correlated randomness and switching phenomena one challenge of biology medicine and economics is that the systems treated by these serious scientific disciplines have no perfect metronome in time and no perfect spatial architecture crystalline or otherwise nonetheless as if by magic out of nothing but randomness one finds remarkably fine tuned processes in time and remarkably fine tuned structures in space further many of these processes and structures have the remarkable feature of switching from one behavior to another as if by magic the past century has philosophically been concerned with placing aside the human tendency to see the universe as a fine tuned machine here we will address the challenge of uncovering how through randomness albeit as we shall see strongly correlated randomness one can arrive at some of the many spatial and temporal patterns in biology medicine and economics and even begin to characterize the switching phenomena that enables a system to pass from one state to another inspired by principles developed by a nihat berker and scores of other statistical physicists in recent years we discuss some applications of correlated randomness to understand switching phenomena in various fields specifically we present evidence from experiments and from computer simulations supporting the hypothesis that water s anomalies are related to a switching point which is not unlike the tipping point immortalized by malcolm gladwell and that the bubbles in economic phenomena that occur on all scales are not outliers another gladwell immortalization though more speculative we support the idea of disease as arising from some kind of yet to be understood complex switching phenomenon by discussing data on selected examples including heart disease and alzheimer disease

time spans between price maxima and price minima in stock markets abstract we empirically investigate the distribution of time spans between price maxima and price minima in international stock markets where a time span is defined as the time interval between a local price minimum and a local price maximum and local price extrema are identified by a method introduced by preis and stanley preis et al 2011 preis 2011 preis and stanley 2011 2010 preis 2010 preis and stanley 2010 stanley et al 2010 preis and stanley 2009 the empirical results show that both the tail distributions of time spans from local price maxima to local price minima and the tail distributions of time spans from local price minima to local price maxima yield an exponential distribution in addition price rise fall asymmetry is observed by comparing the values of the exponents of the distribution curves these results are robust across eight representative stock markets

the efficient computation of some generalised exponential integrals the accurate and efficient computation of the special functions gk x is discussed where gk x 1 k 1 1 exp xy log y k 1 dy y these functions appear in the computation of the derivatives of the l series of an elliptic curve and in radiative transfer problems from astrophysics by dividing 0 into 3 sub intervals we derive chebyshev polynomial expansions for gk k 1 4 with the coefficients given to an accuracy of 20 decimal places

the difference between the weil height and the canonical height on elliptic curves estimates for the difference of the weil height and the canonical height of points on elliptic curves are used for many purposes both theoretical and computational in this note we give an explicit estimate for this difference in terms of the j invariant and discriminant of the elliptic curve the method of proof suggested by serge lang is to use the decomposition of the canonical height into a sum of local heights we illustrate one use for our estimate by computing generators for the mordell weil group in three examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 x ax b with a and b in the ring of integers of k the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 10 chapter viii 9 or 6 chapter vi the canonical height is determined by this property together with the fact that the difference

conditioned random walks and the rsk correspondence we consider the stochastic evolution of three variants of the rsk algorithm giving both analytic descriptions and probabilistic interpretations symmetric functions play a key role and the probabilistic interpretations are obtained by elementary doob hunt theory in each case the evolution of the shape of the tableau obtained via the rsk algorithm can be interpreted as a conditioned random walk this is intuitively appealing and can be used for example to obtain certain relationships between orthogonal polynomial ensembles in a certain scaling limit there is a continous version of the rsk algorithm which inherits much of the structure exhibited in the discrete settings intertwining relationships between conditioned and unconditioned random walks are also given in the continuous limit these are related to the harish chandra itzyksen zuber integral

complex multiplication cycles and a conjecture of beilinson and bloch a generalization of the conjecture of birch and swinnerton dyer is investigated using complex multiplication cycles on a particular kuga fiber variety a weak finiteness result consistent with the conjecture is proved the image of complex multiplication cycles under the abel jacobi map is computed explicitly the results provide numerical evidence supporting the conjecture they also give evidence for a relationship between complex multiplication cycles and a modular form of weight 5 2 and raise questions for further investigation

large integral points on elliptic curves to ml friend dan shanks abstract we describe several methods which permit one to search for big integral points on certain elliptic curves i e for integral solutions x of certain diophantine equations of the form v2 x3 ax b a b e z in a large range ixi iy i b in time polynomial in loglogb we also give a number of individual examples and of parametric families of examples of specific elliptic curves having a relatively large integral point

schensted type correspondence plactic monoid and jeu de taquin for type cn we use kashiwara s theory of crystal bases to study the plactic monoid for uq sp2n then we describe the corresponding insertion and sliding algorithms the sliding algorithm is essentially the symplectic jeu de taquin defined by sheats and our construction gives the proof of its compatibility with plactic relations

inverse statistics in economics the gain loss asymmetry abstract inverse statistics in economics is considered we argue that the natural candidate for such statistics is the investment horizons distribution this distribution of waiting times needed to achieve a predefined level of return is obtained from often detrended historic asset prices such a distribution typically goes through a maximum at a time called the optimal investment horizon since this defines the most likely waiting time for obtaining a given return by considering equal positive and negative levels of return we report on a quantitative gain loss asymmetry most pronounced for short horizons it is argued that this asymmetry reflects the market dynamics and we speculate over the origin of this asymmetry

