Further Results on Exponentially Robust Stability of Uncertain Connection Weights of Neutral-Type Recurrent Neural Networks

Further results on the robustness of the global exponential stability of recurrent neural network with piecewise constant arguments and neutral terms (NPRNN) subject to uncertain connection weights are presented in this paper. Estimating the upper bounds of the two categories of interference factors and establishing a measuring mechanism for uncertain dual connection weights are the core tasks and challenges. Hence, on the one hand, the new sufficient criteria for the upper bounds of neutral terms and piecewise arguments to guarantee the global exponential stability of NPRNN are provided. On the other hand, the allowed enclosed region of dual connection weights is characterized by a four-variable transcendental equation based on the preceding stable NPRNN. In this way, two interference factors and dual uncertain connection weights are mutually restricted in the model of parameter-uncertainty NPRNN, which leads to a dynamic evolution relationship. Finally, the numerical simulation comparisons with stable and unstable cases are provided to verify the effectiveness of the deduced results.

Solution of the Grazing Goat Problem: A Conflict between Beauty and Pragmatism

What is the ideal solution of a problem in mathematics? It depends on your nerd ideology. Pure mathematicians worship the beauty of a mathematics result. Closed form solutions are particularly beautiful. Engineers and applied mathematicians, on the other hand, focus on the result independent of its beauty. If a solution exists and can be calculated, that's enough. The job is done. An example is solution of the grazing goat problem. A recent closed form solution in the form of a ratio of two contour integrals has been found for the grazing goat problem and its beauty has been admired by pure mathematicians. For the engineer and applied mathematician, numerical solution of the grazing goat problem comes from an easily derived transcendental equation. The transcendental equation, known for some time, was not considered a beautiful enough solution for the pure mathematician so they kept on looking until they found a closed form solution. The numerical evaluation of the transcendental equation is not as beautiful. It is not in closed form. But the accuracy of the solution can straightforwardly be evaluated to within any accuracy desired. To illustrate, we derive and solve the transcendental equation for a generalization of the grazing goat problem.

Efficient Biexciton Preparation in a Quantum Dot—Metal Nanoparticle System Using On-Off Pulses

We consider a hybrid nanostructure composed by semiconductor quantum dot coupled to a metallic nanoparticle and investigate the efficient creation of biexciton state in the quantum dot, when starting from the ground state and using linearly polarized laser pulses with on-off modulation. With numerical simulations of the coupled system density matrix equations, we show that a simple on-off-on pulse-sequence, previously derived for the case of an isolated quantum dot, can efficiently prepare the biexciton state even in the presence of the nanoparticle, for various interparticle distances and biexciton energy shifts. The pulse durations in the sequence are obtained from the solution of a transcendental equation.

Bed topography and marine ice-sheet stability

This paper examines the effect of basal topography and strength on the grounding-line position, flux and stability of rapidly-sliding ice streams. It does so by supposing that the buoyancy of the ice stream is small, and of the same order as the longitudinal stress gradient. Making this scaling assumption makes the role of the basal gradient and accumulation rate explicit in the lowest order expression for the ice flux at the grounding line and also provides the transcendental equation for the grounding-line position. It also introduces into the stability condition terms in the basal curvature and accumulation-rate gradient. These expressions revert to well-established expressions in circumstances in which the thickness gradient is large at the grounding line, a result which is shown to be the consequence of the non-linearity of the flow. The behaviour of the grounding-line flux is illustrated for a range of bed topographies and strengths. We show that, when bed topography at a horizontal scale of several tens of ice thicknesses is present, the grounding-line flux and stability have more complex dependencies on bed gradient than that associated with the ‘marine ice-sheet instability hypothesis’, and that unstable grounding-line positions can occur on prograde beds as well as stable positions on retrograde beds.

AN ASSESSMENT OF THE DISPLACEMENT OF A CAM FOLLOWER USING REGULA FALSI METHOD

Cam is a mechanical component that transforms circular motion to reciprocating motion by using mating component, called the follower. The principal aim of this work was to study and analyse the displacement of a cam-follower with Regula Falsi method and verify its input by using MATLAB and FORTRAN simulations. A study was conducted on angle of rotation and the displacement of the follower, which is equal to the radius of the cam given as transcendental equation to find the exact solution. The parameters such as initial guess, final guess, iteration counter and the desired displacement are involved in finding the angular displacement to the cam system in high speed rotation. The analysis was done using a computer programming that enables verification of the results obtained and ascertaining whether the inputs are correct or not for the displacement in cam follower system. The computer output showed results of the two data sets that yielded solutions and two that did not. The results revealed that the programme could be used to find the angular displacement corresponding to a given follower displacement for any cam; if the function CAMF is modified to include the appropriate radius function, r(x). The results further revealed that at a halve cycle of a rotating cam, which is equivalent to (x = 3.142 rad), is a solution that would provide the desired displacement of the follower (opening and closing of valves).

