To each discrete translationally periodic bar-joint framework in , we associate a matrix-valued function defined on the d-torus. The rigid unit mode (RUM) spectrum of is defined in terms of the multi-phases of phase-periodic infinitesimal flexes and is shown to correspond to the singular points of the function and also to the set of wavevectors of harmonic excitations which have vanishing energy in the long wavelength limit. To a crystal framework in Maxwell counting equilibrium, which corresponds to being square, the determinant of gives rise to a unique multi-variable polynomial . For ideal zeolites, the algebraic variety of zeros of on the d-torus coincides with the RUM spectrum. The matrix function is related to other aspects of idealized framework rigidity and flexibility, and in particular leads to an explicit formula for the number of supercell-periodic floppy modes. In the case of certain zeolite frameworks in dimensions two and three, direct proofs are given to show the maximal floppy mode property (order N). In particular, this is the case for the cubic symmetry sodalite framework and some other idealized zeolites.
doi:10.1098/rsta.2012.0030
PMCID: PMC3871295
PMID: 24379422
crystal framework; rigidity operator; crystal polynomial; rigid unit mode
Between the study of small finite frameworks and infinite incidentally periodic frameworks, we find the real materials which are large, but finite, fragments that fit into the infinite periodic frameworks. To understand these materials, we seek insights from both (i) their analysis as large frameworks with associated geometric and combinatorial properties (including the geometric repetitions) and (ii) embedding them into appropriate infinite periodic structures with motions that may break the periodic structure. A review of real materials identifies a number of examples with a local appearance of ‘unit cells’ which repeat under isometries but perhaps in unusual forms. These examples also refocus attention on several new classes of infinite ‘periodic’ frameworks: (i) columns—three-dimensional structures generated with one repeating isometry and (ii) slabs—three-dimensional structures with two independent repeating translations. With this larger vision of structures to be studied, we find some patterns and partial results that suggest new conjectures as well as many additional open questions. These invite a search for new examples and additional theorems.
doi:10.1098/rsta.2012.0032
PMCID: PMC3871296
PMID: 24379423
rigidity; flexibility; finite frameworks; infinite frameworks; fragments
The flexibility or otherwise of periodic tetrahedral TX2 frameworks formed by corner-sharing regular TX4 tetrahedra is discussed. In particular, when T–X–T angle constraints are included, a suitable embedding can often only be found, if at all, in an symmetry less than the maximum possible for that topology. Examples illustrating this are adduced.
doi:10.1098/rsta.2012.0034
PMCID: PMC3871297
PMID: 24379424
periodic nets; zeolites; symmetry
Group theory has been used to study various problems in physics and chemistry for many years. Relatively recently, applications have emerged in engineering, where problems of the vibration, bifurcation and stability of systems exhibiting symmetry have been studied. From an engineering perspective, the main attraction of group-theoretic methods has been their potential to reduce computational effort in the analysis of large-scale problems. In this paper, we focus on vibration problems in structural mechanics and reveal some of the insights and qualitative benefits that group theory affords. These include an appreciation of all the possible symmetries of modes of vibration, the prediction of the number of modes of a given symmetry type, the identification of modes associated with the same frequencies, the prediction of nodal lines and stationary points of a vibrating system, and the untangling of clustered frequencies.
doi:10.1098/rsta.2012.0037
PMCID: PMC3871298
PMID: 24379427
group theory; symmetry; structural mechanics; vibration; natural frequency; mode
We summarize results for two exactly soluble classes of bond-diluted models for rigidity percolation, which can serve as a benchmark for numerical and approximate methods. For bond dilution problems involving rigidity, the number of floppy modes F plays the role of a free energy. Both models involve pathological lattices with two-dimensional vector displacements. The first model involves hierarchical lattices where renormalization group calculations can be used to give exact solutions. Algebraic scaling transformations produce a transition of the second order, with an unstable critical point and associated scaling laws at a mean coordination 〈r〉=4.41, which is above the ‘mean field’ value 〈r〉=4 predicted by Maxwell constraint counting. The order parameter exponent associated with the spanning rigid cluster geometry is β=0.0775 and that associated with the divergence of the correlation length and the anomalous lattice dimension d is dν=3.533. The second model involves Bethe lattices where the rigidity transition is massively first order by a mean coordination 〈r〉=3.94 slightly below that predicted by Maxwell constraint counting. We show how a Maxwell equal area construction can be used to locate the first-order transition and how this result agrees with simulation results on larger random-bond lattices using the pebble game algorithm.
