On characteristic equations, trace identities and Casimir operators of simple Lie algebras

Two approaches are developed to exploit, for simple complex or compact real Lie algebras g, the information that stems from the characteristic equations of representation matrices and Casimir operators. These approaches are selected so as to be viable not only for `small' Lie algebras and suitable for treatment by computer algebra. A very large body of new results emerges in the forms, a) of identities of a tensorial nature, involving structure constants etc. of g, b) of trace identities for powers of matrices of the adjoint and defining representations of g, c) of expressions of non-primitive Casimir operators of g in terms of primitive ones. The methods are sufficiently tractable to allow not only explicit proof by hand of the non-primitive nature of the quartic Casimir of g2, f4, e6, but also e.g. of that of the tenth order Casimir of f4.Comment: 39 pages, 8 tables, late

Alternatieve vormen van financiering : kansen voor het multifunctionele landbouwbedrijf voor niet-bancaire financieringsvormen

Uit onderzoek blijkt dat de meeste multifunctionele ondernemers vrij makkelijk een lening bij de bank krijgen, zeker met een goed bedrijfsplan. Maar banken kunnen ook wat terughoudend zijn, omdat de ondernemer geen gegarandeerde afzet heeft, geen grond of gebouwen als onderpand heeft of er minder bekend is over multifunctionele bedrijfsvormen bij de bank. In deze brochure staan voorbeelden van verschillende vormen van alternatieve financiering, vergezeld van een stappenplan voor een goede waarborging

Harmonic Superspace, Minimal Unitary Representations and Quasiconformal Groups

We show that there is a remarkable connection between the harmonic superspace (HSS) formulation of N=2, d=4 supersymmetric quaternionic Kaehler sigma models that couple to N=2 supergravity and the minimal unitary representations of their isometry groups. In particular, for N=2 sigma models with quaternionic symmetric target spaces of the form G/HXSU(2) we establish a one-to-one mapping between the Killing potentials that generate the isometry group G under Poisson brackets in the HSS formulation and the generators of the minimal unitary representation of G obtained by quantization of its geometric realization as a quasiconformal group. Quasiconformal extensions of U-duality groups of four dimensional N=2, d=4 Maxwell-Einstein supergravity theories (MESGT) had been proposed as spectrum generating symmetry groups earlier. We discuss some of the implications of our results, in particular, for the BPS black hole spectra of 4d, N=2 MESGTs.Comment: 20 pages; Latex file: references added; minor cosmetic change

Representations of the exceptional and other Lie algebras with integral eigenvalues of the Casimir operator

The uniformity, for the family of exceptional Lie algebras g, of the decompositions of the powers of their adjoint representations is well-known now for powers up to the fourth. The paper describes an extension of this uniformity for the totally antisymmetrised n-th powers up to n=9, identifying (see Tables 3 and 6) families of representations with integer eigenvalues 5,...,9 for the quadratic Casimir operator, in each case providing a formula (see eq. (11) to (15)) for the dimensions of the representations in the family as a function of D=dim g. This generalises previous results for powers j and Casimir eigenvalues j, j<=4. Many intriguing, perhaps puzzling, features of the dimension formulas are discussed and the possibility that they may be valid for a wider class of not necessarily simple Lie algebras is considered.Comment: 16 pages, LaTeX, 1 figure, 9 tables; v2: presentation improved, typos correcte

Generalized spacetimes defined by cubic forms and the minimal unitary realizations of their quasiconformal groups

We study the symmetries of generalized spacetimes and corresponding phase spaces defined by Jordan algebras of degree three. The generic Jordan family of formally real Jordan algebras of degree three describe extensions of the Minkowskian spacetimes by an extra "dilatonic" coordinate, whose rotation, Lorentz and conformal groups are SO(d-1), SO(d-1,1) XSO(1,1) and SO(d,2)XSO(2,1), respectively. The generalized spacetimes described by simple Jordan algebras of degree three correspond to extensions of Minkowskian spacetimes in the critical dimensions (d=3,4,6,10) by a dilatonic and extra (2,4,8,16) commuting spinorial coordinates, respectively. The Freudenthal triple systems defined over these Jordan algebras describe conformally covariant phase spaces. Following hep-th/0008063, we give a unified geometric realization of the quasiconformal groups that act on their conformal phase spaces extended by an extra "cocycle" coordinate. For the generic Jordan family the quasiconformal groups are SO(d+2,4), whose minimal unitary realizations are given. The minimal unitary representations of the quasiconformal groups F_4(4), E_6(2), E_7(-5) and E_8(-24) of the simple Jordan family were given in our earlier work hep-th/0409272.Comment: A typo in equation (37) corrected and missing titles of some references added. Version to be published in JHEP. 38 pages, latex fil

