Following an idea by Joyner et al. [Europhys. Lett. 107, 50004 (2014)], a microwave graph with an antiunitary symmetry T obeying T^{2}=-1 is realized. The Kramers doublets expected for such systems are clearly identified and can be lifted by a perturbation which breaks the antiunitary symmetry. The observed spectral level spacings distribution of the Kramers doublets is in agreement with the predictions from the Gaussian symplectic ensemble expected for chaotic systems with such a symmetry.

We investigate spectral quantities of quantum graphs by expanding them as sums over pseudo orbits, sets of periodic orbits. Only a finite collection of pseudo orbits which are irreducible and where the total number of bonds is less than or equal to the number of bonds of the graph appear, analogous to a cut off at half the Heisenberg time. The calculation simplifies previous approaches to pseudo orbit expansions on graphs. We formulate coefficients of the characteristic polynomial and derive a secular equation in terms of the irreducible pseudo orbits. From the secular equation, whose roots provide the graph spectrum, the zeta function is derived using the argument principle. The spectral zeta function enables quantities, such as the spectral determinant and vacuum energy, to be obtained directly as finite expansions over the set of short irreducible pseudo orbits.

Energy levels statistics following the Gaussian Symplectic Ensemble (GSE) of Random Matrix Theory have been predicted theoretically and observed numerically in numerous quantum chaotic systems. However in all these systems there has been one unifying feature: the combination of halfinteger spin and time-reversal invariance. Here we provide an alternative mechanism for obtaining GSE statistics that is based on geometric symmetries of a quantum system which alleviates the need for spin. As an example, we construct a quantum graph with a particular discrete symmetry given by the quaternion group Q8. GSE statistics is then observed within one of its subspectra.In the 1950s and 1960s Wigner and Dyson pioneered the use of random matrices in modelling the statistical properties of the energy eigenvalues belonging to complicated quantum systems [1,2]. The techniques they developed spawned a new field of mathematics which has since become known as Random Matrix Theory (RMT) and its application has spread far and wide to many areas of Mathematics and Physics [3]. In particular it was later conjectured [4] that the high-lying quantum energy levels of classically chaotic systems are faithful to random matrix averages.One of the cornerstones of RMT is Dyson's three-fold way [2], which groups quantum systems without geometric symmetries into three distinct types. The first occurs if time-reversal invariance is broken, for example by a magnetic field, meaning the quantum Hamiltonian H is inherently complex. The remaining two appear if there is an antiunitary time-reversal operator T which leaves H invariant, i.e. [T , H] = 0. They are then distinguished by either T 2 = 1 or T 2 = −1, in which case H is real symmetric or quaternion-real respectively. For chaotic systems, RMT makes predictions in all three instances by averaging over an ensemble of Hermitian matrices with the appropriate internal structure and Gaussian weighted elements. These are referred to as the Gaussian Unitary, Orthogonal and Symplectic ensembles (GUE, GOE and GSE). We note that the number of symmetry classes can be extended to ten if additional anti-commuting symmetries are present [5,6] but this is beyond the scope of this letter.In systems without geometrical symmetries timereversal invariance with T 2 = −1, and hence GSE statistics, can only arise if the wavefunctions have an even number of components, commonly associated with halfinteger spin. For such systems GSE statistics have been predicted and/or observed numerically in examples such as quantum billiards [7], maps [8] and quantum graphs [9], and explained using periodic-orbit theory [10,11]. However to date there has been no experimental observation.For systems with geometric symmetries the situation becomes more involved. Here the Hilbert space decomposes into subspaces invariant under symmetry transformations, and the spectral statistics inside these subspaces depends both on the system's behaviour under time reversal and on the nature of the subspace. For example 3-fold rotationally invariant...

Abstract. We investigate the eigenvalue statistics of random Bernoulli matrices, where the matrix elements are chosen independently from a binary set with equal probability. This is achieved by initiating a discrete random walk process over the space of matrices and analysing the induced random motion of the eigenvalues -an approach which is similar to Dyson's Brownian motion model but with important modifications. In particular, we show our process is described by a Fokker-Planck equation, up to an error margin which vanishes in the limit of large matrix dimension. The stationary solution of which corresponds to the joint probability density function of certain well-known fixed trace Gaussian ensembles.

Abstract. We show how to obtain the joint probability distribution of the first two spectral moments for the GβE random matrix ensembles of any matrix dimension N . This is achieved via a simple method which utilises two complementary invariants of the domain of the spectral moments. Our approach is significantly different from those employed previously to answer related questions and potentially offers new insights. We also discuss the problems faced when attempting to include higher spectral moments.The probability distribution of spectral moments for the GβE 2

Abstract. We investigate the spectral statistics of Hermitian matrices in which the elements are chosen uniformly from U (1), called the uni-modular ensemble (UME), in the limit of large matrix size. Using three complimentary methods; a supersymmetric integration method, a combinatorial graph-theoretical analysis and a Brownian motion approach, we are able to derive expressions for 1/N corrections to the mean spectral moments and also analyse the fluctuations about this mean. By addressing the same ensemble from three different point of view, we can critically compare their relative advantages and derive some new results.

Background Low back pain (LBP) is an increasingly burdensome condition for patients and health professionals alike, with consistent demonstration of increasing persistent pain and disability. Previous decision support tools for LBP management have focused on a subset of factors owing to time constraints and ease of use for the clinician. With the explosion of interest in machine learning tools and the commitment from Western governments to introduce this technology, there are opportunities to develop intelligent decision support tools. We will do this for LBP using a Bayesian network, which will entail constructing a clinical reasoning model elicited from experts. Objective This paper proposes a method for conducting a modified RAND appropriateness procedure to elicit the knowledge required to construct a Bayesian network from a group of domain experts in LBP, and reports the lessons learned from the internal pilot of the procedure. Methods We propose to recruit expert clinicians with a special interest in LBP from across a range of medical specialties, such as orthopedics, rheumatology, and sports medicine. The procedure will consist of four stages. Stage 1 is an online elicitation of variables to be considered by the model, followed by a face-to-face workshop. Stage 2 is an online elicitation of the structure of the model, followed by a face-to-face workshop. Stage 3 consists of an online phase to elicit probabilities to populate the Bayesian network. Stage 4 is a rudimentary validation of the Bayesian network. Results Ethical approval has been obtained from the Research Ethics Committee at Queen Mary University of London. An internal pilot of the procedure has been run with clinical colleagues from the research team. This showed that an alternating process of three remote activities and two in-person meetings was required to complete the elicitation without overburdening participants. Lessons learned have included the need for a bespoke online elicitation tool to run between face-to-face meetings and for careful operational definition of descriptive terms, even if widely clinically used. Further, tools are required to remotely deliver training about self-identification of various forms of cognitive bias and explain the underlying principles of a Bayesian network. The use of the internal pilot was recognized as being a methodological necessity. Conclusions We have proposed a method to construct Bayesian networks that are representative of expert clinical reasoning for a musculoskeletal condition in this case. We have tested the method with an internal pilot to refine the process prior to deployment, which indicates the process can be successful. The internal pilot has also revealed the software support requirements for the elicitation process to model clinical reasoning for a range of conditions. International Registered Report Identifier (IRRID) DERR1-...

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