Confidence: Its role in dependability cases for risk assessment

Society is increasingly requiring quantitative assessment of risk and associated dependability cases. Informally, a dependability case comprises some reasoning, based on assumptions and evidence, that supports a dependability claim at a particular level of confidence. In this paper we argue that a quantitative assessment of claim confidence is necessary for proper assessment of risk. We discuss the way in which confidence depends upon uncertainty about the underpinnings of the dependability case (truth of assumptions, correctness of reasoning, strength of evidence), and propose that probability is the appropriate measure of uncertainty. We discuss some of the obstacles to quantitative assessment of confidence (issues of composability of subsystem claims; of the multi-dimensional, multi-attribute nature of dependability claims; of the difficult role played by dependence between different kinds of evidence, assumptions, etc). We show that, even in simple cases, the confidence in a claim arising from a dependability case can be surprisingly low

Hopf algebras and characters of classical groups

Schur functions provide an integral basis of the ring of symmetric functions. It is shown that this ring has a natural Hopf algebra structure by identifying the appropriate product, coproduct, unit, counit and antipode, and their properties. Characters of covariant tensor irreducible representations of the classical groups GL(n), O(n) and Sp(n) are then expressed in terms of Schur functions, and the Hopf algebra is exploited in the determination of group-subgroup branching rules and the decomposition of tensor products. The analysis is carried out in terms of n-independent universal characters. The corresponding rings, CharGL, CharO and CharSp, of universal characters each have their own natural Hopf algebra structure. The appropriate product, coproduct, unit, counit and antipode are identified in each case.Comment: 9 pages. Uses jpconf.cls and jpconf11.clo. Presented by RCK at SSPCM'07, Myczkowce, Poland, Sept 200

Toward a Formalism for Conservative Claims about the Dependability of Software-Based Systems

In recent work, we have argued for a formal treatment of confidence about the claims made in dependability cases for software-based systems. The key idea underlying this work is "the inevitability of uncertainty": It is rarely possible to assert that a claim about safety or reliability is true with certainty. Much of this uncertainty is epistemic in nature, so it seems inevitable that expert judgment will continue to play an important role in dependability cases. Here, we consider a simple case where an expert makes a claim about the probability of failure on demand (pfd) of a subsystem of a wider system and is able to express his confidence about that claim probabilistically. An important, but difficult, problem then is how such subsystem (claim, confidence) pairs can be propagated through a dependability case for a wider system, of which the subsystems are components. An informal way forward is to justify, at high confidence, a strong claim, and then, conservatively, only claim something much weaker: "I'm 99 percent confident that the pfd is less than 10-5, so it's reasonable to be 100 percent confident that it is less than 10-3." These conservative pfds of subsystems can then be propagated simply through the dependability case of the wider system. In this paper, we provide formal support for such reasoning

SU(3) Quantum Interferometry with single-photon input pulses

We develop a framework for solving the action of a three-channel passive optical interferometer on single-photon pulse inputs to each channel using SU(3) group-theoretic methods, which can be readily generalized to higher-order photon-coincidence experiments. We show that features of the coincidence plots vs relative time delays of photons yield information about permanents, immanants, and determinants of the interferometer SU(3) matrix

Parafermions, parabosons and representations of so(\infty) and osp(1|\infty)

The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra so(\infty) and of the Lie superalgebra osp(1|\infty). A complete solution to the problem is presented, in which the Fock spaces have basis vectors labelled by certain infinite but stable Gelfand-Zetlin patterns, and the transformation of the basis is given explicitly. We also present expressions for the character of the Fock space representations