HAG1 and SWI3A/B control of male germ line development in P. patens suggests conservation of epigenetic reproductive control across land plants

Bryophytes as models to study the male germ line: loss-of-function mutants of epigenetic regulators HAG1 and SWI3a/b demonstrate conserved function in sexual reproduction

Reverse Engineering Time Discrete Finite Dynamical Systems: A Feasible Undertaking?

With the advent of high-throughput profiling methods, interest in reverse engineering the structure and dynamics of biochemical networks is high. Recently an algorithm for reverse engineering of biochemical networks was developed by Laubenbacher and Stigler. It is a top-down approach using time discrete dynamical systems. One of its key steps includes the choice of a term order, a technicality imposed by the use of Gröbner-bases calculations. The aim of this paper is to identify minimal requirements on data sets to be used with this algorithm and to characterize optimal data sets. We found minimal requirements on a data set based on how many terms the functions to be reverse engineered display. Furthermore, we identified optimal data sets, which we characterized using a geometric property called “general position”. Moreover, we developed a constructive method to generate optimal data sets, provided a codimensional condition is fulfilled. In addition, we present a generalization of their algorithm that does not depend on the choice of a term order. For this method we derived a formula for the probability of finding the correct model, provided the data set used is optimal. We analyzed the asymptotic behavior of the probability formula for a growing number of variables n (i.e. interacting chemicals). Unfortunately, this formula converges to zero as fast as , where and . Therefore, even if an optimal data set is used and the restrictions in using term orders are overcome, the reverse engineering problem remains unfeasible, unless prodigious amounts of data are available. Such large data sets are experimentally impossible to generate with today's technologies

Teaching undergraduate students gynecological and obstetrical examination skills: the patient's opinion

Introduction Our study assesses the patients’ opinion about gynecological examination performed by undergraduate students (UgSts). This assessment will be used in improving our undergraduate training program. A positive opinion would mean a lower chance of a patient refusing to be examined by a tutor or student, taking into account vaginal examination (VE). Materials and methods We performed a prospective cross-sectional survey on 1194 patients, consisting of outpatient and inpatient at the departments of obstetrics and gynecology from November 2015 to May 2016. The questionnaire consisted of 46 questions. Besides demographic data, we assessed the mindset of patients regarding the involvement of undergradu ate student (UgSt) in gynecological and obstetrical examinations. We used SPSS version 23 for the statistical analysis. For reporting the data, we followed the STROBE statement of reporting observational studies. Results The median age was 38 years having a median of one child. 34% presented due to obstetrical problems, 38% due to gynecological complaints, and 19% due to known gynecological malignancies. Generally, we retrieved a positive opinion of patients towards the involvement of students in gynecological and obstetrical examination under supervision in 2/3 of the cases. Conclusions There is no reason to exclude medical UgSts from gynecological and obstetrical examinations after obtaining a written or oral consent

Topological wave functions and heat equations

It is generally known that the holomorphic anomaly equations in topological string theory reflect the quantum mechanical nature of the topological string partition function. We present two new results which make this assertion more precise: (i) we give a new, purely holomorphic version of the holomorphic anomaly equations, clarifying their relation to the heat equation satisfied by the Jacobi theta series; (ii) in cases where the moduli space is a Hermitian symmetric tube domain $G/K$, we show that the general solution of the anomaly equations is a matrix element \IP{\Psi | g | \Omega} of the Schr\"odinger-Weil representation of a Heisenberg extension of $G$, between an arbitrary state $\bra{\Psi}$ and a particular vacuum state $\ket{\Omega}$. Based on these results, we speculate on the existence of a one-parameter generalization of the usual topological amplitude, which in symmetric cases transforms in the smallest unitary representation of the duality group $G'$ in three dimensions, and on its relations to hypermultiplet couplings, nonabelian Donaldson-Thomas theory and black hole degeneracies.Comment: 50 pages; v2: small typos fixed, references added; v3: cosmetic changes, published version; v4: typos fixed, small clarification adde