Experimental data on the time evolution of value of Russian rouble are approximated with elementary functions. Range since 2014.02.20 to 2015 February is considered. Data that are already available for the moment of submission are used. Four 3-parametric approximations are compared. The slow dependence of these approximation on the date of its preparation is interpreted as high predictive ability of such a tting. The approximations are suggested as a guide to establish the time-dependent currency band. Such a band could be useful for the stabilisation of short time uctuations of the currency exchange rate. Mathematics Subject Classication: 65D99 Numerical approximation

Holomorphic extension of the Ackermann function is suggested.Algorithms of evaluation of tetration and pentation are discussed and illustrated with explicit plots and complex maps.

An amplifier is characterized by its transfer function T , which expresses the dependence of the output signal on the input signal. This signal may be related to power, intensity, energy of a pulse, or its fluence, or any similar physical quantity. The internal structure of the amplified signal (e.g., its spectral content, polarization, temporal behavior, and spatial distribution) is not taken into account. The amplifier is considered to be spatially homogeneous and uniformly pumped. The transfer function is supposed to be known (measured in an experiment). The problem of reconstruction of the behavior of the signal inside the amplifier is formulated. For a given transfer function T , the evolution of the signal inside is interpreted as the superfunction F , satisfying the transfer equation F ( z + 1) T(F(z)) , where z is of coordinate along the propagation direction, while the length of the amplifier is used as a unit of measurement. (For simplicity, distances are measured in units of the length of the amplifier.) Two examples of realistic transfer function T are considered; they correspond to amplification of continuous wave and to amplification of pulses. In these examples, the transfer function and the distribution of the signal along the amplifier can be expressed in terms of special functions. The iterative procedure is suggested as a general method of reconstructing the signal along the amplifier, if neither the transfer function T , nor the superfunction F can be expressed with a simple combination of special functions. The examples show that the iterations converge to a physically meaningful solution. This method is expected to be useful for the characterization of laser materials from the measurement of the transfer function of a bulk sample.

For the transfer function T(z)=z+exp(z), the entire superfunction f is constructed as solution of the transfer equation f(z+1)=T(f(z)). The ecient algorithm for evaluation of Superfunction f is suggested. Its logarithmic asymptotic behaviour is detected. The application for emulation of the electric eld of a charged wire in empty space is discussed. Mathematics Subject Classication: 30D15, 30C30

TORI refers to the Tools for Outstanding Research and Investigation; the site with this name has been available since March 2011 to February 2013 at http://tori.ils.uec.ac.jp; the clone is available at http://mizugadro.mydns.jp/t. TORI is based on 6 axioms; any scientific concept is postulated to have the following properties: applicability, verifiability, refutability, self-consistency, principle of correspondence and pluralism. The examples of application in physics are suggested.

Propertes (absorption, gain) of a medium are supposed to be functions of the intensity of light. For the straightforward characterization at a given intensity, either the physical model should be used (to extract from the measurements only the parameters of the model), or the sample should be thin, that allows to neglect variation of intensity within the example. The measurement of the transfer function of the bulk sample is suggested as an alternative. The transfer function of a sample determines the output intensity in terms of the input intensity. For the given transfer function, the distribution of intensity within the sample appears as the superfunction. Method for the recovery of the superfunction from the transfer function is suggested.

Guiding of waves between parallel absorbing walls is considered. The principal mode is constructed; its absorption is estimated. The agreement with previous results about reflection of waves from absorbing walls is discussed. Roughly, the effective absorption of the principal mode is proportional the minus third power of the distance between walls, minus 1.5 d power of the wavenumber and minus 0.5 power of the local absorption of the wave in the wall. This estimate is suggested as hint for the design of the atomic waveguides, and also as tool for optimization of attenuation of the amplified spontaneous emission (and suppression of parasitic oscillations) in high power lasers.

The two regular super-exponentials to base exp(1/e) are constructed. An efficient algorithm for the evaluation of these super-exponentials and their inverse functions is suggested and compared to the already published results.

We give a simple uniqueness criterion (and some derived crite- ria) for holomorphic Abel functions and show that Kneser's real analytic Abel function of the exponential is subject to the criterion. Mathematics Subject Classification (2010). Primary 30D05.

We feature different (mostly underexplored) methods for non-integer iteration that accumulated over the past decades and centuries. We show their relation if known and test them by application to exponentials. We prove that the matrix power iteration, when applied to a fixed point, is the (classic) regular iteration at that fixed point.

