Products, coproducts and singular value decomposition

Products and coproducts may be recognized as morphisms in a monoidal tensor category of vector spaces. To gain invariant data of these morphisms, we can use singular value decomposition which attaches singular values, ie generalized eigenvalues, to these maps. We show, for the case of Grassmann and Clifford products, that twist maps significantly alter these data reducing degeneracies. Since non group like coproducts give rise to non classical behavior of the algebra of functions, ie make them noncommutative, we hope to be able to learn more about such geometries. Remarkably the coproduct for positive singular values of eigenvectors in $A$ yields directly corresponding eigenvectors in A\otimes A.Comment: 17 pages, three eps-figure

A Hopf laboratory for symmetric functions

An analysis of symmetric function theory is given from the perspective of the underlying Hopf and bi-algebraic structures. These are presented explicitly in terms of standard symmetric function operations. Particular attention is focussed on Laplace pairing, Sweedler cohomology for 1- and 2-cochains, and twisted products (Rota cliffordizations) induced by branching operators in the symmetric function context. The latter are shown to include the algebras of symmetric functions of orthogonal and symplectic type. A commentary on related issues in the combinatorial approach to quantum field theory is given.Comment: 29 pages, LaTeX, uses amsmat

On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form

Clifford algebras are naturally associated with quadratic forms. These algebras are Z_2-graded by construction. However, only a Z_n-gradation induced by a choice of a basis, or even better, by a Chevalley vector space isomorphism Cl(V) \bigwedge V and an ordering, guarantees a multi-vector decomposition into scalars, vectors, tensors, and so on, mandatory in physics. We show that the Chevalley isomorphism theorem cannot be generalized to algebras if the Z_n-grading or other structures are added, e.g., a linear form. We work with pairs consisting of a Clifford algebra and a linear form or a Z_n-grading which we now call 'Clifford algebras of multi-vectors' or 'quantum Clifford algebras'. It turns out, that in this sense, all multi-vector Clifford algebras of the same quadratic but different bilinear forms are non-isomorphic. The usefulness of such algebras in quantum field theory and superconductivity was shown elsewhere. Allowing for arbitrary bilinear forms however spoils their diagonalizability which has a considerable effect on the tensor decomposition of the Clifford algebras governed by the periodicity theorems, including the Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Cl_{p,q} which can be decomposed in the symmetric case into a tensor product Cl_{p-1,q-1} \otimes Cl_{1,1}. The general case used in quantum field theory lacks this feature. Theories with non-symmetric bilinear forms are however needed in the analysis of multi-particle states in interacting theories. A connection to q-deformed structures through nontrivial vacuum states in quantum theories is outlined.Comment: 25 pages, 1 figure, LaTeX, {Paper presented at the 5th International Conference on Clifford Algebras and their Applications in Mathematical Physics, Ixtapa, Mexico, June 27 - July 4, 199

Management of infertility in a patient presenting with ovarian dysfunction and McCune-Albright syndrome

Persistent autonomous ovarian dysfunction in McCune-Albright syndrome (MAS) patients is associated with the development of multiple dominant follicles, premature luteinization, cyst formation, and anovulatory infertility. Due to the mosaic distribution of the mutation, ovaries may be unequally affected. In the current patient, the least affected ovary became quiescent upon GnRH agonist-induced gonadotropin suppression. Normoovulatory cycles were restored after subsequent removal of the affected right ovary, and a pregnancy was established within 3 months. A healthy unaffected girl was born at term after an uneventful pregnancy. The placental tissue was normal, and the mutation was not detected in the placenta, umbilical cord structures, or umbilical cord blood. GnRH analog administration may help to identify those MAS patients who might benefit from unilateral ovariectomy. Because a healthy baby was born, evidence is provided suggesting that MAS is not passed on to the children from the parents