Energy evolution in time-dependent harmonic oscillator

The theory of adiabatic invariants has a long history, and very important implications and applications in many different branches of physics, classically and quantally, but is rarely founded on rigorous results. Here we treat the general time-dependent one-dimensional harmonic oscillator, whose Newton equation $\ddot{q} + \omega^2(t) q=0$ cannot be solved in general. We follow the time-evolution of an initial ensemble of phase points with sharply defined energy $E_0$ at time $t=0$ and calculate rigorously the distribution of energy $E_1$ after time $t=T$, which is fully (all moments, including the variance $\mu^2$) determined by the first moment $\bar{E_1}$. For example, $\mu^2 = E_0^2 [(\bar{E_1}/E_0)^2 - (\omega (T)/\omega (0))^2]/2$, and all higher even moments are powers of $\mu^2$, whilst the odd ones vanish identically. This distribution function does not depend on any further details of the function $\omega (t)$ and is in this sense universal. In ideal adiabaticity $\bar{E_1} = \omega(T) E_0/\omega(0)$, and the variance $\mu^2$ is zero, whilst for finite $T$ we calculate $\bar{E_1}$, and $\mu^2$ for the general case using exact WKB-theory to all orders. We prove that if $\omega (t)$ is of class ${\cal C}^{m}$ (all derivatives up to and including the order $m$ are continuous) $\mu \propto T^{-(m+1)}$, whilst for class ${\cal C}^{\infty}$ it is known to be exponential $\mu \propto \exp (-\alpha T)$.Comment: 26 pages, 5 figure

Hermitian Young Operators

Starting from conventional Young operators we construct Hermitian operators which project orthogonally onto irreducible representations of the (special) unitary group.Comment: 15 page

Integrity bases for local invariants of composite quantum systems

Unitary group branchings appropriate to the calculation of local invariants of density matrices of composite quantum systems are formulated using the method of $S$-function plethysms. From this, the generating function for the number of invariants at each degree in the density matrix can be computed. For the case of two two-level systems the generating function is $F(q) = 1 + q + 4q^{2} + 6 q^{3} + 16 q^{4} + 23 q^{5} + 52 q^{6} + 77 q^{7} + 150 q^{8} + 224 q^{9} + 396 q^{10} + 583 q^{11}+ O(q^{12})$. Factorisation of such series leads in principle to the identification of an integrity basis of algebraically independent invariants. This note replaces Appendix B of our paper\cite{us} J Phys {\bf A33} (2000) 1895-1914 (\texttt{quant-ph/0001076}) which is incorrect.Comment: Latex, 4 pages, correcting Appendix B of quant-ph/0001076 Error in $F(q)$ corrected and conclusions modified accordingl

Elasticity-driven Nanoscale Texturing in Complex Electronic Materials

Finescale probes of many complex electronic materials have revealed a non-uniform nanoworld of sign-varying textures in strain, charge and magnetization, forming meandering ribbons, stripe segments or droplets. We introduce and simulate a Ginzburg-Landau model for a structural transition, with strains coupling to charge and magnetization. Charge doping acts as a local stress that deforms surrounding unit cells without generating defects. This seemingly innocuous constraint of elastic `compatibility', in fact induces crucial anisotropic long-range forces of unit-cell discrete symmetry, that interweave opposite-sign competing strains to produce polaronic elasto-magnetic textures in the composite variables. Simulations with random local doping below the solid-solid transformation temperature reveal rich multiscale texturing from induced elastic fields: nanoscale phase separation, mesoscale intrinsic inhomogeneities, textural cross-coupling to external stress and magnetic field, and temperature-dependent percolation. We describe how this composite textured polaron concept can be valuable for doped manganites, cuprates and other complex electronic materials.Comment: Preprin

Distribution of roots of random real generalized polynomials

The average density of zeros for monic generalized polynomials, $P_n(z)=\phi(z)+\sum_{k=1}^nc_kf_k(z)$, with real holomorphic $\phi ,f_k$ and real Gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative. We obtain compact expressions for both the regular component (generated by the complex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicinity of the real axis like $|\hbox{\rm Im}\,z|$. We present the low and high disorder asymptotic behaviors. Then we particularize to the large $n$ limit of the average density of complex roots of monic algebraic polynomials of the form $P_n(z) = z^n +\sum_{k=1}^{n}c_kz^{n-k}$ with real independent, identically distributed Gaussian coefficients having zero mean and dispersion $\delta = \frac 1{\sqrt{n\lambda}}$. The average density tends to a simple, {\em universal} function of $\xi={2n}{\log |z|}$ and $\lambda$ in the domain $\xi\coth \frac{\xi}{2}\ll n|\sin \arg (z)|$ where nearly all the roots are located for large $n$.Comment: 17 pages, Revtex. To appear in J. Stat. Phys. Uuencoded gz-compresed tarfile (.66MB) containing 8 Postscript figures is available by e-mail from [email protected]