On the Products of Three Theta Functions

In this paper, we re-examine several beautiful identities of Ramanujan from the perspective of the products of three theta functions.     

The new science of protein mimetics

New chemistries have been developed for de novo protein design. Protein mimetics of different structural and functional properties such as synthetic peptide ligases and Dn symmetrical helical bundles have been reported. The Template-Assembled Synthetic Protein (TASP) method (as well as the ßMolecular Kit' approach) has also been utilized to prepare protein-like molecules. Here we report the synthesis of single chain, scaffold (TRIS)- and dendrimer-assembled collagen mimetics composed of the Gly-Nleu-Pro sequence where Nleu denotes N-isobutyl glycine. From the CD spectra and the thermal denaturation studies it can be seen that the collagen mimetics prepared form stable triple helices except the single chain structure. Furthermore, the 162-residue collagen mimetic dendrimer exhibits enhanced triple helical stability compared to the equivalent scaffold-terminated structure by an increase in the melting temperature in both H₂O and 2:1 ethylene glycol/H₂O (4°C and 12°C respectively). The concentration dependence for the melting transition of the collagen mimetic dendrimer was measured from which it was determined that the stabilization effect arises from the intramolecular clustering of the triple helical arrays about the core structure. This ensemble excludes solvent from the interior portion of the array which stabilizes the triple helix cluster.     

因子von Neumann代数上的非线性混合Lie三重可导映射

本文通过经典的可导映射,运用矩阵分块的方法,证明了因子von Neumann代数■上的每一个非线性混合Lie三重可导映射都是可加的*-导子.     

Further results on large set of disjoint pure Mendelsohn triple systems

In this article, we study a large set of disjoint pure Mendelsohn triple systems “with holes” (briefly LPHMTS), which is a generalization of large set of disjoint pure Mendelsohn triple systems (briefly LPMTS), and give some recursive constructions on LPHMTS. Using these constructions, we show that there exists LPMTS(2ⁿ + 2) for any n ≠ 2. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 274–290, 2000     

Tris-(2-pyridyl)-pyrazolyl Borate Zinc(II) Complexes: Synthesis,DNA/Protein Binding and In Vitro Cytotoxicity Studies

Zn(II) complexes bearing tris[3-(2-pyridyl)-pyrazolyl] borate (Tp^py) ligand (1–3) was synthesized and examined by spectroscopic and analytical tools. Mononuclear [Tp^pyZnCl] (1) has a Zn(II) centre with one arm (pyrazolyl-pyridyl) dangling outside the coordination sphere which is a novel finding in Tp^pyZn(II) chemistry. In complex [Tp^pyZn(H₂O)][BF₄] (2) hydrogen bonding interaction of aqua moiety stabilizes the dangling arm. In addition, solution state behaviour of complex 1 confirms the tridentate binding mode and reactivity studies show the exogenous axial substituents used to form the [Tp^pyZnN₃] (3). The complexes (1–3) were tested for their ability to bind with Calf thymus (CT) DNA and Bovine serum albumin (BSA) wherein they revealed to exhibit good binding constant values with both the biomolecules in the order of 10⁴–10⁵ M⁻¹. The intercalative binding mode with CT DNA was confirmed from the UV-Visible absorption, viscosity, and ethidium bromide (EB) DNA displacement studies. Further, the complexes were tested for in vitro cytotoxic ability on four triple-negative breast cancer (TNBC) cell lines (MDA-MB-231, MDA-MB-468, HCC1937, and Hs 578T). All three complexes (1–3) exhibited good IC₅₀ values (6.81 to 16.87 μM for 24 h as seen from the MTS assay) results which indicated that these complexes were found to be potential anticancer agents against the TNBC cells.     

Edge and surface antiferromagnetism in ABO3 perovskite-type nanoparticle within the effective field theory

ABSTRACT

The edge and surface antiferromagnetic properties of the ABO₃ perovskite-type nanoparticle and their components (A, B and O) are investigated through the Effective Field Theory developed by Kaneyoshi. It is found that the Edge AntiFerroMagnetism (EAFM) causes a decrease in the Curie temperature and separation in the magnetizations. However, the Surface AntiFerroMagnetism (SAFM) causes a compensation behaviour (T_Comp?=?1.23) in the total magnetisation of the ABO₃ perovskite-type nanoparticle. The magnetisation of the central B component has been affected by the edge antiferromagnetism more strongly than by the surface antiferromagnetism. Moreover, coercivity (Hc) and remanence magnetizations (Mr) decrease for the EAFM, but they increase for the SAFM. Triple hysteresis behaviour is obtained for the SAFM. Therefore, it is suggested that there is a strong relationship between the compensation behaviour and the triple hysteresis behaviour.     

