TR Expression Research Articles

On the preparation of pure states in resonant microcavities

We consider the time evolution of the radiation field (R) and a two-level atom (A) in a resonant microcavity in terms of the Jaynes-Cummings model with an initial general pure quantum state for the radiation field. It is then shown, using the Cauchy-Schwarz inequality and also a Poisson resummation technique, that perfect coherence of the atom can in general never be achieved. The atom and the radiation field are, however, to a good approximation in a pure state ∥ψ⟩A⊗R=∥ψ⟩A×∥ψ⟩R in the middle of what has been traditionally called the ‘collapse region’, independent of the initial state of the atoms, provided that the initial pure state of the radiation field has a photon number probability distribution which is sufficiently peaked and phase differences that do not vary significantly around this peak. An approximate analytic expression for the quantity Tr[ρ2 A(t)], where ρA (t) is the reduced density matrix for the atom, is derived. We also show that under quite general circumstances an initial entangled pure state will be disentangled to the pure state ∥ψ⟩PSA⊗R.

Molecular cloning and developmental expression patterns of thyroid hormone receptors and T3 target genes in the turbot ( Scophtalmus maximus) during post-embryonic development

Thyroid hormones (TH) are pleiotropic factors important for many developmental and physiological functions in vertebrates and particularly in amphibian metamorphosis. Their effects are mediated by two specific receptors (TRα and TRβ), which are ligand-dependent transcription factors, members of the nuclear hormone receptor superfamily. Besides their pivotal role in amphibian metamorphosis, TH are also critical for fish metamorphosis. As this later role of TH is less studied, we analyzed their action in the turbot ( Scophtalmus maximus), a metamorphosing flat fish. We describe the isolation of sequences for the turbot orthologs of a number of Xenopus genes, which are induced during amphibian metamorphosis. Developmental expression of these genes during turbot metamorphosis was studied by several methods and the expression patterns of these genes compared with those in Xenopus and flounder. We find that the period between the onset and the end of eye migration (day 22 to day 30 post-hatching) most likely corresponds to the metamorphic climax with either high TRα or high TH levels. Our results show that in contrast to amphibians, it is TRα and not TRβ mRNA that is up-regulated during metamorphosis. Our results highlight the notion that TH regulates, through a rise of TR expression, a genetic cascade during turbot metamorphosis. The fact that TH regulates metamorphosis in amphibian and teleost fishes suggests that TH-regulated metamorphosis is a post-embryonic process conserved in most vertebrates.

Localization and expression of thyroid hormone receptors normal and neoplastic human thyroid.

The aim of this study was to investigate the regional expression of thyroid hormone nuclear receptor forms (TR(alpha) and TR(beta)) and isoform (TR(alpha1) and TR(beta2)) mRNAs in normal and neoplastic (benignant and malignant) human thyroid tissue. Tumor specimens from patients with thyroid carcinomas (papillary: 5 cases; follicular: 5 cases; anaplastic: 2 cases), thyroid follicular adenomas (7 cases) and tissue from normal thyroid glands (12 cases) were analyzed by in situ hybridization and semiquantitative RT-PCR for the expression of TR(alpha1) and beta, as well as for the isoform alpha2 that does not bind the hormone. In normal tissues, TR(alpha2) was expressed at lower levels compared to TR(alpha1) (alpha1/alpha2 = 4.3). In papillary and follicular carcinomas, the expression of TR(alpha1) and TR(beta) did not change as compared with normal thyroid tissue and adenomas (0.87 +/- 0.15 SD vs 0.89 +/- 0.17 densitometric units, DU, and 0.15 +/- 0.02 vs 0.14 +/- 0.03 DU, respectively). However, the expression of TR(alpha2) was significantly higher in differentiated carcinomas compared to normal thyroid tissue and adenomas (0.47 +/- 0.05 vs 0.20 +/- 0.05 DU, p < 0.05) with alpha1/alpha2 = 1.4. In anaplastic carcinoma all TRs were absent. We concluded that both normal and pathological thyroid tissues, with the exception of anaplastic carcinoma, express all TRs in thyreocites and that differentiated thyroid carcinomas are associated in enhancing the expression of TR(alpha2) mRNA.

Quantum hydrodynamic equations and quantum-hierarchy decoupling scheme.

