduced in L. Borsten, D. Dahanayake, M. J. Duff and W. Rubens, Phys.Rev. D80, 026003 (2009), and defined as an anti-involution on the dyonic charge vector in d = 4 space-time dimensions for those dualities admitting a quartic invariant, is proved to be a symmetry not only of the classical Bekenstein-Hawking entropy but also of the critical points of the black hole potential. Furthermore, Freudenthal duality is extended to any generalized special geometry, thus encompassing all N > 2 supergravities, as well as N = 2 generic special geometry, not necessarily having a coset space structure.Freudenthal duality, introduced in Borsten et al. (2009) [1] and defined as an anti-involution on the dyonic charge vector in d=4 space–time dimensions for those dualities admitting a quartic invariant, is proved to be a symmetry not only of the classical Bekenstein–Hawking entropy but also of the critical points of the black hole potential.Freudenthal duality, introduced in L. Borsten, D. Dahanayake, M. J. Duff and W. Rubens, Phys.Rev. D80, 026003 (2009), and defined as an anti-involution on the dyonic charge vector in d = 4 space-time dimensions for those dualities admitting a quartic invariant, is proved to be a symmetry not only of the classical Bekenstein-Hawking entropy but also of the critical points of the black hole potential. Furthermore, Freudenthal duality is extended to any generalized special geometry, thus encompassing all N > 2 supergravities, as well as N = 2 generic special geometry, not necessarily having a coset space structure
Derivations and local derivations on Freudenthal almost f-algebras
No full textAbstractLet A be an Archimedean f-algebra and let N(A) be the set of all nilpotent elements of A. Colville et al. [4] proved that a positive linear map d:A→A is a derivation if and only if d(A)⊂N(A) and d(A2)={0}, where A2 is the set of all products ab in A.In this paper, we establish a result corresponding to the Colville–Davis–Keimel theorem for arbitrary derivation d on Freudenthal almost f-algebras. Moreover, we prove that any local derivation on a Freudenthal almost f-algebra A, such that N(A)={a∈A;a2=0}, is a derivation
Hyperkähler manifolds from the Tits–Freudenthal magic square
No full textInternational audienceWe suggest a way to associate to each Lie algebra of type G 2 , D 4 , F 4 , E 6 , E 7 , E 8 a family of polarized hyperkähler fourfolds, constructed as parametrizing certain families of cycles of hyperplane sections of certain homogeneous or quasi-homogeneous varieties. These cycles are modeled on the Legendrian varieties studied by Freudenthal in his geometric approach to the celebrated Tits-Freudenthal magic square of Lie algebras
'Practicing place value': How children interpret and use virtual representations and features
No full textSchulz A, Walter D. 'Practicing place value': How children interpret and use virtual representations and features. In: Thomas Jankvist U, van den Heuvel-Panhuizen M, Veldhuis M, eds. {Eleventh Congress of the European Society for Research in Mathematics Education}. Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (CERME11). Vol TWG16. Utrecht, Netherlands: {Freudenthal Group}; 2019
On a weak Freudenthal spectral theorem
No full textsummary:Let be an Archimedean Riesz space and \Cal P(X) its Boolean algebra of all band projections, and put \Cal P_{e}=\{P e:P\in \Cal P(X)\} and \Cal B_{e}=\{x\in X: x\wedge (e-x)=0\}, . is said to have Weak Freudenthal Property (\text{}) provided that for every the lattice lin\, \Cal P_{e} is order dense in the principal band . This notion is compared with strong and weak forms of Freudenthal spectral theorem in Archimedean Riesz spaces, studied by Veksler and Lavrič, respectively. \text{} is equivalent to -denseness of \Cal P_{e} in \Cal B_{e} for every , and every Riesz space with sufficiently many projections has \text{} (THEOREM)
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