An elementary approach to toy models for D. H. Lehmer's conjecture

In 1947, Lehmer conjectured that the Ramanujan's tau function $\tau (m)$ never vanishes for all positive integers $m$, where $\tau (m)$ is the $m$-th Fourier coefficient of the cusp form $\Delta_{24}$ of weight 12. The theory of spherical $t$-design is closely related to Lehmer's conjecture because it is shown, by Venkov, de la Harpe, and Pache, that $\tau (m)=0$ is equivalent to the fact that the shell of norm $2m$ of the $E_{8}$-lattice is a spherical 8-design. So, Lehmer's conjecture is reformulated in terms of spherical $t$-design. Lehmer's conjecture is difficult to prove, and still remains open. However, Bannai-Miezaki showed that none of the nonempty shells of the integer lattice \ZZ^2 in \RR^2 is a spherical 4-design, and that none of the nonempty shells of the hexagonal lattice $A_2$ is a spherical 6-design. Moreover, none of the nonempty shells of the integer lattices associated to the algebraic integers of imaginary quadratic fields whose class number is either 1 or 2, except for \QQ(\sqrt{-1}) and \QQ(\sqrt{-3}) is a spherical 2-design. In the proof, the theory of modular forms played an important role. Recently, Yudin found an elementary proof for the case of \ZZ^{2}-lattice which does not use the theory of modular forms but uses the recent results of Calcut. In this paper, we give the elementary (i.e., modular form free) proof and discuss the relation between Calcut's results and the theory of imaginary quadratic fields.Comment: 18 page

Quantum Fields in Hyperbolic Space-Times with Finite Spatial Volume

The one-loop effective action for a massive self-interacting scalar field is investigated in $4$-dimensional ultrastatic space-time $R \times H^3/\Gamma$, $H^3/\Gamma$ being a non-compact hyperbolic manifold with finite volume. Making use of the Selberg trace formula, the $\zeta$-function related to the small disturbance operator is constructed. For an arbitrary gravitational coupling, it is found that $\zeta(s)$ has a simple pole at $s=0$. The one-loop effective action is analysed by means of proper-time regularisations and the one-loop divergences are explicitly found. It is pointed out that, in this special case, also $\zeta$-function regularisation requires a divergent counterterm, which however is not necessary in the free massless conformal invariant coupling case. Finite temperature effects are studied and the high-temperature expansion is presented. A possible application to the problem of the divergences of the entanglement entropy for a free massless scalar field in a Rindler-like space-time is briefly discussed.Comment: 13 pages, LaTex. The contribution of hyperbolic elements has been added. Other minor corrections and reference

The trace of the heat kernel on a compact hyperbolic 3-orbifold

The heat coefficients related to the Laplace-Beltrami operator defined on the hyperbolic compact manifold H^3/\Ga are evaluated in the case in which the discrete group \Ga contains elliptic and hyperbolic elements. It is shown that while hyperbolic elements give only exponentially vanishing corrections to the trace of the heat kernel, elliptic elements modify all coefficients of the asymptotic expansion, but the Weyl term, which remains unchanged. Some physical consequences are briefly discussed in the examples.Comment: 11 page

Support varieties for selfinjective algebras

Support varieties for any finite dimensional algebra over a field were introduced by Snashall-Solberg using graded subalgebras of the Hochschild cohomology. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, periodic modules are lines and for symmetric algebras a generalization of Webb's theorem is true

Selberg Supertrace Formula for Super Riemann Surfaces III: Bordered Super Riemann Surfaces

This paper is the third in a sequel to develop a super-analogue of the classical Selberg trace formula, the Selberg supertrace formula. It deals with bordered super Riemann surfaces. The theory of bordered super Riemann surfaces is outlined, and the corresponding Selberg supertrace formula is developed. The analytic properties of the Selberg super zeta-functions on bordered super Riemann surfaces are discussed, and super-determinants of Dirac-Laplace operators on bordered super Riemann surfaces are calculated in terms of Selberg super zeta-functions.Comment: 43 pages, amste