There is a need to extract the relevant device physics from exact quantum transport formulations, and hence to reduce the complexity of quantum transport simulations for practical applications. We use a Hermite polynomial expansion to drastically reduce the number of degrees of freedom associated with the momentum variables. The result is a quantum hierarchy in real space. We also give a general procedure for the quantum-hierarchy decoupling scheme to derive the quantum hydrodynamic (QHD) and quantum drift-diffusion transport equations. We present some numerical results for the quantum hierarchy. A rigorous foundation of a decoupling procedure is given whereby the lower-order equations are renormalized in terms of a self-consistent effective potential, quantum diffusion coefficient, and moments, endowed with all the quantum corrections to order variant Planck's over h(2). Our decoupling scheme is based on the general expression of Tr H(n) to order variant Planck's over h(2), valid at all temperatures without the need for expansion in terms of the small parameter and high temperature assumption. This is very important conceptually since existing QHD formulations, using expansion to order h(2), are based on a Boltzmann distribution with the restrictive assumption of a small parameter, which is not valid in abrupt heterojunction semiconductor devices. They also fail to account for important quantum nonlinearity in the form of nonequilibrium quantum corrections. These nonequilibrium quantum corrections are expected to play a major role in approximating the coherence manifested by the highly nonlinear current-voltage characteristics of resonant tunneling structures.

TR(beta)1 protein is preferentially expressed in the pericentral zone of rat liver and exhibits marked diurnal variation.

We investigated the distribution and diurnal variation of TR(beta)1 protein expression in liver with specific antibodies against TR(beta)1. Immunohistochemistry showed a zonal distribution of TR(beta)1 with maximum expression in the pericentral zone matching some known T(3)-responsive enzyme activities in the liver, such as glutamine synthetase, cholesterol 7alpha- hydroxylase, and spot 14. Combining immunohistochemistry and image analysis we found and quantified the same zonal distribution for 5'-deiodinase type 1 as for TR(beta)1. Western blot analysis revealed a profound diurnal variation for TR(beta)1 protein expression, with highest levels at the beginning of the dark period. TR(beta)1 diurnal variation partly overlaps with the T(3)-responsive genes, cholesterol 7alpha-hydroxylase and spot 14. Furthermore, TR(beta)1 distribution along the porto-central axis does not change during the day, indicating that the zonal expression of TR(beta)1 is stable. This is the first time that zonal distribution in liver has been demonstrated for a member of the nuclear receptor family. This finding together with the observed diurnal rhythm has major implications for interpreting and timing experiments concerning the TR and its downstream actions in liver.

TR expression and function in human bone marrow stromal and osteoblast-like cells.

Thyroid hormones influence both bone formation and bone resorption. In vitro studies demonstrate direct effects of thyroid hormones on cells of the osteoblast lineage. Transcriptional regulation by thyroid hormones is mediated by ligand-dependent transcription factors called TRs. The three main T(3)-binding TR isoforms are TRalpha1, TRbeta1, and TRbeta2. TRs have been identified in cells of the osteoblast lineage, but it is still not known whether TR isoform expression differs in primary cultures of human osteoblasts. We used immunocytochemistry, Western blotting, nuclear binding assays, and transient transfection studies to examine the expression of functional TR isoforms in primary cultures of osteoblasts (hOb) derived from explants of trabecular bone, in human bone marrow stromal cells (hBMS), which are believed to be the source of osteoblast progenitor cells, and for comparison in the transformed human osteosarcoma cell lines MG63 and SaOs-2. TRalpha1, TRbeta1, and TRbeta2 proteins were expressed in all cells, although expression was greatest in MG63 > hBMS > SaOs-2 > hOb. Differences between isoforms were also apparent, with TRalpha1> TRbeta1 > TRbeta2 in all cell types. Incubation with [(125)I]T(3) confirmed reversible T(3) binding to cell nuclei. Specific binding was greatest in MG63 > hBMS > SaOs-2 > hOb. Finally, endogenous TR activity was determined in transfections using a thyroid hormone response element derived from the rat GH gene linked to the luciferase reporter gene. In MG63 and hBMS cells T(3) treatment increased luciferase activity 5.5 +/- 0.7-fold (P < 0.05), confirming the presence of endogenous receptors. In SaOs-2 and hOb cells, T(3) treatment had no effect on thyroid hormone response element-thymidine kinase-luciferase expression, suggesting that in these cells TR expression was too low to be detected. These results indicate that three main TR isoforms are expressed in cells of the human osteoblast lineage, but that expression and endogenous TR activity are predominantly present in hBMS cells. Whether there are distinct mechanisms of thyroid hormone action mediated by TRalpha1, TRbeta1, and TRbeta2 in hOb and hBMS cells remains to be shown.

