Rational Minimax Iterations for Computing the Matrix pth Root

It was revealed that liquid water surfaces, including water drop’s surface might generate Negative Water Ions/molecules to air with probability of (1/11)⁸ among the water surface molecules by the author’s previous paper. Three experiments by spraying water from shower nozzle had been performed on a bathtub covered by soft plastic film with thickness of 10 mm during Aug.-Sep. 2019. The densities of negative ion (DNI) and the atmospheric parameters had been recorded. The experimental results showed that DNI increased immediately after spraying waters in orders with big variations. High DNI up to -190000 ion/cm³ were recorded under spraying water temperatures about 45°C. The sliding averaged (for 10 instant values) DNI were direct proportion to differences of saturated vapor pressure of spraying water and air vapor pressure, and increasing of relative humidity (under constant air temperature). The liquid water drops become positively charged after evaporation with negative ions. After stopping water spray DNI might keep at high level in a short time interval, then decrease gradually with vibrations by processes of mixing and water phase exchange in a cloud-fog system with -/+ air ions. These experiments proved that neutral liquid water drops may generate air negative water ions/molecules by evaporation and positive charged water drops.

In a previous paper [1] we considered the possibility that (within the\nearly-radiation epoch) there has been (also) a short period of a significant\npresence of cosmic strings. During this radiation-plus-strings stage the\nUniverse matter-energy content can be modelled by a two-component fluid,\nconsisting of radiation (dominant) and a cosmic-string fluid (subdominant). It\nwas found that, during this stage, the cosmological gravitational waves (CGWs)\n- that had been produced in an earlier (inflationary) epoch - with comoving\nwave-numbers below a critical value (which depends on the physics of the\ncosmic-string network) were filtered, leading to a distorsion in the expected\n(scale-invariant) CGW power spectrum. In any case, the cosmological evolution\ngradually results in the scaling of any long-cosmic-string network and, hence,\nafter a short time-interval, the Universe enters into the late-radiation era.\nHowever, along the transition from an early-radiation epoch to the\nlate-radiation era through the radiation-plus-strings stage, the\ntime-dependence of the cosmological scale factor is modified, something that\nleads to a discontinuous change of the corresponding scalar curvature, which,\nin turn, triggers the quantum-mechanical creation of gravitons. In this paper\nwe discuss several aspects of such a process, and, in particular, the\nobservational consequences on the expected gravitational-wave (GW) power\nspectrum.\n

Abstract. For solving a linear ill-posed problem, a combination of Tikhonov regularization and a finite dimensional projection method is considered. In the present paper we treat the dimension of the projection as the second parameter of regularization. The method using the discrepancy set of all regularization parameter pairs which satisfy a certain discrepancy principle is investigated. For the case of truncated SVD and LSQ projection methods, under the standard source conditions, an order of convergence is derived. This order of convergence is optimal under the same conditions as those for Tikhonov regularization with the discrepancy principle and without additional discretization.

An alternative method for local surveillance of coolant flow in BWRs fuel channels

We revisit our investigation of the diffusion Monte Carlo (DMC) simulation of para-diiodobenzene (p-DIB) molecular crystal polymorphism. [See J. Phys. Chem. Lett. 2010, 1, 1789-1794.] We perform, for the first time, a rigorous study of finite-size effects and choice of nodal surface on the prediction of polymorph stability in molecular crystals using fixed-node DMC. Our calculations are the largest that are currently feasible using the resources of the K-computer and provide insights into the formidable challenge of predicting such properties from first principles. In particular, we show that finite-size effects can influence the trial nodal surface of a small (1 × 1 × 1) simulation cell considerably. Therefore, we repeated our DMC simulations with a 1 × 3 × 3 simulation cell, which is the largest such calculation to date. We used a density functional theory (DFT) nodal surface generated with the PBE functional, and we accumulated statistical samples with ∼6.4 × 10(5) core hours for each polymorph. Our final results predict a polymorph stability that is consistent with experiment, but they also indicate that the results in our previous paper were somewhat fortuitous. We analyze the finite-size errors using model periodic Coulomb (MPC) interactions and kinetic energy corrections, according to the CCMH scheme of Chiesa, Ceperley, Martin, and Holzmann. We investigate the dependence of the finite-size errors on different aspect ratios of the simulation cell (k-mesh convergence) in order to understand how to choose an appropriate ratio for the DMC calculations. Even in the most expensive simulations currently possible, we show that the finite size errors in the DMC total energies are much larger than the energy difference between the two polymorphs, although error cancellation means that the polymorph prediction is accurate. Finally, we found that the T-move scheme is essential for these massive DMC simulations in order to circumvent population explosions and large time-step biases.

In the development of adaptive systems in control theory and signal processing, it frequently occurs that the problem of orthonormalization must be addressed. This paper explored the underlying mathematical framework of developing orthonormalization methods that are free of computing matrix square roots. These algorithms are easily modified so that minor and principal component analysis methods are developed. The proposed methods have several important features: 1) higher order convergence can be achieved by choosing a specific stepsize, 2) the methods can be used to compute square root of positive definite matrices.

