Input Design for System Identification via Convex Relaxation

This paper proposes a new framework for the optimization of excitation inputs for system identification. The optimization problem considered is to maximize a reduced Fisher information matrix in any of the classical D-, E-, or A-optimal senses. In contrast to the majority of published work on this topic, we consider the problem in the time domain and subject to constraints on the amplitude of the input signal. This optimization problem is nonconvex. The main result of the paper is a convex relaxation that gives an upper bound accurate to within $2/\pi$ of the true maximum. A randomized algorithm is presented for finding a feasible solution which, in a certain sense is expected to be at least $2/\pi$ as informative as the globally optimal input signal. In the case of a single constraint on input power, the proposed approach recovers the true global optimum exactly. Extensions to situations with both power and amplitude constraints on both inputs and outputs are given. A simple simulation example illustrates the technique.Comment: Preprint submitted for journal publication, extended version of a paper at 2010 IEEE Conference on Decision and Contro

Precision Determination of Invisible-Particle Masses at the CERN LHC: II

We further develop the constrained mass variable techniques to determine the mass scale of invisible particles pair-produced at hadron colliders. We introduce the constrained mass variable M_3C which provides an event-by-event lower bound and upper bound to the mass scale given the two mass differences between the lightest three new particle states. This variable is most appropriate for short symmetric cascade decays involving two-body decays and on-shell intermediate states which end in standard-model particles and two dark-matter particles. An important feature of the constrained mass variables is that they do not rely simply on the position of the end point but use the additional information contained in events which lie far from the end point. To demonstrate our method we study the supersymmetric model SPS 1a. We select cuts to study events with two Neutralino_2 each of which decays to Neutralino_1, and two opposite-sign same-flavor (OSSF) charged leptons through an intermediate on-shell slepton. We find that with 300 fb^-1 of integrated luminosity the invisible-particle mass can be measured to M=96.4 +/- 2.4 GeV. Combining fits to the shape of the M_3C constrained mass variable distribution with the max m_ll edge fixes the mass differences to +/- 0.2 GeV.Comment: 13 pages, 10 figure

P?=NP as minimization of degree 4 polynomial, integration or Grassmann number problem, and new graph isomorphism problem approaches

While the P vs NP problem is mainly approached form the point of view of discrete mathematics, this paper proposes reformulations into the field of abstract algebra, geometry, fourier analysis and of continuous global optimization - which advanced tools might bring new perspectives and approaches for this question. The first one is equivalence of satisfaction of 3-SAT problem with the question of reaching zero of a nonnegative degree 4 multivariate polynomial (sum of squares), what could be tested from the perspective of algebra by using discriminant. It could be also approached as a continuous global optimization problem inside $[0,1]^n$, for example in physical realizations like adiabatic quantum computers. However, the number of local minima usually grows exponentially. Reducing to degree 2 polynomial plus constraints of being in $\{0,1\}^n$, we get geometric formulations as the question if plane or sphere intersects with $\{0,1\}^n$. There will be also presented some non-standard perspectives for the Subset-Sum, like through convergence of a series, or zeroing of $\int_0^{2\pi} \prod_i \cos(\varphi k_i) d\varphi$ fourier-type integral for some natural $k_i$. The last discussed approach is using anti-commuting Grassmann numbers $\theta_i$, making $(A \cdot \textrm{diag}(\theta_i))^n$ nonzero only if $A$ has a Hamilton cycle. Hence, the P$\ne$NP assumption implies exponential growth of matrix representation of Grassmann numbers. There will be also discussed a looking promising algebraic/geometric approach to the graph isomorphism problem -- tested to successfully distinguish strongly regular graphs with up to 29 vertices.Comment: 19 pages, 8 figure