Induction of thioredoxin and thioredoxin reductase gene expression in lungs of newborn primates by oxygen.

Thioredoxin (TRX) is a potent protein disulfide oxidoreductase important in antioxidant defense and regulation of cell growth and signal transduction processes, among them the production of nitric oxide. We report that lung TRX and its reductase, TR, are specifically upregulated at birth by O2. Throughout the third trimester, mRNAs for TRX and TR were expressed constitutively at low levels in fetal baboon lungs. However, after premature birth (125 or 140 of 185 days gestation), lung TRX and TR mRNAs increased rapidly with the onset of O2 or air breathing. Lung TRX mRNA also increased in lungs of term newborns with air breathing. Premature animals (140 days) breathing 100% O2 develop chronic lung disease within 7-14 days. These animals had greater TRX and TR mRNAs after 1, 6, or 10 days of life than fetal control animals. In 140-day animals given lesser O2 concentrations (as needed) who do not develop chronic lung disease, lung TRX and TR mRNAs were also increased on days 1 and 6 but not significantly on day 10. In fetal distal lung explant culture, mRNAs for TRX and TR were elevated within 4 h in 95% O2 relative to 1% O2, and the response was similar at various gestations. In contrast, TRX protein did not increase in lung explants from premature animals (125 or 140 days) but did in those from near-term (175-day) fetal baboons after exposure to hyperoxia. However, lung TRX protein and activity, as well as TR activity, eventually did increase in vivo in response to hyperoxia (6 days). Increases in TRX and TR mRNAs in response to 95% O2 also were observed in adult baboon lung explants. When TRX redox status was determined, increased O2 tension shifted TRX to its oxidized form. Treatment of lung explants with actinomycin D inhibited TRX and TR mRNA increases in 95% O2, indicating transcriptional regulation by O2. The acute increase in gene expression for both TRX and TR in response to O2 suggests an important role for these proteins during the transition from relatively anaerobic fetal life to O2 breathing at birth.

Inclusion of a Legume in a Saturated Fat-Rich Diet Affects the Cholesterol Status but not the Expression of Triiodothyronine and Retinoic Acid Receptors in Rat Liver

In a previous study, we showed that a high-fat diet, rich in saturated fatty acids and cholesterol, induced hypercholesterolemia and, in rat liver, a decreased expression of TR and RAR, nuclear receptors which are transcription factors for genes involved in cell growth, differentiation and cellular homeostasis. The aim of this study was to determine whether hypercholesterolemia plays a role in the onset of this decreased expression. Rats were fed a high-fat diet, rich in saturated fatty acids and cholesterol, for 3 weeks and then received in addition Vicia faba for 2 weeks. The inclusion of the legume induced a decrease in hypercholesterolemia (in agreement with numerous data) but did not affect the underexpression of TR or RAR (expression evaluated as the maximum binding capacity of receptors and as the abundance of mRNA of these receptors). Thus, it is suggested that, at least in this experimental model, hypercholesterolemia plays a minor role in the decreased expression of nuclear receptors in rat liver.

Transferrin Receptor–Independent Uptake of Differic Transferrin by Human Hepatoma Cells With Antisense Inhibition of Receptor Expression

The hepatic uptake of transferrin–bound iron by a nontransferrin receptor (NTR)–mediated process was investigated using the human hepatoma cell line HuH7. Because HuH7 cells also acquire iron from transferrin by a receptor (TR)–mediated process, TR expression was inhibited by transfecting the cells with a plasmid containing human TR complementary DNA in antisense orientation relative to a human cytomegalovirus promoter/enhancer element. Cell clones were obtained that expressed a 50% to 60% reduction in cell surface TR, leading to a corresponding decrease in transferrin and iron uptake compared with wild–type cells. Uptake of transferrin by a second process was nonsaturable and not inhibited by a 100–fold excess of unlabeled transferrin. The amounts of transferrin taken up by the wild–type and antisense cells by this process were similar, showing that it did not involve TR. The proteolytic enzyme Pronase reduced the uptake of transferrin, suggesting that the NTR–mediated process entailed the nonsaturable binding of transferrin to plasma membrane proteins. This process, like the TR–mediated one, involved the internalization and recycling of transferrin, leading to accumulation of iron with time. Iron uptake mediated by NTR process was saturable and displaced by 100–fold excess unlabeled transferrin and reduced by weak bases and metabolic inhibitors. Therefore, the NTR–mediated process entailed transferrin adsorption to membrane–bound proteins, internalization, and release of iron from transferrin by a pH–dependent step followed by the intracellular transport of iron into ferritin and heme by a saturable carrier–mediated mechanism.

