1,721,315 research outputs found
Overlapping Stokes smoothings: survival of the error function and canonical catastrophe integrals
No full textWe derive doubly uniform approximations for the remainder in the optimally truncated saddle-point expansion for an integral containing a large parameter. Double uniformity means that the formulae remain valid while distant saddles responsible for the divergence of the expansion coalesce and separate (as described by catastrophe theory) and while the subdominant exponentials they contribute switch on and off (as described by the error-function smoothing of the Stokes phenomenon). Two sorts of asymptotic singularity are thereby united in a common framework. The formula for the remainder incorporates both the Stokes error function and the canonical catastrophe integrals. A numerical illustration is given, in which the distant cluster contains two saddles; the asymptotic theory gives an accurate description of the details of the fractional remainder, even when this is of order exp ( –36)
Fake Airy functions and the asymptotics of reflectionlessness
No full textTwo classes of analytic refractive-index profile P2(z,ε ), whose reflection coefficients r are zero for all values of a parameter in , are studied as in to 0. The aim is to understand why r=0 rather than r varies as exp(-1/ε ) as for generic profiles. The authors find that reflectionlessness is a consequence of the fact that transition points of P2 (zeros or poles in the complex z plane) form tight clusters (whose size vanishes with in ) which can be regarded neither as coalesced nor well separated. Expansion near a cluster yields the local wave not as the usual Airy function, whose Stokes phenomenon generates reflection, but as Bessel functions of half-integer order (fake Airy functions) which are exactly trigonometric functions with no Stokes phenomenon and so no reflection
High orders of the Weyl expansion for quantum billiards: resurgence of periodic orbits, and the Stokes phenomenon
No full textA formalism is developed for calculating high coefficients cr of the Weyl (high energy) expansion for the trace of the resolvent of the Laplace operator in a domain B with smooth boundary ∂B. The cr are used to test the following conjectures. (a) The sequence of cr diverges factorially, controlled by the shortest accessible real or complex periodic geodesic. (b) If this is a 2-bounce orbit, it corresponds to the saddle of the chord length function whose contour is first crossed when climbing from the diagonal of the Möbius strip which is the space of chords of B. (c) This orbit gives an exponential contribution to the remainder when the Weyl series, truncated at its least term, is subtracted from the resolvent; the exponential switches on smoothly (according to an error function) where it is smallest, that is across the negative energy axis (Stokes line). These conjectures are motivated by recent results in asymptotics. They survive tests for the circle billiard, and for a family of curves with 2 and 3 bulges, where the dominant orbit is not always the shortest and is sometimes complex. For some systems which are not smooth billiards (e. g. a particle on a ring, or in a billiard where ∂B is a polygon), the Weyl series terminates and so no geodesics are accessible; for a particle on a compact surface of constant negative curvature, only the complex geodesics are accessible from the Weyl series
Hyperasymptotics
No full textWe develop a technique for systematically reducing the exponentially small (‘superasymptotic’) remainder of an asymptotic expansion truncated near its least term, for solutions of ordinary differential equations of Schrödinger type where one transition point dominates. This is achieved by repeatedly applying Borel summation to a resurgence formula discovered by Dingle, relating the late to the early terms of the original expansion. The improvements form a nested sequence of asymptotic series truncated at their least terms. Each such ‘hyperseries’ involves the terms of the original asymptotic series for the particular function being approximated, together with terminating integrals that are universal in form, and is half the length of its predecessor. The hyperasymptotic sequence is therefore finite, and leads to an ultimate approximation whose error is less than the square of the original superasymptotic remainder. The Stokes phenomenon is automatically and exactly incorporated into the scheme. Numerical computations confirm the efficacy of the technique
Infinity interpreted
No full textDespite the denunciations of the mathematician Abel, if the devil did invent divergent series it was because his creator counterpart chose to build our physical universe so that they are among the more useful ways to describe its finite properties
Determining the Difference Between Active and Inactive Caries White Spot Lesions Using Sodium Iodide
