This paper addresses and establishes some of the fundamental barriers in the theory of computations and finally settles the long standing computational spectral problem.Due to the barriers presented in this paper, there are many problems, some of them at the heart of computational theory, that do not fit into the classical frameworks of complexity theory. Hence, we are in need for a new extended theory of complexity, capable of handling these new issues. Such a theory is presented in this paper. Many computational problems can be solved as follows: a sequence of approximations is created by an algorithm, and the solution to the problem is the limit of this sequence (think about computing eigenvalues of a matrix for example). However, as we demonstrate, for several basic problems in computations (computing spectra of infinite dimensional operators, solutions to linear equations or roots of polynomials using rational maps) such a procedure based on one limit is impossible. Yet, one can compute solutions to these problems, but only by using several limits. This may come as a surprise, however, this touches onto the definite boundaries of computational mathematics. To analyze this phenomenon we use the Solvability Complexity Index (SCI). The SCI is the smallest number of limits needed in order to compute a desired quantity. In several cases (spectral problems, inverse problems) we provide sharp results on the SCI, thus we establish the absolute barriers for what can be achieved computationally. For example, we show that the SCI of spectra and essential spectra of infinite matrices is equal to three, and that the SCI of spectra of self-adjoint infinite matrices is equal to two, thus providing the lower bound barriers and the first algorithms to compute such spectra in two and three limits. This finally settles the long standing computational spectral problem. We also show that the SCI of solutions to infinite linear systems is two.Moreover, we establish barriers on error control. We prove that no algorithm can provide error control on the computational spectral problem or solutions to infinite-dimensional linear systems. In particular, one can get arbitrarily close to the solution, but never knowing when one is "epsilon" away. This is universal for all algorithms regardless of operations allowed. In addition, we provide bounds for the SCI of spectra of classes of Schrödinger operators, thus we affirmatively answer the long standing question on whether or not these spectra can actually be computed. Finally, we show how the SCI provides a natural framework for understanding barriers in computations. It has a direct link to the Arithmetical Hierarchy, and in particular, we demonstrate how the impossibility result of McMullen on polynomial root finding with rational maps in one limit, and the framework of Doyle and McMullen on solving the quintic in several limits, can be put in the SCI framework. CONTENTS Remark 1.1 (Polynomial time algorithms and implementation).Note that all upper bounds on the SCI provided in this pap...

We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite-difference equations with random, possibly non-symmetric coefficients. Under the assumption that the coefficients are stationary and ergodic in the quantitative form of a Logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions d ≥ 2. Similar estimates have recently been obtained in the special case of diagonal coefficients making extensive use of the maximum principle and scalar techniques. Our new method only invokes arguments that are also available for elliptic systems and does not use the maximum principle. In particular, our proof relies on the LSI to quantify ergodicity and on regularity estimates on the derivative of the discrete Green's function in weighted spaces. In the critical case d = 2 our argument for the estimate on the gradient of the elliptic Green's function uses a Calderón-Zygmund estimate in discrete weighted spaces, which we state and prove. As applications, we provide a quantitative two-scale expansion and a quantitative approximation of the homogenized coefficients. 2 Main results and sketch of proof 2.1 General framework Discrete functions and derivatives. Let {e i } d i=1 denote the canonical basis of R d . For a scalar function u : Z d → R and a vector field g : Z d → R d with components

We consider the question of linear instability of an equilibrium of the Relativistic Vlasov-Maxwell (RVM) System that has a strong magnetic field. Standard instability results deal with systems where there are fewer particles with higher energies. In this paper we extend those results to the class of equilibria for which the number of particles does not depend monotonically on the energy. Without the standard sign assumptions, the analysis becomes significantly more involved.

The relativistic Vlasov-Maxwell system describes the evolution of a collisionless plasma. The problem of linear instability of this system is considered in two physical settings: the so-called "one and onehalf" dimensional case, and the three dimensional case with cylindrical symmetry. Sufficient conditions for instability are obtained in terms of the spectral properties of certain Schrödinger operators that act on the spatial variable alone (and not in full phase space). An important aspect of these conditions is that they do not require any boundedness assumptions on the domains, nor do they require monotonicity of the equilibrium.

We consider the 1 1 2 -dimensional relativistic Vlasov-Maxwell system that describes the time-evolution of a plasma. We find a relatively simple criterion for spectral instability of a wide class of equilibria. This class includes non-homogeneous equilibria that need not satisfy any additional symmetry properties (as was the case in previous results), nor should they be monotone in the particle energy. The criterion is given in terms of the spectral properties of two Schrödinger operators that arise naturally from Maxwell's equations. The spectral analysis of these operators is quite delicate, and some general functional analytic tools are developed to treat them. These tools can be applied to similar systems in higher dimensions, as long as their domain is finite or periodic. C