Transferrin receptor-independent uptake of differic transferrin by human hepatoma cells with antisense inhibition of receptor expression

The hepatic uptake of transferrin-bound iron by a nontransferrin receptor (NTR)-mediated process was investigated using the human hepatoma cell line HuH7. Because HuH7 cells also acquire iron from transferrin by a receptor (TR)-mediated process, TR expression was inhibited by transfecting the cells with a plasmid containing human TR complementary DNA in antisense orientation relative to a human cytomegalovirus promoter/enhancer element. Cell clones were obtained that expressed a 50% to 60% reduction in cell surface TR, leading to a corresponding decrease in transferrin and iron uptake compared with wild-type cells. Uptake of transferrin by a second process was nonsaturable and not inhibited by a 100-fold excess of unlabeled transferrin. The amounts of transferrin taken up by the wild-type and antisense cells by this process were similar, showing that it did not involve TR. The proteolytic enzyme Pronase reduced the uptake of transferrin, suggesting that the NTR-mediated process entailed the nonsaturable binding of transferrin to plasma membrane proteins. This process, like the TR-mediated one, involved the internalization and recycling of transferrin, leading to accumulation of iron with time. Iron uptake mediated by NTR process was saturable and displaced by 100-fold excess unlabeled transferrin and reduced by weak bases and metabolic inhibitors. Therefore, the NTR-mediated process entailed transferrin adsorption to membrane-bound proteins, internalization, and release of iron from transferrin by a pH-dependent step followed by the intracellular transport of iron into ferritin and heme by a saturable carrier-mediated mechanism. (Hepatology 1996 Jun;23(6):1512-20)

Exactness in the Wentzel–Kramers–Brillouin approximation for some homogeneous spaces

Analysis of the Wentzel–Kramers–Brillouin (WKB) exactness in some homogeneous spaces is attempted. CPN as well as its noncompact counterpart DN,1 is studied. U(N+1) or U(N,1) based on the Schwinger bosons leads us to CPN or DN,1 path integral expression for the quantity tr e−iHT, with the aid of coherent states. The WKB approximation terminates in the leading order and yields the exact result provided that the Hamiltonian is given by a bilinear form of the creation and the annihilation operators. An argument on the WKB exactness to more general cases is also made.

Reply to González-Calvet and Sánchez-Chóoliz

My 1989 paper, on which the comments of Gonzalez-Calvet and Sdnchez-Ch6liz focus, had a fairly simple goal. It attempted to show that partially endogenous credit and effective demand might combine to produce self-sustained growth cycles. Using a dynamical model of interest and capital stock growth rates, sufficient conditions for self-sustained cycles were derived. Based on these conditions, it was argued that ceteris paribus stronger firn response to cash flow could, in some cases, make cycles more likely. It also was pointed out that increased endogeneity in the supply of credit would make it more likely that the sufficient conditions would be met. As the authors of these comments have recognized, that paper contained some algebra errors. The expression for Tr(J), the trace of the Jacobian of the model, was incorrectly written in the text. The expression for r2, the value of the r = 0 isocline when r = 0, contains an error. Also the labeling in the phase diagram in figure 2 is at variance with the appendix, where the proof of the existence of cycles is stated.' None of these mistakes should have been made. I apologize for any inconvenience they may have caused readers.

Effect of chronic energy deprivation on cardiac thyroid hormone receptor and myosin isoform expression