doi:10.1098/rsta.2012.0038
PMCID: PMC3871299
PMID: 24379428
rigidity; percolation; lattice; tree; hierarchical; renormalization
In kinematics, a framework is called overconstrained if its continuous flexibility is caused by particular dimensions; in the generic case, a framework of this type is rigid. Famous examples of overconstrained structures are the Bricard octahedra, the Bennett isogram, the Grünbaum framework, Bottema's 16-bar mechanism, Chasles’ body–bar framework, Burmester's focal mechanism or flexible quad meshes. The aim of this paper is to present some examples in detail and to focus on their symmetry properties. It turns out that only for a few is a global symmetry a necessary condition for flexibility. Sometimes, there is a hidden symmetry, and in some cases, for example, at the flexible type-3 octahedra or at discrete Voss surfaces, there is only a local symmetry. However, there remain overconstrained frameworks where the underlying algebraic conditions for flexibility have no relation to symmetry at all.
doi:10.1098/rsta.2012.0040
PMCID: PMC3871300
PMID: 24379430
overconstrained mechanism; Grünbaum framework; Kokotsakis mesh; flexible bipartite framework; Burmester's focal mechanism
It is well known that (i) the flexibility and rigidity of proteins are central to their function, (ii) a number of oligomers with several copies of individual protein chains assemble with symmetry in the native state and (iii) added symmetry sometimes leads to added flexibility in structures. We observe that the most common symmetry classes of protein oligomers are also the symmetry classes that lead to increased flexibility in certain three-dimensional structures—and investigate the possible significance of this coincidence. This builds on the well-developed theory of generic rigidity of body–bar frameworks, which permits an analysis of the rigidity and flexibility of molecular structures such as proteins via fast combinatorial algorithms. In particular, we outline some very simple counting rules and possible algorithmic extensions that allow us to predict continuous symmetry-preserving motions in body–bar frameworks that possess non-trivial point-group symmetry. For simplicity, we focus on dimers, which typically assemble with twofold rotational axes, and often have allosteric function that requires motions to link distant sites on the two protein chains.
doi:10.1098/rsta.2012.0041
PMCID: PMC3871301
PMID: 24379431
rigidity of frameworks; flexibility; symmetry; proteins; allostery; pebble game algorithms
Beat-to-beat variations in heart period provide information on cardiovascular control and are closely linked to variations in arterial pressure and respiration. Joint symbolic analysis of heart period, systolic arterial pressure and respiration allows for a simple description of their shared short-term dynamics that are governed by cardiac baroreflex control and cardiorespiratory coupling. In this review, we discuss methodology and research applications. Studies suggest that analysis of joint symbolic dynamics provides a powerful tool for identifying physiological and pathophysiological changes in cardiovascular and cardiorespiratory control.
doi:10.1098/rsta.2014.0097
PMCID: PMC4281868
PMID: 25548272
symbolic dynamics; heart rate; blood pressure; respiration; baroreflex; respiratory sinus arrhythmia
doi:10.1098/rsta.2014.0099
PMCID: PMC4281870
PMID: 25548265
symbolic dynamics; pattern classification; multivariate signal processing; entropy; transfer entropy; complexity
doi:10.1098/rsta.2014.0115
PMCID: PMC4275930
PMID: 25512591
offshore engineering; naval engineering; oil and gas engineering; fluid mechanics; modelling
We argue that the freezing transition scenario, previously conjectured to occur in the statistical mechanics of 1/f-noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials pN(θ) of large N×N random unitary (circular unitary ensemble) matrices UN; i.e. the extreme value statistics of pN(θ) when . In addition, we argue that it leads to multi-fractal-like behaviour in the total length μN(x) of the intervals in which |pN(θ)|>Nx,x>0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta function ζ(s) over stretches of the critical line of given constant length and present the results of numerical computations of the large values of ). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.
doi:10.1098/rsta.2012.0503
PMCID: PMC3866466
PMID: 24344336
random matrix theory; Riemann zeta function; extreme values
Sturm's oscillation theorem states that the nth eigenfunction of a Sturm–Liouville operator on the interval has n−1 zeros (nodes) (Sturm 1836 J. Math. Pures Appl.
1, 106–186; 373–444). This result was generalized for all metric tree graphs (Pokornyĭ et al. 1996 Mat. Zametki
60, 468–470 (doi:10.1007/BF02320380); Schapotschnikow 2006 Waves Random Complex Media
16, 167–178 (doi:10.1080/1745530600702535)) and an analogous theorem was proved for discrete tree graphs (Berkolaiko 2007 Commun. Math. Phys.
278, 803–819 (doi:10.1007/S00220-007-0391-3); Dhar & Ramaswamy 1985 Phys. Rev. Lett.
54, 1346–1349 (doi:10.1103/PhysRevLett.54.1346); Fiedler 1975 Czechoslovak Math. J.