The holomorphic function h is constructed such that h h z = z!; this function is interpreted as square root of Factorial.

We introduce the concept; of regular super-functions at a fixed point. It is derived from the concept of regular iteration. A super-function F of h is a solution of F(z+1)=h(F(z)). We provide a condition for F being entire, we also give two uniqueness criteria for regular super-functions.In the particular case h(x)=b boolean AND x we call F super-exponential. h has two real fixed points for b between 1 and e boolean AND(1/e). Exemplary we choose the base b=sqrt(2) and portray the four classes of real regular super-exponentials in the complex plane. There are two at fixed point 2 and two at fixed point 4. Each class is given by the translations along the x-axis of a suitable representative.Both super-exponentials at fixed point 4-one strictly increasing and one strictly decreasing-are entire. Both super-exponentials at fixed point 2-one strictly increasing and one strictly decreasing-are holomorphic on a right half-plane. All four super-exponentials are periodic along the imaginary axis. Only the strictly increasing super-exponential at 2 can satisfy F(0)=1 and can hence be called tetrational.We develop numerical algorithms for the precise evaluation of these functions and their inverses in the complex plane. We graph the two corresponding different half-iterates of h(z)=sqrt(2)boolean AND z. An apparent symmetry of the tetrational to base sqrt(2) disproved.

Tetration F as the analytic solution of equations F(z - 1) = ln(F(z)), F(0) = 1 is considered. The representation is suggested through the integral equation for values of F at the imaginary axis. Numerical analysis of this equation is described. The straightforward iteration converges within tens of cycles; with double precision arithmetics, the residual of order of 1.e-14 is achieved. The numerical solution for F remains finite at the imaginary axis, approaching fixed points L, L* of logarithm (L = In L). Robustness of the convergence and smallness of the residual indicate the existence of unique tetration F(z), that grows along the real axis and approaches L along the imaginary axis, being analytic in the whole complex z-plane except for singularities at integer the z < -1 and the cut at z < -2. Application of the same method for other cases of the Abel equation is discussed.

The quantum mechanical system with continuum of modes is considered as model of the decaying atom. The exponential decay and the Lorentian shape of distribution of the emitted photon are deduced as the non-perturbative solution of the time-dependent Schrodinger equation. The possible application of the formalism to the decay of trapped atoms is discussed.

Limit of power from amplified spontaneous emission (ASE), round-trip loss, and overheating is considered within simple model. The optimization of the output power with respect to size of the pumped region, its thickness and round-trip gain leads to the scaling laws for these parameters. In vicinity of the limit of power scaling, the round-trip loss should scale inversely proportional to the cubic root of the desired output power. This prediction agrees with the experimental data. The use of the thick undoped layer (anti-ASE cap) allows to increase for an order of magnitude the maximal power achievable at the same round-trip loss.

Thin disk laser is analyzed, assuming that the heat is drained in the same direction, as the optical pulse, withdrawing the stored energy, comes. Amplified spontaneous emission, the background loss, and overheating are taken into account with simple model. The scaling laws of the basic parameters are deduced. For the case of fixed repetition rate, the upper bound of thickness is obtained. Key parameters are suggested. The key energy parameter is promoted as criterion for evaluation of different laser materials for the high energy, high mean power disk lasers. The maximum energy per active element is estimated. For the scaling up the power and/or energy withdrawn from a single active element, the background loss should scale down inversely proportional to the cube of the background loss. This scaling law gives the criterion whether the heat should be drained in the direction orthogonal to the beam that withdraws the energy.

Existence of analytic extension of the fourth Ackermann function A(4,z) to the complex z plane is supposed. This extension is assumed to remain finite at the imaginary axis. On the base of this assumption, the algorithm is suggested for evaluation of this function. The numerical implementation with double-precision arithmetics leads to residual at the level of rounding errors. Application of the algorithm to more general cases of the Abel equation is discussed.

The optimum design of a powerful thin-disk laser implies a compromise between amplified spontaneous emission (ASE), overheating, and the round-trip losses. The power enhancement of a composite thin-disk laser made of an undoped layer bonded over a thin active layer to reduce ASE losses is estimated analytically. Scaling laws for the parameters of a disk laser are suggested for cases both with and without an anti-ASE cap. Predictions of the maximal power achievable for a given laser material are compared to the published experimental data. The anti-ASE cap allows an increase of the maximal output power proportional to the square of the logarithm of the round-trip loss.