Thyroid hormone (3,5,3'-triiodothyronine; T3) and its receptor (TR) play an important regulatory role for in vivo and in vitro cardiac myosin heavy chain (MHC) isoform gene expression by activating the alpha- and inhibiting the beta-MHC genes. Previous studies have shown that chronic energy deprivation (CED; 50% of normal caloric intake) in the rat impacts cardiac MHC protein expression and hemodynamic parameters in a pattern typically seen with hypothyroidism; yet, unlike hypothyroidism, circulating T3 levels are not depressed. Therefore, the goal of this study was to determine if the altered MHC isoform expression observed in CED is associated with altered TR expression, both at the mRNA and protein levels. Female rats weighing approximately 250 g were allocated into two groups, designated as normal control (NC) and CED. After 5 wk, the relative content of alpha-MHC protein and mRNA levels decreased in CED ventricles by 20% (P < 0.05). In contrast, the relative content of both beta-MHC protein and mRNA levels increased five- to sixfold in CED (P < 0.05). Although there were no changes in TR mRNA levels relative to 18S rRNA in CED, the total number of nuclear TRs decreased 3.5-fold in the CED group (P < 0.05), from a maximum binding capacity of 840 +/- 130 fmol/mg DNA in NC to 241 +/- 118 fmol/mg DNA in CED, with no change in the affinity of the receptor.(ABSTRACT TRUNCATED AT 250 WORDS)

Mapping of a B-cell epitope present in the neuraminidase of Trypanosoma cruzi

We have previously shown that a polyclonal (rabbit anti-TCNA) and a mouse monoclonal antibody (TCN-2) against the neuraminidase of Trypanosoma cruzi (TCNA) inhibit enzyme activity, immunoprecipitate active enzyme, enhance in vitro infection, and identify a subpopulation of extracellular trypomastigotes. We now report on the identification of a synthetic peptide that contains the epitope recognized by these antibodies. The synthetic peptide (TR) is a dodecamer (D-S-S-A-H-G-T-P-S-T-P-A) deduced from the DNA sequence of the long tandem repeat (LTR) domain present in the TCNA carboxy-terminus. By ELISA, rabbit anti-TCNA bound to TR coupled to ovalbumin, and the binding was inhibited by soluble TR but not by BR (Y-S-V-D-D-G-E-T-W-E), a peptide derived from the N-terminal domain of the enzyme. TCN-2 recognized TR, and this reaction as well as TCN-2 binding to endogenous TCNA could be inhibited by soluble TR but not by BR. These results indicate that the rabbit anti-TCNA and TCN-2 react with the LTR region of TCNA. Antibodies to TR reacted by immunoblot with the TCNA of the Silvio X-10/4, MV-13 and Y-H6 strains, identifying the same molecular polymorphism previously observed with the rabbit anti-TCNA and TCN-2. Furthermore, anti-TR antibodies immunoprecipitated active enzyme and immunofluorescence analysis revealed that anti-TR and TCN-2 antibodies detected equally well the differential expression of their epitopes in intra- and extracellular trypomastigotes. Moreover, expression of TR and TCN-2 epitopes on the different stages of T. cruzi paralleled the stage-specificity of TCNA activity. TCN-2 prevented desialylation by TCNA of intact cells but not of soluble glycoconjugates, indicating that TCN-2 epitope is probably not associated with the enzyme catalytic site, in agreement with the predicted sequence of the TCNA gene. Finally, analysis of the humoral response of a Chagasic patient to different areas of the TCNA molecule indicated that the antibody response is predominantly against TR suggesting that the tandem repeat is the immunodominant domain of TCNA.

Retinoic acid treatment alters the distribution of retinoic acid receptor-beta transcripts in the embryonic chick face.

Retinoic acid is a metabolite of vitamin A that can act as a signalling molecule in a number of developmental systems. Retinoic acid is also known to be teratogenic in mammals, causing a range of defects including abnormalities in craniofacial development. Exposure of the developing chick face to retinoic acid released slowly from a bead implanted in the wing bud results in a specific facial defect, in which outgrowth of the frontonasal mass is inhibited. This results in clefting of the primary palate and absence of the upper beak. To investigate the role of nuclear retinoic acid receptors in normal and abnormal chick face morphogenesis, we isolated chick retinoic acid receptor-beta (RAR-beta) cDNA clones and probed northern blots of RNA isolated from chick embryos at stages 22, 24 and 25 and from adults. RAR-beta transcripts of 2.8 and 3.5 kb were present in several regions of the embryo, including the facial primordia, and were also present at much lower levels in adult tissues. In situ hybridisation showed that RAR-beta transcripts were present in all of the facial primordia at embryonic stages 20, 24 and 28, but that their distribution was not uniform. Transcripts were abundant in the lateral nasal processes, at the edges and corners of the frontonasal mass and in the anterior part of the maxillary primordia. Lower levels were present elsewhere. Treatment of stage 20 embryos with retinoic acid altered the distribution of RAR-beta transcripts in the maxillary primordia, such that high levels of transcripts were present throughout, rather than being confined to the anterior part. This change was detectable at stage 24, before any alterations in the morphology of the facial primordia were apparent. By stage 28, when the morphology of the facial primordia was clearly abnormal, there were more widespread changes in the distribution of RAR-beta transcripts. These results show that RAR-beta transcripts are particularly concentrated in regions of the primordia that give rise to the upper beak, the development of which is specifically affected by retinoic acid. In addition, they demonstrate that retinoic acid can induce changes in the pattern of expression of RAR-beta transcripts in vivo.