Dynamic manipulation of probe pulse and coherent generation of beating signals based on tunneling-induced inference in triangular quantum dot molecules

We investigate the dynamic propagation of a probe field via the tunneling-induced interference effect in a triple model of quantum dot molecules. By theoretical analysis and numerical simulation, we find that the number of transparency window relate to the energy splitting and the group velocity of probe field can be effectively controlled by the tunneling coupling intensity. In addition, in the process of light storage and retrieval, when the excited states have no energy splitting in the storage stage but opposite values of the energy splitting in the retrieval stage, the beating signals can be generated.     

判定广义Birkhoff系统稳定性的三重组合梯度方法

研究判定广义Birkhoff系统稳定性的三重组合梯度方法．首先,给出4类三重组合梯度系统的定义和微分方程;其次,得到广义Birkhoff系统成为三重组合梯度系统的条件,从而将广义Birkhoff系统化成三重组合梯度系统;最后,利用组合梯度系统的性质来研究系统的稳定性,举例说明结果的应用．     

Products of two 1ϕ1 series and implications

An expansion formula into bivariate formal power series is established that implies, as consequences, three remarkable identities of partial theta functions due to Warnaar (2003), Schilling–Warnaar (2002) and Andrews–Warnaar (2007).     

Weyl families of transformed boundary pairs

Let $(\mathfrak{L},\mathrm{\Gamma })$ be an isometric boundary pair associated with a closed symmetric linear relation T in a Krein space $\mathfrak{H}$. Let $M_{\mathrm{\Gamma }}$ be the Weyl family corresponding to $(\mathfrak{L},\mathrm{\Gamma })$. We cope with two main topics. First, since $M_{\mathrm{\Gamma }}$ need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation $M_{\mathrm{\Gamma }}(z)$, for some $z\in \mathbb{C}\setminus \mathbb{R}$, becomes a nontrivial task. Regarding $M_{\mathrm{\Gamma }}(z)$ as the (Shmul'yan) transform of $zI$ induced by Γ, we give conditions for the equality in $\overline{M_{\mathrm{\Gamma }}(z)}\subseteq \overline{M_{\overline{\mathrm{\Gamma }}}(z)}$ to hold and we compute the adjoint $M_{\overline{\mathrm{\Gamma }}}{(z)}^{\ast }$. As an application, we ask when the resolvent set of the main transform associated with a unitary boundary pair for $T^{+}$ is nonempty. Based on the criterion for the closeness of $M_{\mathrm{\Gamma }}(z)$, we give a sufficient condition for the answer. From this result it follows, for example, that, if T is a standard linear relation in a Pontryagin space, then the Weyl family $M_{\mathrm{\Gamma }}$ corresponding to a boundary relation Γ for $T^{+}$ is a generalized Nevanlinna family; a similar conclusion is already known if T is an operator. In the second topic, we characterize the transformed boundary pair $({\mathfrak{L}}^{\prime },{\mathrm{\Gamma }}^{\prime })$ with its Weyl family $M_{{\mathrm{\Gamma }}^{\prime }}$. The transformation scheme is either ${\mathrm{\Gamma }}^{\prime }=\mathrm{\Gamma }{V}^{-1}$ or ${\mathrm{\Gamma }}^{\prime }=V\mathrm{\Gamma }$ with suitable linear relations V. Results in this direction include but are not limited to: a 1-1 correspondence between $(\mathfrak{L},\mathrm{\Gamma })$ and $({\mathfrak{L}}^{\prime },{\mathrm{\Gamma }}^{\prime })$; the formula for $M_{{\mathrm{\Gamma }}^{\prime }}-M_{\mathrm{\Gamma }}$, for an ordinary boundary triple and a standard unitary operator V (first scheme); construction of a quasi boundary triple from an isometric boundary triple $(\mathfrak{L},{\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_1)$ with $ker\mathrm{\Gamma }=T$ and $T_0=T_0^{\ast }$ (second scheme, Hilbert space case).