the use of cornu spirals in drawing planar curves of controlled curvature abstract cornu spirals or clothoids have been used in highway design for many years in the past the spirals have been found manually by draftsmen the purpose of this paper is to show that it is practical to find these spirals with a micro computer the design curve will be made up of arcs of circles and segments of cornu spirals joined in such a way that the curvature is continuous throughout and takes its largest values on the arcs of circles thus the radii of the circles used will limit and control the curvature of the whole design curve

generating functions for weighted hurwitz numbers double hurwitz numbers enumerating weighted n sheeted branched coverings of the riemann sphere or equivalently weighted paths in the cayley graph of sn generated by transpositions are determined by an associated weight generating function a uniquely determined 1 parameter family of 2d toda functions of hypergeometric type is shown to consist of generating functions for such weighted hurwitz numbers four classical cases are detailed in which the weighting is uniform okounkov s double hurwitz numbers for which the ramification is simple at all but two specified branch points the case of belyi curves with three branch points two with specified profiles the general case with a specified number of branch points two with fixed profiles the rest constrained only by the genus and the signed enumeration case with sign determined by the parity of the number of branch points using the exponentiated quantum dilogarithm function as a weight generator three new types of weighted enumerations are intr

using max entropy ratio of recurrence plot to measure electrocorticogram changes in epilepsy patients abstract a maximum entropy ratio mer method is firstly adapted to investigate the high dimensional electrocorticogram ecog data from epilepsy patients mer is a symbolic analysis approach for the detection of recurrence domains of complex dynamical systems from time series data were chosen from eight patients undergoing pre surgical evaluation for drug resistant temporal lobe epilepsy mers for interictal and ictal data were calculated and compared a statistical test was performed to evaluate the ability of mer to separate the interictal state from the ictal state mer showed significant changes from the interictal state into the ictal state where mer was low at the ictal state and is significantly different with that at the interictal state these suggest that mer is able to separate the ictal state from the interictal state based on ecog data it has the potential of detecting the transition between normal brain activity and the ictal state

computing fresnel integrals via modified trapezium rules in this paper we propose methods for computing fresnel integrals based on truncated trapezium rule approximations to integrals on the real line these trapezium rules modified to take into account poles of the integrand near the real axis our starting point is a method for computation of the error function of complex argument due to matta and reichel j math phys 34 298 307 1956 and hunter and regan math comp 26 539 541 1972 we construct approximations which we prove are exponentially convergent as a function of n n the number of quadrature points obtaining explicit error bounds which show that accuracies of 10 15 10 15 uniformly on the real line are achieved with n 12 n 12 this confirmed by computations the approximations we obtain are attractive additionally in that they maintain small relative errors for small and large argument are analytic on the real axis echoing the analyticity of the fresnel integrals and are straightforward to implement

multiscale recurrence quantification analysis of order recurrence plots abstract in this paper we propose a new method of multiscale recurrence quantification analysis msrqa to analyze the structure of order recurrence plots the msrqa is based on order patterns over a range of time scales compared with conventional recurrence quantification analysis rqa the msrqa can show richer and more recognizable information on the local characteristics of diverse systems which successfully describes their recurrence properties both synthetic series and stock market indexes exhibit their properties of recurrence at large time scales that quite differ from those at a single time scale some systems present more accurate recurrence patterns under large time scales it demonstrates that the new approach is effective for distinguishing three similar stock market systems and showing some inherent differences

an arc spline approximation to a clothoid the clothoid is a spiral used in highway and railway route design clothoids are transcendental functions and so have been approximated by polynomials by power series and continued fractions and by rational functions here the clothoid is approximated by an arc spline the chief advantage in doing so is that arc splines are very easy to lay out and to offset examples show that the approximation is of extremely high accuracy it is proved that if the arc spline has n arcs then the error in the approximation is of order o 1 n2

interpolation of two dimensional curves with euler spirals we propose an algorithm for the interpolation of two dimensional curves using euler spirals the method uses a lower order reconstruction to approximate solution derivatives at each sample point the computed tangents are then used to connect consecutive points with segments of euler spirals the resulting interpolation is g 1 in regions where the curve being interpolated is smooth the algorithm uses an adaptive stencil which allows it to construct an approximation free of oscillations near discontinuities in the function or its derivatives the approximation is based on geometrical shapes which makes it particularly suitable for two dimensional curves

