Abstract:
A family of exact sum rules for the one-polaron spectral function in the low-density limit is derived. An algorithm to calculate energy moments of arbitrary order of the spectral function is presented. Explicit expressions are given for the first two moments of a model with general electron-phonon interaction, and for the first four moments of the Holstein polaron. The sum rules are linked to experiments on momentum-resolved photoemission spectroscopy. The bare electronic dispersion and the electron-phonon coupling constant can be extracted from the first and second moments of spectrum. The sum rules could serve as constraints in analytical and numerical studies of electron-phonon models.

Abstract:
We present an analytical expression for the static many-body local field factor $G_{+}(q)$ of a homogeneous two-dimensional electron gas, which reproduces Diffusion Monte Carlo data and embodies the exact asymptotic behaviors at both small and large wave number $q$. This allows us to also provide a closed-form expression for the exchange and correlation kernel $K_{xc}(r)$, which represents a key input for density functional studies of inhomogeneous systems.

Abstract:
We present an analytical expression for the local field factor G(q) of the homogeneous electron gas which reproduces recently published Quantum Monte--Carlo data by S. Moroni, D.M. Ceperley, and G. Senatore [Phys. Rev. Lett. 75, 689 (1995)], reflects the theoretically known asymptotic behaviours for both small and large q limits, and allows to express the exchange-correlation kernel K_{xc} analytically in both the direct and reciprocal spaces. The last property is particularly useful in numerical applications to real solids

Abstract:
We present some new analytical expressions for the so-called Parrondo effect, where simple coin-flipping games with negative expected value are combined into a winning game. Parrondo games are state-dependent. By identifying the game state variable, we can compute the stationary game state probabilities. Mixing losing games increases the probability of entering into a game state with positive value, as can be seen clearly from our analytical expressions for the Parrondo game value.

Abstract:
We apply both analytical and ab-initio methods to explore heterostructures composed of a threedimensional topological insulator (3D TI) and an ultrathin normal insulator (NI) overlayer as a proof ground for the principles of the topological phase engineering. Using the continual model of a semi-infinite 3D TI we study the surface potential (SP) effect caused by an attached ultrathin layer of 3D NI on the formation of topological bound states at the interface. The results reveal that spatial profile and spectrum of these near-surface states strongly depend on both the sign and strength of the SP. Using ab-initio band structure calculations to take materials specificity into account, we investigate the NI/TI heterostructures formed by a single tetradymite-type quintuple or septuple layer block and the 3D TI substrate. The analytical continuum theory results relate the near-surface state evolution with the SP variation and are in good qualitative agreement with those obtained from density-functional theory (DFT) calculations. We predict also the appearance of the quasi-topological bound state on the 3D NI surface caused by a local band gap inversion induced by an overlayer.

Abstract:
We present an analytical expression for the static many-body local field factor $G_{-}(q)$ of a homogeneous two-dimensional electron gas, which reproduces Diffusion Monte Carlo data and embodies the exact asymptotic behaviors at both small and large wave number $q$. This allows us to also provide a closed-form expression for the spin-antisymmetric exchange and correlation kernel $K^{-}_{xc}(r)$ which represents a key input for spin-density functional studies of inhomogeneous electronic systems.

Abstract:
The Sersic model has become the standard to parametrize the surface brightness distribution of early-type galaxies and bulges of spiral galaxies. A major problem is that the deprojection of the Sersic surface brightness profile to a luminosity density cannot be executed analytically for general values of the Sersic index. Mazure & Capelato (2002) used the Mathematica computer package to derive an expression of the Sersic luminosity density in terms of the Meijer G function for integer values of the Sersic index. We generalize this work using analytical means and use Mellin integral transforms to derive an exact, analytical expression for the luminosity density in terms of the Fox H function for all values of the Sersic index. We derive simplified expressions for the luminosity density, cumulative luminosity and gravitational potential in terms of the Meijer G function for all rational values of the Sersic index and we investigate their asymptotic behaviour at small and large radii. As implementations of the Meijer G function are nowadays available both in symbolic computer algebra packages and as high-performance computing code, our results open up the possibility to calculate the density of the Sersic models to arbitrary precision.

Abstract:
We present analytical expressions for the polarizability $P_\mu(q_x,\omega)$ of graphene modeled by the hexagonal tight-binding model for small wave number $q_x$, but arbitrary chemical potential $\mu$. Generally, we find $P_\mu(q_x,\omega)=P_\mu^<(\omega/\omega_q)+q_x^2P_\mu^>(\omega)$ with $\omega_q=v_Fq_x$ the Dirac energy, where the first term is due to intra-band and the second due to inter-band transitions. Explicitly, we derive the analytical expression for the imaginary part of the polarizability including intra-band contributions and recover the result obtained from the Dirac cone approximation for $\mu\rightarrow0$. For $\mu<\sqrt{3}t$, there is a square-root singularity at $\omega_q=v_Fq_x$ independent of $\mu$. For doping levels close to the van Hove singularity, $\mu=t\pm\delta\mu$, $ImP_\mu(q_x,\omega)$ is constant for $\delta\mu/t<\omega/\omega_q\ll1$.

Abstract:
For the calculation of charge excitations as those observed in, e.g., photo-emission spectroscopy or in electron-energy loss spectroscopy, a correct description of ground-state charge properties is essential. In strongly correlated systems like the undoped cuprates this is a highly non-trivial problem. In this paper we derive a non-perturbative analytical approximation for the ground state of the three-band Hubbard model on an infinite, half filled CuO_2 plane. By comparison with Projector Quantum Monte Carlo calculations it is shown that the resulting expressions correctly describe the charge properties of the ground state. Relations to other approaches are discussed. The analytical ground state preserves size consistency and can be generalized for other geometries, while still being both easy to interpret and to evaluate.

Abstract:
A mathematical model of potentiometric and amperometric enzyme electrodes is discussed. The model is based on the system of non-linear steady-state coupled reaction diffusion equations for Michaelis-Menten formalism that describe the concentrations of substrate and product within the enzymatic layer. Analytical expressions for the concentration of substrate, product and corresponding flux response have been derived for all values of parameters using Homotopy analysis method. The obtained solution allow a full characterization of the response curves for only two kinetic parameters (The Michaelis constant and the ratio of overall reaction and the diffusion rates). A simple relation between the concentration of substrate and products for all values of parameter is also reported. All the analytical results are compared with simulation results (Scilab/Matlab program). The simulated results are agreed with the appropriate theories. The obtained theoretical results are valid for the whole solution domain.