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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences RSS feed -- current issue1471-2946June, 2018Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences1364-5021<![CDATA[Exact eigenstates of a nanometric paraboloidal emitter and field emission quantities]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20170692?rss=1
The progress in field emission theory from its initial Fowler–Nordheim form is centred on the transmission coefficient. For the supply (of electrons) function one still uses the constant value due to a supply of plane-waves states. However, for emitting tips of apex radius of 1–5 nm this is highly questionable. To address this issue, we have solved the Schrödinger equation in a sharp paraboloidally shaped quantum box. The Schrödinger equation is separable in the rotationally parabolic coordinate system and we hence obtain the exact eigenstates of the system. Significant differences from the usual Cartesian geometry are obtained. (1) Both the normally incident and parallel electron fluxes are functions of the angle to the emitter axis and affect the emission angle. (2) The WKB approximation fails for this system. (3) The eigenfunctions of the nanoemitter form a continuum only in one dimension while complete discretization occurs in the other two directions. (4) The parallel electron velocity vanishes at the apex which may explain the recent spot-size measurements in near-field scanning electron microscopy. (5) Competing effects are found as the tip radius decreases to 1 nm: The electric field increases but the total supply function decreases so that possibly an optimum radius exists.
]]>2018-06-13T00:09:52-07:00info:doi/10.1098/rspa.2017.0692hwp:master-id:royprsa;rspa.2017.06922018-06-13Research articles47422142017069220170692<![CDATA[Surface theory in discrete projective differential geometry. I. A canonical frame and an integrable discrete Demoulin system]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20170770?rss=1
We present the first steps of a procedure which discretizes surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi equations adopt compact forms. Based on a scaling symmetry which injects a parameter into the linear Gauss-Weingarten equations, we set down an algebraic classification scheme of discrete projective minimal surfaces which turns out to admit a geometric counterpart formulated in terms of discrete notions of Lie quadrics and their envelopes. In the case of discrete Demoulin surfaces, we derive a Bäcklund transformation for the underlying discrete Demoulin system and show how the latter may be formulated as a two-component generalization of the integrable discrete Tzitzéica equation which has originally been derived in a different context. At the geometric level, this connection leads to the retrieval of the standard discretization of affine spheres in affine differential geometry.
]]>2018-06-13T00:17:57-07:00info:doi/10.1098/rspa.2017.0770hwp:master-id:royprsa;rspa.2017.07702018-06-13Research articles47422142017077020170770<![CDATA[How realistic are painted lightnings? Quantitative comparison of the morphology of painted and real lightnings: a psychophysical approach]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20170859?rss=1
Inspired by the pioneer work of the nineteenth century photographer, William Nicholson Jennings, we studied quantitatively how realistic painted lightnings are. In order to answer this question, we examined 100 paintings and 400 photographs of lightnings. We used our software package to process and evaluate the morphology of lightnings. Three morphological parameters of the main lightning branch were analysed: (i) number of branches N_{b}, (ii) relative length r, and (iii) number of local maxima (peaks) N_{p} of the turning angle distribution. We concluded: (i) Painted lightnings differ from real ones in N_{b} and N_{p}. (ii) The r-values of painted and real lightnings vary in the same range. (iii) 67 and 22% of the studied painted and real lightnings were non-bifurcating (N_{b} = 1, meaning only the main branch), the maximum of N_{b} of painted and real lightnings is 11 and 51, respectively, and painted bifurcating lightnings possess mostly 2–4 branches, while real lightnings have mostly 2–10 branches. To understand these findings, we performed two psychophysical experiments with 10 test persons, whose task was to guess N_{b} on photographs of real lightnings which were flashed for short time periods t = 0.5, 0.75 and 1 s (characteristic to lightnings) on a monitor. We obtained that (i) test persons can estimate the number of lightning branches quite correctly if N_{b} ≤ 11. (ii) If N_{b} > 11, its value is strongly underestimated with exponentially increasing difference between the real and estimated numbers. (iii) The estimation is independent of the flashing period t of lightning photos/pictures. (iv) The estimation is more accurate, if skeletonized lightning pictures are flashed, rather than real lightning photos. These findings explain why artists usually illustrate lightnings with branches not larger than 11.
]]>2018-06-06T00:09:54-07:00info:doi/10.1098/rspa.2017.0859hwp:master-id:royprsa;rspa.2017.08592018-06-06Research47422142017085920170859<![CDATA[Force appropriation of nonlinear structures]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20170880?rss=1
Nonlinear normal modes (NNMs) are widely used as a tool for developing mathematical models of nonlinear structures and understanding their dynamics. NNMs can be identified experimentally through a phase quadrature condition between the system response and the applied excitation. This paper demonstrates that this commonly used quadrature condition can give results that are significantly different from the true NNM, in particular, when the excitation applied to the system is limited to one input force, as is frequently used in practice. The system studied is a clamped–clamped cross-beam with two closely spaced modes. This paper shows that the regions where the quadrature condition is (in)accurate can be qualitatively captured by analysing transfer of energy between the modes of the system, leading to a discussion of the appropriate number of input forces and their locations across the structure.