recurrence plot statistics and the effect of embedding abstract recurrence plots provide a graphical representation of the recurrent patterns in a timeseries the quantification of which is a relatively new field here we derive analytical expressions which relate the values of key statistics notably determinism and entropy of line length distribution to the correlation sum as a function of embedding dimension these expressions are obtained by deriving the transformation which generates an embedded recurrence plot from an unembedded plot a single unembedded recurrence plot thus provides the statistics of all possible embedded recurrence plots if the correlation sum scales exponentially with embedding dimension we show that these statistics are determined entirely by the exponent of the exponential this explains the results of iwanski and bradley j s iwanski e bradley recurrence plots of experimental data to embed or not to embed chaos 8 1998 861 871 who found that certain recurrence plot statistics are apparently invariant to embedding dimension for certain low dimensional systems we also examine the relationship between the mutual information content of two timeseries and the common recurrent structure seen in their recurrence plots this allows time localized contributions to mutual information to be visualized this technique is demonstrated using geomagnetic index data we show that the au and al geomagnetic indices share half their information and find the timescale on which mutual features appear

weighted hurwitz numbers and topological recursion an overview multiparametric families of hypergeometric functions of kp or toda type serve as generating functions for weighted hurwitz numbers providing weighted enumerations of branched covers of the riemann sphere a graphical interpretation of the weighting is given in terms of constellations mapped onto the covering surface the theory is placed within the framework of topological recursion with the baker function at t 0 shown to satisfy the quantum spectral curve equation whose classical limit is rational a basis for the space of formal power series in the spectral variable is generated that is adapted to the grassmannian element associated with the function multicurrent correlators are defined in terms of the function and shown to provide an alternative generating function for weighted hurwitz numbers fermionic vacuum state expectation value representations are provided for the adapted bases pair correlators and multicurrent correlators choosing the weight generating function as a polynomial and restricting the number of nonzero second kp flow parameters in the toda function to be finite implies a finite rank covariant derivative equation with rational coefficients satisfied by a finite window of adapted basis elements the pair correlator is shown to provide a christoffel darboux type finite rank integrable kernel and the wkb series coefficients of the associated adjoint system are computed recursively leading to topological recursion relations for the generators of the weighted hurwitz numbers

geometric hermite interpolation with tschirnhausen cubics explicit formulae are found that give the unique tschirnhausen cubic that solves a geometric hermite interpolation problem that solution is used to create a planar g1 spline by joining segments of tschirnhausen cubics if the geometric hermite data is from a smooth function the tschirnhausen cubic approximates the smooth function the error in the approximation of a short segment of length h can be expressed as a power series in h the error is o h4 and the coefficient of the leading term is found

clothoid spline transition spirals highway and railway designers use clothoid splines planar g2 curves consisting of straight line segments circular arcs and clothoid segments as center lines in route location this paper considers the problem of finding a clothoid spline transition spiral which joins two given points and matches given curvatures and unit tangents at the two points conditions are given for the existence and uniqueness of the clothoid spline transition spirals and algorithms for finding them are outlined

rational approximations for the fresnel integrals a class of simple rational polynomial approximations for the fresnel integrals is given with maximum errors from 1 7 x 10 3 down to 4 x 10 8 the domain 0 oo is not subdivided the format is particularly convenient for programmable hand calculators and microcomputer subroutines the best known algorithms for computer evaluation of the fresnel integrals 1 c x is x j exp ip t2 dt 0 are those of hastings 1 41 and of boersma 2 the former is of very limited precision the latter is relatively cumbersome and is subject to nonobvious underflow subtraction errors as is direct evaluation of the standard maclaurin and asymptotic series an extensive set of rational polynomial approximations has been given by cody 3 these are efficient for very high precision work but tedious for intermediate levels of precision because the domain is subdivided into five intervals our approach follows hastings but recasts the formulas into the polar form 2 c x 2ri x sinf1pr ajk x x2 i 3 s x 2 rim x cos 7t ajk x x2 where the rim and ajk functions are rational approximations of the form i m2 4 r 01 l c1xi e dix i o i 0 i k s ajk e a x e b x o i o the domain of x is positive real numbers from zero to infinity we consider error functions of the form 6 sr rim ro 7 sa 7tro ak ao where r0 and ao are exact values these functions represent orthogonal errors in the plane of the cornu spiral with the coefficient inserted in 7 to make the magnitudes received may 2 1983 revised january 9 1984 1980 mathematics subject classification primary 65d20 secondary 33a20 41a50 78a50

