Effective actions on the squashed three-sphere

The effective actions of a scalar and massless spin-half field are determined as functions of the deformation of a symmetrically squashed three-sphere. The extreme oblate case is particularly examined as pertinant to a high temperature statistical mechanical interpretation that may be relevant for the holographic principle. Interpreting the squashing parameter as a temperature, we find that the effective `free energies' on the three-sphere are mixtures of thermal two-sphere scalars and spinors which, in the case of the spinor on the three-sphere, have the `wrong' thermal periodicities. However the free energies do have the same leading high temperature forms as the standard free energies on the two-sphere. The next few terms in the high-temperature expansion are also explicitly calculated and briefly compared with the Taub-Bolt-AdS bulk result.Comment: 23 pages, JyTeX. Conclusion slightly amended, one equation and minor misprints correcte

The renormalization group and spontaneous compactification of a higher-dimensional scalar field theory in curved spacetime

The renormalization group (RG) is used to study the asymptotically free $\phi_6^3$-theory in curved spacetime. Several forms of the RG equations for the effective potential are formulated. By solving these equations we obtain the one-loop effective potential as well as its explicit forms in the case of strong gravitational fields and strong scalar fields. Using zeta function techniques, the one-loop and corresponding RG improved vacuum energies are found for the Kaluza-Klein backgrounds $R^4\times S^1\times S^1$ and $R^4\times S^2$. They are given in terms of exponentially convergent series, appropriate for numerical calculations. A study of these vacuum energies as a function of compactification lengths and other couplings shows that spontaneous compactification can be qualitatively different when the RG improved energy is used.Comment: LaTeX, 15 pages, 4 figure

Spectral analysis and zeta determinant on the deformed spheres

We consider a class of singular Riemannian manifolds, the deformed spheres $S^N_k$, defined as the classical spheres with a one parameter family $g[k]$ of singular Riemannian structures, that reduces for $k=1$ to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian $\Delta_{S^N_k}$, we study the associated zeta functions $\zeta(s,\Delta_{S^N_k})$. We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in $\zeta(s,\Delta_{S^N_k})$. An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular $\zeta(0,\Delta_{S^N_k})$ and $\zeta'(0,\Delta_{S^N_k})$. We give explicit formulas for the zeta regularized determinant in the low dimensional cases, $N=2,3$, thus generalizing a result of Dowker \cite{Dow1}, and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter $k$.Comment: 1 figur

A Conformally Invariant Holographic Two-Point Function on the Berger Sphere

We apply our previous work on Green's functions for the four-dimensional quaternionic Taub-NUT manifold to obtain a scalar two-point function on the homogeneously squashed three-sphere (otherwise known as the Berger sphere), which lies at its conformal infinity. Using basic notions from conformal geometry and the theory of boundary value problems, in particular the Dirichlet-to-Robin operator, we establish that our two-point correlation function is conformally invariant and corresponds to a boundary operator of conformal dimension one. It is plausible that the methods we use could have more general applications in an AdS/CFT context.Comment: 1+49 pages, no figures. v2: Several typos correcte

Eff ect of modifi er nature on the preconcentration effi ciency of rutin and quercetin on the magnetite nanoparticles

Submitted 28 March 2019, received in revised form 24 May 2019Поступила в редакцию 28 марта 2019 г., после исправления 24 мая 2019 г.Flavonoids belong to a wide group of polyphenols present in many plants, flowers and seeds, vegetables and fruits. Their antioxidant action helps protect the human body from the oxidative stress, cardiovascular illnesses, inflammation, cancer and many other diseases. Researchers pay the most attention to quercetin and its glycoside rutin, which are present in many plant and food objects. One of the problems of their determination in various objects is preconcentration, which should be quick and quantitative. In the last decade, the method of magnetic solid-phase extraction (MSPE) was proposed for the preconcentration of many biologically active substances. This method is based on the phenomenon of superparamagnetism, in which magnetic nanoparticles with adsorbed analyte are separated during several tens of second from the matrix solution by a permanent magnet. In our study the magnetic nanoparticles (MNPs) of magnetite, the surface of which was modified with SiO2, SiO2 and polyethylenimine (PEI) and only PEI, are synthesized by the chemical co-precipitation method. The synthesized MNPs were characterized by the dynamic light scattering and transmission electron microscopy methods. It was shown that the magnitude and sign of the zeta potential of the MNPs were influenced by the nature of the modifier and pH of the solution. The effect of pH, the amount of sorbent, the sorption time, and the method of mixing the solution were studied and the optimal conditions for the sorption of quercetin and rutin were found. It was established that the sorption of flavonoids quantitatively occurs on magnetite, modified both with SiO2@PEI and only PEI, but the degree of extraction is higher on MNPs modified with PEI, which for quercetin and rutin was 98% and 86%, respectively. The highest degree of extraction of quercetin and rutin from the volume of 4 ml at the concentration of 10-6 -10-5 M was achieved at the pH of 10-11, the mechanical stirring time was 10 min and the mass of sorbent was 10 mg, the desorption time was 20 minutes. The modification of magnetite by PEI and the preconcentration of flavonoids were fast and could be used for their determination in objects.Методом химического соосаждения синтезированы магнитные наночастицы (МНЧ) магнетита, поверхность которых модифицирована диоксидом кремния, диоксидом кремния и полиэтиленимином и только полиэтиленимином. Полученные МНЧ охарактеризованы методами электрофоретического рассеяния света и просвечивающей электронной микроскопии. Показано, что на величину и знак дзета-потенциала МНЧ влияют природа модификатора и рН раствора. Изучено влияние рН, количества сорбента, времени сорбции, способа перемешивания раствора и найдены оптимальные условия сорбции кверцетина и рутина. Установлено, что сорбция указанных флавоноидов количественно происходит на магнетите, модифицированном как SiO2@ПЭИ так и только ПЭИ, протекает за 10 мин, однако степень извлечения выше на МНЧ, модифицированных ПЭИ, которая для кверцетина и рутина составляет 98 % и 86 %, соответственно. Показано, что степень извлечения кверцетина и рутина при десорбции 4 мл 0.1 М NaOH в течение 20 минут составляет 62 и 56 процентов, соответственно.This work was supported by the Russian Foundation for Fundamental Research, project no. 18-03-01029a.Работа выполнена при поддержке Российского фонда фундаментальных исследований, проект № 18-03-01029а