Asymptotics of the principal eigenvalue of the Laplacian in 2D periodic domains with small traps

We derive and numerically implement various asymptotic approximations for the lowest or principal eigenvalue of the Laplacian with a periodic arrangement of localised traps of small \[\mathcal{O}(\varepsilon )\] spatial extent that are centred at the lattice points of an arbitrary Bravais lattice in \[{\mathbb{R}^2}\] . The expansion of this principal eigenvalue proceeds in powers of \[\nu \equiv - 1/\log (\varepsilon {d_c})\] , where d c is the logarithmic capacitance of the trap set. An explicit three-term approximation for this principal eigenvalue is derived using strong localised perturbation theory, with the coefficients in this series evaluated numerically by using an explicit formula for the source-neutral periodic Green’s function and its regular part. Moreover, a transcendental equation for an improved approximation to the principal eigenvalue, which effectively sums all the logarithmic terms in powers of v, is derived in terms of the regular part of the periodic Helmholtz Green’s function. By using an Ewald summation technique to first obtain a rapidly converging infinite series representation for this regular part, a simple Newton iteration scheme on the transcendental equation is implemented to numerically evaluate the improved ‘log-summed’ approximation to the principal eigenvalue. From a numerical computation of the PDE eigenvalue problem defined on the fundamental Wigner–Seitz (WS) cell for the lattice, it is shown that the three-term asymptotic approximation for the principal eigenvalue agrees well with the numerical result only for a rather small trap radius. In contrast, the log-summed asymptotic result provides a very close approximation to the principal eigenvalue even when the trap radius is only moderately small. For a circular trap, the first few transcendental correction terms that further improves the log-summed approximation for the principal eigenvalue are derived. Finally, it is shown numerically that, amongst all Bravais lattices with a fixed area of the primitive cell, the principal eigenvalue is maximised for a regular hexagonal arrangement of traps.

Analytical Solutions of the Fractional Mathematical Model for the Concentration of Tumor Cells for Constant Killing Rate

Two generalized mathematical models with memory for the concentration of tumor cells have been analytically studied using the cylindrical coordinate and the integral transform methods. The generalization consists of the formulating of two mathematical models with Caputo-time fractional derivative, models that are suitable to highlight the influence of the history of tumor evolution on the present behavior of the concentration of cancer cells. The time-oscillating concentration of cancer cells has been considered on the boundary of the domain. Analytical solutions of the fractional differential equations of the mathematical models have been determined using the Laplace transform with respect to the time variable and the finite Hankel transform with respect to the radial coordinate. The positive roots of the transcendental equation with Bessel function J0(r)=0, which are needed in our study, have been determined with the subroutine rn=root(J0(r),r,(2n−1)π/4,(2n+3)π/4),n=1,2,… of the Mathcad 15 software. It is found that the memory effects are stronger at small values of the time, t. This aspect is highlighted in the graphical illustrations that analyze the behavior of the concentration of tumor cells. Additionally, the concentration of cancer cells is symmetric with respect to radial angle, and its values tend to be zero for large values of the time, t.

Ferromagnetic Hysteresis by Heisenberg Partition Function and Its Processing Methods in Nanoparticle Magnetization Modeling

The Heisenberg {\it ab initio} theory of magnetization is developed to apply for multilayer nanoparticles. The theory is based on distribution and partition functions modification with account the difference between exchange integral and closest neighbour numbers, that change the system of resulting transcendental equation for magnetization and its reversal to form either a paramagnetic type curve or hysteresis loops patterns. The equations are obtained within the Heisenberg partition function construction by Heitler diagonalization of energy matrix via irreducible representations of permutation symmetry group. A combination with the Gauss distribution gives the explicit expression for the partition function in the asymptotic limit] at large spin range in terms of transcendent function. The exchange integral, as a parameter of the equation of state (material equation) is evaluated from Curie temperature value by means of a formula derived within the presented theory. Methods of data processing from the simultaneous solution of the material equation system are proposed. The multi-valued function of hysteresis loop is found by combination of graphical approach and special procedure for elimination of mistaken peaks and prolapses of the patterns. The theory and computation methods are applied to spherical particles with separate surface layers consideration. The contribution of the surface layers, that are specified by number of closest neighbors and exchange integrals into overall magnetization, is studied for two-layer and three-layer models, that are discussed and compared graphically.