]]>2018-06-13T00:09:52-07:00info:doi/10.1098/rspa.2017.0880hwp:master-id:royprsa;rspa.2017.08802018-06-13Research articles47422142017088020170880<![CDATA[An asymptotic higher-order theory for rectangular beams]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20180001?rss=1
A direct asymptotic integration of the full three-dimensional problem of elasticity is employed to derive a consistent governing equation for a beam with the rectangular cross section. The governing equation is consistent in the sense that it has the same long-wave low-frequency behaviour as the exact solution of the original three-dimensional problem. Performance of the new beam equation is illustrated by comparing its predictions against the results of direct finite-element computations. Limiting behaviours for beams with large (and small) aspect ratios, which can be established using classical plate theories, are recovered from the new governing equation to illustrate its consistency and also to illustrate the importance of using plate theories with the correctly refined boundary conditions. The implications for the correct choice of the shear correction factor in Timoshenko's beam theory are also discussed.
]]>2018-06-13T00:09:52-07:00info:doi/10.1098/rspa.2018.0001hwp:master-id:royprsa;rspa.2018.00012018-06-13Research articles47422142018000120180001<![CDATA[An algorithm to explore entanglement in small systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20180023?rss=1
A quantum state’s entanglement across a bipartite cut can be quantified with entanglement entropy or, more generally, Schmidt norms. Using only Schmidt decompositions, we present a simple iterative algorithm to maximize Schmidt norms. Depending on the choice of norm, the optimizing states maximize or minimize entanglement, possibly across several bipartite cuts at the same time and possibly only among states in a specified subspace. Recognizing that convergence but not success is certain, we use the algorithm to explore topics ranging from fermionic reduced density matrices and varieties of pure quantum states to absolutely maximally entangled states and minimal output entropy of channels.
]]>2018-06-13T00:17:57-07:00info:doi/10.1098/rspa.2018.0023hwp:master-id:royprsa;rspa.2018.00232018-06-13Research articles47422142018002320180023<![CDATA[Tension-dependent transverse buckles and wrinkles in twisted elastic sheets]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20180062?rss=1
We investigate with experiments the twist-induced transverse buckling instabilities of an elastic sheet of length L, width W and thickness t, that is clamped at two opposite ends while held under a tension T. Above a critical tension T_{} and critical twist angle _{tr}, we find that the sheet buckles with a mode number n≥1 transverse to the axis of twist. Three distinct buckling regimes characterized as clamp-dominated, bendable and stiff are identified, by introducing a bendability length L_{B} and a clamp length L_{C}(<L_{B}). In the stiff regime (L>L_{B}), we find that mode n=1 develops above _{tr}_{S}~(t/W)T^{–1/2}, independent of L. In the bendable regime L_{C}<L<L_{B}, n=1 as well as n>1 occur above trB~t/LT–1/4. Here, we find the wavelength B~LtT–1/4, when n>1. These scalings agree with those derived from a covariant form of the Föppl-von Kármán equations, however, we find that the n=1 mode also occurs over a surprisingly large range of L in the bendable regime. Finally, in the clamp-dominated regime (L<L_{C}), we find that _{tr} is higher compared to _{B} due to additional stiffening induced by the clamped boundary conditions.
]]>2018-06-13T00:09:52-07:00info:doi/10.1098/rspa.2018.0062hwp:master-id:royprsa;rspa.2018.00622018-06-13Research articles47422142018006220180062<![CDATA[The nature of Earth's correlation wavefield: late coda of large earthquakes]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/474/2214/20180082?rss=1
The seismic correlation wavefield constructed from the stacked cross-correlograms of the late coda of earthquake signals at stations across the globe provides a wealth of observed pulses as a function of inter-station distance. The interval from 3 to 10 h after the onset of major earthquakes is employed for the period range from 15 to 50 s. The observations can be well matched by synthetic seismograms for a radially stratified Earth. Many of the correlation phases have similar time behaviour to those in the regular wavefield, but others have no correspondence. All such correlation phases can be explained by the interaction of arrivals with a common slowness at the each of the stations being correlated. Using a generalized ray description of the seismic wavefield, the time-distance behaviour of these correlation phases arises from differences in accumulated phase on different propagation paths through the Earth. Distinct arrivals emerge from the correlation field when there are many ways in which combinations of seismic phases can arise with the same difference in propagation legs. The constituents of the late coda are dominated by steeply travelling waves, and in consequence features associated with multiple passages through the whole Earth emerge distinctly, such as high-order multiples of PKIKP.
]]>2018-06-13T00:09:52-07:00info:doi/10.1098/rspa.2018.0082hwp:master-id:royprsa;rspa.2018.00822018-06-13Research articles47422142018008220180082