Quantum field theory and Hopf algebra cohomology

We exhibit a Hopf superalgebra structure of the algebra of field operators of quantum field theory (QFT) with the normal product. Based on this we construct the operator product and the time-ordered product as a twist deformation in the sense of Drinfeld. Our approach yields formulas for (perturbative) products and expectation values that allow for a significant enhancement in computational efficiency as compared to traditional methods. Employing Hopf algebra cohomology sheds new light on the structure of QFT and allows the extension to interacting (not necessarily perturbative) QFT. We give a reconstruction theorem for time-ordered products in the spirit of Streater and Wightman and recover the distinction between free and interacting theory from a property of the underlying cocycle. We also demonstrate how non-trivial vacua are described in our approach solving a problem in quantum chemistry.Comment: 39 pages, no figures, LaTeX + AMS macros; title changed, minor corrections, references update

Nonsupplemented luteal phase characteristics after the administration of recombinant human chorionic gonadotropin, recombinant luteinizing hormone, or gonadotropin-releasing hormone (GnRH) agonist to induce final oocyte maturation in in vitro fertilization patients after ovarian stimulation with recombinant follicle-stimulating hormone and GnRH antagonist cotreatment

Replacing GnRH agonist cotreatment for the prevention of a premature rise in LH during ovarian stimulation for in vitro fertilization (IVF) by the late follicular phase administration of GnRH antagonist may render supplementation of the luteal phase redundant, because of the known rapid recovery of pituitary function after antagonist cessation. This randomized two-center study was performed to compare nonsupplemented luteal phase characteristics after three different strategies for inducing final oocyte maturation. Forty patients underwent ovarian stimulation using recombinant (r-)FSH (150 IU/d, fixed) combined with a GnRH antagonist (antide; 1 mg/d) during the late follicular phase. When at least one follicle above 18 mm was observed, patients were randomized to induce oocyte maturation by a single injection of either r-human (h)CG (250 microg) (n = 11), r-LH (1 mg) (n = 13), or GnRH agonist (triptorelin; 0.2 mg) (n = 15). Retrieved oocytes were fertilized by either IVF or intracytoplasmatic sperm injection, depending on sperm quality. Embryo transfer was performed 3-4 d after oocyte retrieval. No luteal support was provided. Serum concentrations of FSH, LH, estradiol (E(2)), progesterone (P), and hCG were assessed at fixed intervals during the follicular and luteal phase. The median duration of the luteal phase was 13, 10, and 9 d for the r-hCG, the r-LH, and the GnRH agonist group, respectively (P = 0.005). The median area under the curve per day (from 4 d post randomization until the onset of menses) for LH was 0.50, 2.34, and 1.07 for the r-hCG, the r-LH, and the GnRH agonist group, respectively (P = 0.001). The median area under the curve per day for P was 269 vs. 41 and 16 for the r-hCG, the r-LH, and the GnRH agonist group, respectively (P < 0.001). Low pregnancy rates (overall, 7.5%; range, 0-18% per started cycle) were observed in all groups. In conclusion, the nonsupplemented luteal phase was insufficient in all three groups. In the patients receiving r-hCG, the luteal phase was less disturbed, compared with both other groups, presumably because of prolonged clearance of hCG from the circulation and the resulting extended support of the corpus luteum. Despite high P and E(2) concentrations during the early luteal phase in all three groups, luteolysis started prematurely, presumably because of excessive negative steroid feedback resulting in suppressed pituitary LH release. Hence, support of corpus luteum function remains mandatory after ovarian stimulation for IVF with GnRH antagonist cotreatment

The structure of Green functions in quantum field theory with a general state

In quantum field theory, the Green function is usually calculated as the expectation value of the time-ordered product of fields over the vacuum. In some cases, especially in degenerate systems, expectation values over general states are required. The corresponding Green functions are essentially more complex than in the vacuum, because they cannot be written in terms of standard Feynman diagrams. Here, a method is proposed to determine the structure of these Green functions and to derive nonperturbative equations for them. The main idea is to transform the cumulants describing correlations into interaction terms.Comment: 13 pages, 6 figure