quantum hurwitz numbers and macdonald polynomials parametric families in the center z c sn of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for macdonald polynomials as commuting jucys murphy elements their eigenvalues provide coefficients in the double schur function expansion of 2d toda functions of hypergeometric type expressing these in the basis of products of power sum symmetric functions the coefficients may be interpreted geometrically as parametric families of quantum hurwitz numbers enumerating weighted branched coverings of the riemann sphere combinatorially they give quantum weighted sums over paths in the cayley graph of sn generated by transpositions dual pairs of bases for the algebra of symmetric functions with respect to the scalar product in which the macdonald polynomials are orthogonal provide both the geometrical and combinatorial significance of these quantum weighted enumerative invariants

fermionic approach to weighted hurwitz numbers and topological recursion a fermionic representation is given for all the quantities entering in the generating function approach to weighted hurwitz numbers and topological recursion this includes kp and 2d toda tau functions of hypergeometric type which serve as generating functions for weighted single and double hurwitz numbers the baker function which is expanded in an adapted basis obtained by applying the same dressing transformation to all vacuum basis elements the multipair correlators and the multicurrent correlators multiplicative recursion relations and a linear differential system are deduced for the adapted bases and their duals and a christoffel darboux type formula is derived for the pair correlator the quantum and classical spectral curves linking this theory with the topological recursion program are derived as well as the generalized cut and join equations the results are detailed for four special cases the simple single and double hurwitz numbers the weakly monotone case corresponding to signed enumeration of coverings the strongly monotone case corresponding to belyi curves and the simplest version of quantum weighted hurwitz numbers

a degeneration formula of gw invariants this is the sequel to the paper 23 in this paper we construct the virtual moduli cycles of the degeneration of the moduli of stable morphisms constructed in 23 we also construct the virtual moduli cycles of the moduli of relative stable morphisms of a pair of a smooth divisor in a smooth variety based on these we prove a degeneration formula of the gromov witten invariants

hurwitz numbers and intersections on moduli spaces of curves 1 department of mathematics university of stockholm s 10691 stockholm sweden e mail teke matematik su se 2 higher college of mathematics independent university of moscow and institute for system research ras moscow russia e mail lando mccme ru 3 department of mathematics royal institute of technology s 10044 stockholm sweden e mail mshapiro math kth se 4 department of mathematics and department of computer science university of haifa haifa 31905 israel e mail alek mathcs haifa ac il

a note on finding clothoids the clothoid is a spiral that is used as transition curve in highway and railway route design although its defining formulas are transcendental functions recent work has shown that it can be used fairly easily on small computers a single nonlinear equation must be solved in each of four common arrangements of the clothoid the purpose of this work is to give starting intervals for which newton s root finding method applied to those equations is guaranteed to converge

measuring nonstationarity by analyzing the loss of recurrence in dynamical systems we propose a measure for nonstationarity which is based on the analysis of distributions of temporal distances of neighboring vectors in state space as an extension of previous techniques our method does not require a partitioning of the time series moreover the deviation of mean recurrence times from frequency distributions that would be expected under stationary conditions allows us to estimate the statistical significance of the method

rationally weighted hurwitz numbers meijer g functions and matrix integrals the quantum spectral curve equation associated with kp functions of hypergeometric type serving as generating functions for rationally weighted hurwitz numbers is solved by generalized hypergeometric series the basis elements spanning the corresponding sato grassmannian element are shown to be meijer g functions or their asymptotic series using their mellin integral representation the function evaluated at the trace invariants of an externally coupled matrix is expressed as a matrix integral

quasifinite highest weight modules over the lie algebra of differential operators on the circle we classify positive energy representations with finite degeneracies of the lie algebraw 1 and construct them in terms of representation theory of the lie algebra hat gl infty r m of infinites matrices with finite number of non zero diagonals over the algebrar m t t m 1 the unitary ones are classified as well similar results are obtained for the sin algebras