ABSTRACTThis paper aims to obtain decompositions of higher dimensional functions into sums of non-tangential boundary limits of the corresponding Hardy space functions on tubes for the index range . In the one-dimensional case, Deng and Qian recently obtained such a Hardy space decomposition result: for any function , there exist functions and such that , where and are, respectively, the non-tangential boundary limits of some Hardy space functions in the upper-half and lower-half planes. In the present paper, we generalize the one-dimensional Hardy space decomposition result to the higher dimensions and discuss the uniqueness issue of such decomposition.

A method for rational approximation of linear fractional order systems (LFOS) is presented in the present paper. The method is computationally efficient, flexible and effective, as is illustrated by numerous examples. The proposed approach can also be used as an intermediate stage in designing indirect discrete rational approximations.

Mathematical modelling of fluidised bed dryers with horizontal dispersion

In the present paper, a spectral meshless radial point interpolation (SMRPI) technique is applied to solve the Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative in two dimensional case. The time fractional derivative is described in the Riemann–Liouville sense. The applied approach is based on combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of SMRPI. It is proved the scheme is unconditionally stable with respect to the time variable in \(H^1\) and convergent with the order of convergence \(\mathcal {O}(\delta t^{1+\beta })\), \(0<\beta <1\). In the current work, the thin plate splines (TPS) are used as the basis functions. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme. Two numerical examples show that the SMRPI has reliable accuracy in general shape domains.

In the present paper, by approximating the derivatives in the Newton-Steffensen third-order method by central difference quotient, we obtain a new modification of this method free from derivatives. We prove that the method obtained preserves their order of convergence, without calculating any derivative. Finally, numerical tests confirm that our method give the better performance as compare to the other well known derivative free Steffensen type methods.

In the general case a real irrational number cannot be approximated by infinitely many rationalsp/q involving error terms less than q-2 when the denominatorsq are taken from a given thin set of positive integers. The distribution of irrationals which are situated in close neighborhoods of infinitely many fractionsp/q, whereq is restricted to the elements of a thin set, depends on the asymptotic behaviour of theq’s and on their arithmetic properties.

This paper is devoted to studying the asymptotic behaviour of solutions to generalized incommensurate fractional systems. To this end, we first consider fractional systems with rational orders and introduce a criterion that is necessary and sufficient to ensure the stability of such systems. Next, from the fractional order pseudospectrum definition proposed by Šanca et al., we formulate the concept of a rational approximation for the fractional spectrum of incommensurate fractional systems with general, not necessarily rational, orders. Our first important new contribution is to show the equivalence between the fractional spectrum of a incommensurate linear system and its rational approximation. With this result in hand, we use ideas developed in our earlier work to demonstrate the stability of an equilibrium point to nonlinear systems in arbitrary finite-dimensional spaces. A second novel aspect of our work is the fact that the approach is constructive. It is effective and widely applicable in studying the asymptotic behaviour of solutions to linear incommensurate fractional differential systems with constant coefficient matrices and linearized stability theory for nonlinear incommensurate fractional differential systems. Finally, we give numerical simulations to illustrate the merit of the proposed theoretical results.

The possibility is raised of adding a randomly fluctuating interaction term to the Schrodinger equation, so that the new equation “reduces” the state vector. The exact event that occurs is predicted by the equation and depends upon the precise time dependence of the interaction term. The uncertainty in nature is attributed to the random behavior of this term. A class of such terms is found. This class includes terms whose nonlinear dependence on the wave function is identical to that of terms introduced in a previous paper for a similar purpose. In the previous paper, the exact event predicted depends upon the initial phase factors in the superposition making up the state vector: the uncertainty in nature is attributed to random initial phase factors. Another derivation of the results in the previous paper is given in an appendix: the calculations in that paper and in this appendix are of second order in perturbation theory. On the other hand, the calculations in the present paper are exact. A possible answer is given to the question, raised in the previous paper, of the nature of the “observable” states to which the state vector reduces.

In a previous paper, I proved the consistency of a non-finitary system of logic based on the theory of types, which was shown to contain the axiom of reducibility in a form which seemed not to interfere with the classical construction of real numbers. A form of the system containing a strong axiom of choice was also proved consistent.It seems to me now that the real-number approach used in that paper, though valid, was not the most fruitful one. We can, on the lines therein suggested, prove the consistency of axioms closely resembling Tarski's twenty axioms for the real numbers; but this, from the standpoint of mathematical practice, is a pitifully small fragment of analysis. The consistency of a fairly strong set-theory can be proved, using the results of my previous paper, with little more difficulty than that of the Tarski axioms; this being the case, it would seem a saving in effort to derive the consistency of such a theory first, then to strengthen that theory (if possible) in such ways as can be shown to preserve consistency; and finally to derive from the system thus strengthened, if need be, a more usable real-number theory. The present paper is meant to achieve the first part of this program. The paragraphs of this paper are numbered consecutively with those of my previous paper, of which it is to be regarded as a continuation.