No full textIdentifying the differences between active and inactive white spot caries lesions is presently a challenging task in clinical dentistry. White spot lesions are normally associated with demineralization that has happened (inactive) or is happening (active) to the tooth at a given time. In inactive white spot lesions, the demineralization is at rest and the infectious carious process is stopped. It is important to be able to differentiate an inactive lesion from an active one because inactive lesions can be treated without surgical methods. Avoiding unnecessary repair will preserve the natural structure of the tooth and increase its longevity. We have tested the ability of a concentrated, 11 molar Sodium Iodide (NaI) solution as a radiographic contrast agent to differentiate between active and inactive caries white spot lesions. The hypothesis is that concentrated NaI solution is likely to penetrate into the tooth structure of active, but not inactive, lesions, thereby providing a visual method for distinguishing the activity states. To test this hypothesis, two teeth (one human maxillary 1st molar and one maxillary 3rd molar) with visible white spot lesions were collected for analysis. Photomicrography, radiography and Scanning Electron Microscopy (SEM) assessments were used to study these specimens. Consistent with our hypothesis, radiographic images revealed a high penetration of NaI solution through the enamel of the 1st molar, which had an active white spot lesion, as confirmed by photomicrography and SEM analysis. However, minimal to no NaI penetration was observed through the enamel of the 3rd molar, which was confirmed to have an inactive white spot lesion. While this study is preliminary, new methods to identify active carious lesions have the potential to improve the management of caries and reduce unnecessary restorative measures.ProQuest Traditional Publishing Optionvii, 22 page
Unfolding the high orders of asymptotic expansions with coalescing saddles: singularity theory, crossover and duality
No full textWe study the leading behaviour of the late coefficients (high orders r) of asymptotic expansions in a large parameter k, for contour integrals involving a cluster of coalescing saddles, and thereby establish the form of the divergence of the expansions. The two principal cases are: 'saddle-to-cluster’, where the integral is through a simple saddle and its expansion diverges because of a distant cluster; and 'cluster-to-saddle', where the integral is through a cluster and its expansion diverges because of a distant simple saddle. In both, the large-r coefficients are dominated by the 'factorial divided by power' familiar in asymptotics, but this changes its form as the saddles in the cluster are made to coalesce and separate by varying parameters A= {A1,A2....} in the integrand. The 'crossover' between different forms is described by a series of canonical integrals, built from the cuspoid catastrophe polynomials of singularity theory that describe the geometry of the coalescence. The arguments of these integrals involve not only the A but also fractional powers of r, which by a curious duality replace the powers of the original large parameter k which occur in uniform approximations involving these integrals. A by-product of the cluster-to-saddle analysis is a new exact formula for the coefficients of uniform asymptotic expansions
Hyperasymptotics for integrals with saddles
No full textIntegrals involving exp { –kf(z)}, where |k| is a large parameter and the contour passes through a saddle of f(z), are approximated by refining the method of steepest descent to include exponentially small contributions from the other saddles, through which the contour does not pass. These contributions are responsible for the divergence of the asymptotic expansion generated by the method of steepest descent. The refinement is achieved by means of an exact ‘resurgence relation', expressing the original integral as its truncated saddle-point asymptotic expansion plus a remainder involving the integrals through certain ‘adjacent’ saddles, determined by a topological rule. Iteration of the resurgence relation, and choice of truncation near the least term of the original series, leads to a representation of the integral as a sum of contributions associated with ‘multiple scattering paths’ among the saddles. No resummation of divergent series is involved. Each path gives a ‘hyperseries’, depending on the terms in the asymptotic expansions for each saddle (these depend on the particular integral being studied and so are non-universal), and certain ‘hyperterminant’ functions defined by integrals (these are always the same and hence universal). Successive hyperseries get shorter, so the scheme naturally halts. For two saddles, the ultimate error is approximately ∊2.386, where ∊ (proportional to exp (—A│k│) where A is a positive constant), is the error in optimal truncation of the original series. As a numerical example, an integral with three saddles is computed hyperasymptotically.<br/
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