25, 607–618). We prove the converse theorems for both discrete and metric graphs. Namely if for all n, the nth eigenfunction of the graph has n−1 zeros, then the graph is a tree. Our proofs use a recently obtained connection between the graph's nodal count and the magnetic stability of its eigenvalues (Berkolaiko 2013 Anal. PDE
6, 1213–1233 (doi:10.2140/apde.2013.6.1213); Berkolaiko & Weyand 2014 Phil. Trans. R. Soc. A
372, 20120522 (doi:10.1098/rsta.2012.0522); Colin de Verdière 2013 Anal. PDE
6, 1235–1242 (doi:10.2140/apde.2013.6.1235)). In the course of the proof, we show that it is not possible for all (or even almost all, in the metric case) the eigenvalues to exhibit a diamagnetic behaviour. In addition, we develop a notion of ‘discretized’ versions of a metric graph and prove that their nodal counts are related to those of the metric graph.
doi:10.1098/rsta.2012.0504
PMCID: PMC3866467
PMID: 24344337
nodal count; nodal domain; tree graph; inverse problems; diamagnetic
We consider a sequence of finite-dimensional Hilbert spaces of dimensions . Motivating examples are eigenspaces, or spaces of quasi-modes, for a Laplace or Schrödinger operator on a compact Riemannian manifold. The set of Hermitian orthonormal bases of may be identified with U(dN), and a random orthonormal basis of is a choice of a random sequence UN∈U(dN) from the product of normalized Haar measures. We prove that if and if tends to a unique limit state ω(A), then almost surely an orthonormal basis is quantum ergodic with limit state ω(A). This generalizes an earlier result of the author in the case where is the space of spherical harmonics on S2. In particular, it holds on the flat torus if d≥5 and shows that a highly localized orthonormal basis can be synthesized from quantum ergodic ones and vice versa in relatively small dimensions.
doi:10.1098/rsta.2012.0511
PMCID: PMC3866470
PMID: 24344341
quantum ergodcity; laplace eigenfunctions; random orthonormal basis
It was demonstrated in Atas & Bogomolny (2012 Phys. Rev. E
86, 021104) that the ground-state wave functions for a large variety of one-dimensional spin- models are multi-fractals in the natural spin-z basis. We present here the details of analytical derivations and numerical confirmations of this statement.
doi:10.1098/rsta.2012.0520
PMCID: PMC3866471
PMID: 24344342
spin chains; ground-state wave function; multi-fractality
We prove an analogue of the magnetic nodal theorem on quantum graphs: the number of zeros ϕ of the nth eigenfunction of the Schrödinger operator on a quantum graph is related to the stability of the nth eigenvalue of the perturbation of the operator by magnetic potential. More precisely, we consider the nth eigenvalue as a function of the magnetic perturbation and show that its Morse index at zero magnetic field is equal to ϕ−(n−1).
doi:10.1098/rsta.2012.0522
PMCID: PMC3866472
PMID: 24344344
quantum graphs; nodal count; zeros of eigenfunctions; magnetic Schrödinger operator; magnetic-nodal connection
As an alternative to nodal domains in the usual sense, we propose a partition of the domain of a real-valued eigenfunction in by trajectories of the gradient, linking saddle points to extrema. Its most elementary properties are listed and exemplified. The main point is that the problem of avoided crossings is largely eliminated.
doi:10.1098/rsta.2012.0505
PMCID: PMC3866468
PMID: 24344338
eigenfunction; node; saddle point
We compute the mean two-point spectral form factor and the spectral number variance for permutation matrices of large order. The two-point correlation function is expressed in terms of generalized divisor functions, which are frequently discussed in number theory. Using classical results from number theory and casting them in a convenient form, we derive expressions which include the leading and next to leading terms in the asymptotic expansion, thus providing a new point of view on the subject, and improving some known results.
doi:10.1098/rsta.2012.0508
PMCID: PMC3866469
PMID: 24344339
spectral statistics; permutation matrices; random matrix theory
We study a semiclassical asymptotics of the Cauchy problem for a time-dependent Schrödinger equation on metric and decorated graphs with a localized initial function. A decorated graph is a topological space obtained from a graph via replacing vertices with smooth Riemannian manifolds. The main term of an asymptotic solution at an arbitrary finite time is a sum of Gaussian packets and generalized Gaussian packets (localized near a certain set of codimension one). We study the number of packets as time tends to infinity. We prove that under certain assumptions this number grows in time as a polynomial and packets fill the graph uniformly. We discuss a simple example of the opposite situation: in this case, a numerical experiment shows a subexponential growth.