Transferrin receptor expression in tumours of the human nervous system: relation to tumour type, grading and tumour growth fraction

The expression of transferrin receptor (Tr) was investigated by means of immunohistochemistry in 101 tumours of the human central and peripheral nervous system. The results were compared with the proliferative activity of the tumours, determined by immunostaining for the proliferation-associated antigen Ki-67. In addition to immunostaining of normal and proliferated blood vessel endothelium and of a fraction of tumour infiltrating lymphocytes, we observed staining for Tr in a variable fraction of neoplastic cells of all histological tumour types. Immunoreactivity in the majority of tumour cells was found only in anaplastic tumours such as glioblastomas. Furthermore, a positive correlation between Tr expression and the Ki-67 growth fraction was established for gliomas. Non-glial tumours strongly expressing Tr included one metastatic rhabdomyosarcoma, one intracerebral malignant lymphoma, two of four plasmocytomas and seven of nine metastatic carcinomas. Our results indicate that immunohistochemistry for Tr and Ki-67 can provide additional information about the biological behaviour of nervous system tumours, thus complementing conventional histopathological criteria for anaplasia.

The Influence of FK‐506 and Low‐Concentration Ciclosporin on Human Lymphocyte Activation Antigen Expression and Blastogenesis: a Flow Cytometric Analysis

The influence of FK-506 and ciclosporin (CsA), either alone or in combination, on PHA- and alloantigen-induced human blood lymphocyte transformation was investigated. The concentrations of FK-506 and CsA required to cause 50% inhibition of mitogen-induced thymidine incorporation were 1.0 and 200 ng/ml respectively. Evidence of synergistic inhibition of DNA synthesis was obtained when doses of each drug below these concentrations were combined (FK-506, less than or equal to 0.5 ng/ml; CsA less than or equal to 50 ng/ml). In contrast, FK-506 and CsA failed to inhibit PHA-induced increases in cell size and granularity determined by flow cytometric analysis at 1 and 3 days of culture. Moreover, no significant changes in the expression of the interleukin 2 receptor (IL-2R; CD25), transferrin receptor (TR; CD71) or HLA-DR were observed in terms of percentage positive lymphocytes or mean cell surface membrane fluorescence intensity. In primary mixed lymphocyte cultures, 50% inhibition of thymidine incorporation was obtained with FK-506 and CsA concentrations of approximately 0.25 and 25 ng/ml respectively. Evidence of synergy was obtained when lower doses of each drug were combined (FK-506, less than or equal to 0.1 ng/ml; CsA less than or equal to 2 ng/ml). On their own, these concentrations of CsA were ineffective. In contrast to corresponding PHA-stimulated cells, FK-506-treated alloactivated lymphocytes exhibited reductions in IL-2R, TR and HLA-DR antigen expression, which in the cases of IL-2R and TR were further reduced by combination of FK-506 with low-concentration CsA. These data provide further evidence of the potential of combined low-dosage FK-506 and CsA for the control of human T-cell activation and proliferation in response to allostimulation.

On the Onsager problem for Potts models

It is shown how the calculation of partition functions for all Potts models might be reduced to the calculation of Tr( gamma i1 . . . gamma ipn) where the gamma stand for generators of a generalised Clifford algebra. Then the expression for Tr( gamma i1 . . . gamma is) for an arbitrary collection of such gamma matrices is derived.

Barrier penetration and spontaneous fission in the time-dependent mean-field approximation

A mean-field theory is obtained for the spontaneous decay of unstable nuclei by applying the stationary-phase approximation to a functional integral expression for Tr(H-E)/sup -1/. The method is applied first to the lifetime of a metastable state in one-dimensional quantum mechanics and subsequently generalized to many-fermion systems. Solutions to the resulting nonlinear self-consistent field equations are presented for a self-bound, saturating many-body system of fermions in one spatial dimension.

Traces of products of angular momentum operators

A symmetrical and computationally advantageous analytic expression for Tr(Jk-JzlJ+k) is obtained.