Abstract:
An algorithm for generating a class of closed form solutions to the Navier-Stokes equations is suggested, with examples. Of particular interest are those exact solutions that exhibit intermittency, tertiary Hopf bifurcations, flow reversal, and hysteresis.

Abstract:
In this paper, three analytic closed form solutions are introduced for arbitrary Nonuniform Transmission Lines (NTLs). The differential equations of NTLs are written in three suitable matrix equation forms, first. Then the matrix equations are solved to obtain the chain parameter matrix of NTLs. The obtained solutions are applicable to arbitrary lossy and dispersive NTLs. The validation of the proposed solutions is verified using some comprehensive examples.

Abstract:
For one-dimensional vibrating cavity systems appearing in the standard illustration of the dynamical Casimir effect, we propose an approach to the construction of exact closed-form solutions. As new results, we obtain solutions that are given for arbitrary frequencies, amplitudes and time regions. In a broad range of parameters, a vibrating cavity model exhibits the general property of exponential instability. Marginal behavior of the system manifests in a power-like growth of radiated energy.

Abstract:
This paper derives -- considering a Gaussian setting -- closed form solutions of the statistics that Adrian and Brunnermeier and Acharya et al. have suggested as measures of systemic risk to be attached to individual banks. The statistics equal the product of statistic specific Beta-coefficients with the mean corrected Value at Risk. Hence, the measures of systemic risks are closely related to well known concepts of financial economics. Another benefit of the analysis is that it is revealed how the concepts are related to each other. Also, it may be relatively easy to convince the regulators to consider a closed form solution, especially so if the statistics involved are well known and can easily be communicated to the financial community.

Abstract:
We present some results regarding metric perturbations in general relativity and other metric theories of gravity. In particular, using the Newman Penrose variables, we write down and discuss the equations which govern tensor field perturbations of ranks $0, \pm 1$ and $\pm 2$ (scalar,vector,tensor bosonic perturbations) over certain space-times that admit specific background metrics expressible in isotropic coordinates. Armed with these equations, we are able to develop the Hadamard series which can be associated with the fundamental solution of the equations, wherein we introduce an inhomogeneous singularity at the physical space-time point of the perturbing particle. The Hadamard series is then used to generate closed form solutions by making choices for an appropriate ansatz solution. In particular, we solve for the spin-weighted electrostatic potential for the Reissner-Nordstrom black hole and for the fully dynamical potential for the Friedmann-Robertson-Walker cosmological solution.

Abstract:
Partial differential equations (PDEs) combined with suitably chosen boundary conditions are effective in creating free form surfaces. In this paper, a fourth order partial differential equation and boundary conditions up to tangential continuity are introduced. The general solution is divided into a closed form solution and a non-closed form one leading to a mixed solution to the PDE. The obtained solution is applied to a number of surface modelling examples including glass shape design, vase surface creation and arbitrary surface representation.

Abstract:
We study power control in optimization and game frameworks. In the optimization framework there is a single decision maker who assigns network resources and in the game framework users share the network resources according to Nash equilibrium. The solution of these problems is based on so-called water-filling technique, which in turn uses bisection method for solution of non-linear equations for Lagrange multiplies. Here we provide a closed form solution to the water-filling problem, which allows us to solve it in a finite number of operations. Also, we produce a closed form solution for the Nash equilibrium in symmetric Gaussian interference game with an arbitrary number of users. Even though the game is symmetric, there is an intrinsic hierarchical structure induced by the quantity of the resources available to the users. We use this hierarchical structure to perform a successive reduction of the game. In addition, to its mathematical beauty, the explicit solution allows one to study limiting cases when the crosstalk coefficient is either small or large. We provide an alternative simple proof of the convergence of the Iterative Water Filling Algorithm. Furthermore, it turns out that the convergence of Iterative Water Filling Algorithm slows down when the crosstalk coefficient is large. Using the closed form solution, we can avoid this problem. Finally, we compare the non-cooperative approach with the cooperative approach and show that the non-cooperative approach results in a more fair resource distribution.

Abstract:
In this paper, we provide general closed-form solutions to the incomplete-market random-coefficient dynamic optimization problem without the restrictive assumption of exponential or HARA utility function. Moreover, we explicitly express the optimal portfolio as a function of the optimal consumption and show the impact of optimal consumption on the optimal portfolio.

Abstract:
In this paper we show how to find a closed form solution for third order difference operators in terms of solutions of second order operators. This work is an extension of previous results on finding closed form solutions of recurrence equations and a counterpart to existing results on differential equations. As motivation and application for this work, we discuss the problem of proving positivity of sequences given merely in terms of their defining recurrence relation. The main advantage of the present approach to earlier methods attacking the same problem is that our algorithm provides human-readable and verifiable, i.e., certified proofs.