doi:10.1098/rsta.2013.0145
PMCID: PMC3866473
PMID: 24344346
metric graphs; decorated graphs; semiclassical approximation; lattice points
doi:10.1098/rsta.2014.0049
PMCID: PMC4247396
doi:10.1098/rsta.2014.0430
PMCID: PMC4247397
doi:10.1098/rsta.2014.0050
PMCID: PMC4240949
PMID: 25404675
climate engineering; marine-cloud brightening; stratospheric aerosols; social impacts
The general principle behind the marine cloud brightening (MCB) climate engineering technique is that seeding marine stratocumulus clouds with substantial concentrations of roughly monodisperse sub-micrometre-sized seawater particles might significantly enhance cloud albedo and longevity, thereby producing a cooling effect. This paper is concerned with preliminary studies of the possible beneficial application of MCB to three regional issues: (1) recovery of polar ice loss, (2) weakening of developing hurricanes and (3) elimination or reduction of coral bleaching. The primary focus is on Item 1. We focus discussion herein on advantages associated with engaging in limited-area seeding, regional effects rather than global; and the levels of seeding that may be required to address changing current and near-term conditions in the Arctic. We also mention the possibility that MCB might be capable of producing a localized cooling to help stabilize the West Antarctic Ice Sheet.
doi:10.1098/rsta.2014.0053
PMCID: PMC4240952
PMID: 25404682
Arctic; Antarctic; polar sea-ice; recovery; preservation
The large-scale production of vast numbers of suitable salt nuclei and their upward launch is one of the main technological barriers to the experimental testing of marine cloud brightening (MCB). Very promising, though not definitive, results have been obtained using an adapted version of effervescent spray atomization. The process is simple, robust and inexpensive. This form of effervescent spraying uses only pressurized water and air sprayed from small nozzles to obtain very fine distributions. While it is far from optimized, and may not be the best method if full deployment is ever desired, we believe that even in its present form the process would lend itself well to preliminary field test investigations of MCB. Measurements obtained using standard aerosol instrumentation show approximately lognormal distributions of salt nuclei with median diameters of approximately 65 nm and geometric standard deviations slightly less than 2. However, these measurements are not in agreement with those based on scanning electron microscopy imaging of collected particles, an observation that has not yet been explained. Assuming the above distribution, 1015 particles per second could be made with 21 kW of spray power, using approximately 200 nozzles. It is envisioned that existing snow making equipment can be adapted to launch the nuclei 60–100 m into the air, requiring approximately 20 kW of additional power.
doi:10.1098/rsta.2014.0055
PMCID: PMC4240953
PMID: 25404673
salt aerosol production; marine cloud brightening; effervescent spray atomization
We investigate the sensitivity of marine cloud brightening to the properties of the added salt particle distribution using a cloud parcel model, with an aim to address the question of, ‘what is the most efficient particle size distribution that will produce a desired cooling effect?’ We examine the effect that altering the aerosol particle size distribution has on the activation and growth of drops, i.e. the Twomey effect alone, and do not consider macrophysical cloud responses that may enhance or mitigate the Twomey effect. For all four spray generation methods considered, Rayleigh jet; Taylor cone jet; supercritical fluid; and effervescent spray, salt particles within the median dry diameter range Dm=30–100 nm are the most effective range of sizes. The Rayleigh jet method is also the most energy efficient overall. We also find that care needs to be taken when using droplet activation parametrizations: for the concentrations considered, Aitken particles do not result in a decrease in the total albedo, as was found in a recent study, and such findings are likely to be a result of the parametrizations' inability to simulate the effect of swollen aerosol particles. Our findings suggest that interstitial aerosol particles play a role in controlling the albedo rather than just the activated cloud drops, which is an effect that the parametrization methods do not consider.
doi:10.1098/rsta.2014.0056
PMCID: PMC4240954
PMID: 25404674
cloud brightening; aerosols; cloud-drops; albedo; geoengineering
Although solar radiation management (SRM) through stratospheric aerosol methods has the potential to mitigate impacts of climate change, our current knowledge of stratospheric processes suggests that these methods may entail significant risks. In addition to the risks associated with current knowledge, the possibility of ‘unknown unknowns’ exists that could significantly alter the risk assessment relative to our current understanding. While laboratory experimentation can improve the current state of knowledge and atmospheric models can assess large-scale climate response, they cannot capture possible unknown chemistry or represent the full range of interactive atmospheric chemical physics. Small-scale, in situ experimentation under well-regulated circumstances can begin to remove some of these uncertainties. This experiment—provisionally titled the stratospheric controlled perturbation experiment—is under development and will only proceed with transparent and predominantly governmental funding and independent risk assessment. We describe the scientific and technical foundation for performing, under external oversight, small-scale experiments to quantify the risks posed by SRM to activation of halogen species and subsequent erosion of stratospheric ozone. The paper's scope includes selection of the measurement platform, relevant aspects of stratospheric meteorology, operational considerations and instrument design and engineering.
doi:10.1098/rsta.2014.0059
PMCID: PMC4240955
PMID: 25404681
geoengineering; solar radiation management; stratosphere; balloon; ozone depletion