Structural characterization of vanadium oxide catalysts supported on nanostructured silica SBA-15 using X-ray absorption spectroscopy

The local structure of vanadium oxide supported on nanostructured SiO2 (VxOy/SBA-15) was investigated by in situ X-ray absorption spectroscopy (XAS). Because the number of potential parameters in XAS data analysis often exceeds the number of "independent" parameters, evaluating the reliability and significance of a particular fitting procedure is mandatory. The number of independent parameters (Nyquist) may not be sufficient. Hence, in addition to the number of independent parameters, a novel approach to evaluate the significance of structural fitting parameters in XAS data analysis is introduced. Three samples with different V loadings (i.e. 2.7 wt %, 5.4 wt %, and 10.8 wt %) were employed. Thermal treatment in air at 623 K resulted in characteristic structural changes of the V oxide species. Independent of the V loading, the local structure around V centers in dehydrated VxOy/SBA-15 corresponded to an ordered arrangement of adjacent V2O7 units. Moreover, the V2O7 units were found to persist under selective oxidation reaction conditions

Effects of oestradiol and tamoxifen on VEGF, soluble VEGFR-1, and VEGFR-2 in breast cancer and endothelial cells

Angiogenesis is regulated by the balance between pro- and antiangiogenic factors. Vascular endothelial growth factor (VEGF), acting via the receptors VEGFR-1 and VEGFR-2, is a key mediator of tumour angiogenesis. The soluble form of the VEGF receptor-1 (sVEGFR-1) is an important negative regulator of VEGF-mediated angiogenesis. The majority of breast cancers are oestrogen dependent, but it is not fully understood how oestrogen and the antioestrogen, tamoxifen, affect the balance of angiogenic factors. Angiogenesis is a result of the interplay between cancer and endothelial cells, and sex steroids may exert effects on both cell types. In this study we show that oestradiol decreased secreted sVEGFR-1, increased secreted VEGF, and decreased the ratio of sVEGFR-1/VEGF in MCF-7 human breast cancer cells. The addition of tamoxifen opposed these effects. Moreover, human umbilical vein endothelial cells (HUVEC) incubated with supernatants from oestradiol-treated MCF-7 cells exhibited higher VEGFR-2 levels than controls. In vivo, MCF-7 tumours from oestradiol+tamoxifen-treated nude mice exhibited decreased tumour vasculature. Our results suggest that tamoxifen and oestradiol exert dual effects on the angiogenic environment in breast cancer by regulating cancer cell-secreted angiogenic ligands such as VEGF and sVEGFR-1 and by affecting VEGFR-2 expression of endothelial cells

Molecular and pathological signatures of epithelial–mesenchymal transitions at the cancer invasion front

Reduction of epithelial cell–cell adhesion via the transcriptional repression of cadherins in combination with the acquisition of mesenchymal properties are key determinants of epithelial–mesenchymal transition (EMT). EMT is associated with early stages of carcinogenesis, cancer invasion and recurrence. Furthermore, the tumor stroma dictates EMT through intensive bidirectional communication. The pathological analysis of EMT signatures is critically, especially to determine the presence of cancer cells at the resection margins of a tumor. When diffusion barriers disappear, EMT markers may be detected in sera from cancer patients. The detection of EMT signatures is not only important for diagnosis but can also be exploited to enhance classical chemotherapy treatments. In conclusion, further detailed understanding of the contextual cues and molecular mediators that control EMT will be required in order to develop diagnostic tools and small molecule inhibitors with potential clinical implications

Estrogen receptor transcription and transactivation: Basic aspects of estrogen action

Estrogen signaling has turned out to be much more complex and exciting than previously thought; the paradigm shift in our understanding of estrogen action came in 1996, when the presence of a new estrogen receptor (ER), ERβ, was reported. An intricate interplay between the classical ERα and the novel ERβ is of paramount importance for the final biological effect of estrogen in different target cells