quantum sl_2 infinite curvature and pitman s 2m x theorem the classical theorem by pitman states that a brownian motion minus twice its running infimum enjoys the markov property on the one hand biane understood that pitman s theorem is intimately related to the representation theory of the quantum group mathcal u q left mathfrak sl 2 right in the so called crystal regime q rightarrow 0 on the other hand bougerol and jeulin showed the appearance of exactly the same pitman transform in the infinite curvature limit r rightarrow infty of a brownian motion on the hyperbolic space mathbb h 3 sl 2 mathbb c su 2 this paper aims at understanding this phenomenon by giving a unifying point of view in order to do so we exhibit a presentation mathcal u q hbar left mathfrak sl 2 right of the jimbo drinfeld quantum group which isolates the role of curvature r and that of the planck constant hbar the simple relationship between parameters is q e r the semi classical limits hbar rightarrow 0 are the poisson lie groups dual to sl 2 mathbb c with varying curvatures r in mathbb r we also construct classical and quantum random walks drawing a full picture which includes biane s quantum walks and the construction of bougerol jeulin taking the curvature parameter r to infinity leads indeed to the crystal regime at the level of representation theory hbar 0 and to the bougerol jeulin construction in the classical world hbar 0 all these results are neatly in accordance with the philosophy of kirillov s orbit method

conditioned random walks from kac moody root systems random paths are time continuous interpolations of random walks by using littelmann path model we associate to each irreducible highest weight module of a kac moody algebra g a random path w under suitable hypotheses we make explicit the probability of the event e w never exits the weyl chamber of g we then give the law of the random walk defined by w conditioned by the event e and proves this law can be recovered by applying to w the generalized pitmann transform introduced by biane bougerol and o connell this generalizes the main results of 10 and 16 to kac moody root systems and arbitrary highest weight modules moreover we use here a completely new approach by exploiting the symmetry of our construction under the action of the weyl group of g rather than renewal theory and doob s theorem on martin kernels

inverse statistics in the foreign exchange market abstract we investigate intra day foreign exchange fx time series using the inverse statistic analysis developed by simonsen et al eur phys j 27 2002 583 and jensen et al physica a 324 2003 338 specifically we study the time averaged distributions of waiting times needed to obtain a certain increase decrease in the price of an investment the analysis is performed for the deutsch mark dm against the us for the full year of 1998 but similar results are obtained for the japanese yen against the us with high statistical significance the presence of resonance peaks in the waiting time distributions is established such peaks are a consequence of the trading habits of the market participants as they are not present in the corresponding tick business waiting time distributions furthermore a new stylized fact is observed for the normalized waiting time distribution in the form of a power law pdf this result is achieved by rescaling of the physical waiting time by the corresponding tick time thereby partially removing scale dependent features of the market activity

computing heights on elliptic curves we describe how to compute the canonical height of points orn elliptic curves tate has given a rapidly converging series for archimedean local heights over r we describe a modified version of tate s series which also converges over c and give an efficient procedure for calculating local heights at non archimedean places in this way we can calculate heights over number fields having complex embeddings we also give explicit estimates for the tail of our series and present several examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 alxy a3y x3 a2x2 a4x a6 the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 11 viii section 9 or 6 chapter vi the canonical height is an extremely important theoretical tool in the arithmetic theory of elliptic curves being used for such diverse purposes as studying values of l functions 5 numbers of integral points 12 and transcendence theory 9 it is also important as a computational tool such as its use in zagier s algorithm for finding integral points up to large bounds 18 it is thus of interest to have an efficient method for calculating the canonical height of a point the usual definition of h as a limit h p limn 0 4 nh x 2np is not practical for computation instead one uses the fact that the canonical height can be written as a sum of local heights one term for each distinct absolute value on k 2 h p e na p vemk for example if k q then mk can be identified with the set of rational primes together with the usual absolute value on q the multiplicities nv are chosen so that the product formula holds and so that h is independent of the choice of the field k the local height corresponding to a non archimedean absolute value is given by intersection theory in a well known manner see e g 2 4 or 7 chapter 11 section 5 we will describe a quick way to compute non archimedean local heights in section 5 the local height for an archimedean absolute value is given by a transcendental function and so efficient computation is somewhat more difficult j tate 15 1988 american mathematical society 0025 5718 88 1 00 25 per page 339 received august 20 1987 revised october 21 1987 1980 mathematics subject classification 1985 revision primary 11g05 14k07 11d25 this work was partially supported by nsf grant dms 8612393 current address mathematics department brown university providence ri 02912 this content downloaded from 207 46 13 103 on thu 20 oct 2016 04 37 48 utc all use subject to http about jstor org terms 340 joseph h silverman has given an easily computed power series which works for real absolute values precisely for a given curve e and point p x y he gives a sequence of easily computed numbers co cl so that

periods of cusp forms and elliptic curves over imaginary quadratic fields in this paper we explore the arithmetic correspondence between on the one hand isogeny classes of elliptic curves e defined over an imaginary quadratic field k of class number one and on the other hand rational newforms f of weight two for the congruence subgroups where n is an ideal in the ring of integers r of k this continues work of the first author and forms part of the ph d thesis of the second author in each case we compute numerically the value of the l series at and compare with the value of which is predicted by the birch swinnerton dyer conjecture finding agreement to several decimal places in particular we find that whenever has a point of infinite order several examples are given in detail from the extensive tables computed by the authors

empirical evidence for the birch and swinnerton dyer conjectures for modular jacobians of genus 2 curves this paper provides empirical evidence for the birch and swinnerton dyer conjectures for modular jacobians of genus 2 curves the second of these conjectures relates six quantities associated to a jacobian over the rational numbers one of these six quantities is the size of the shafarevich tate group unable to compute that we computed the five other quantities and solved for the last one in all 32 cases the result is very close to an integer that is a power of 2 in addition this power of 2 agrees with the size of the 2 torsion of the shafarevich tate group which we could compute

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

on the multi dimensional renewal theorem abstract let x be a random variable on rn n 2 having a density assume x has a finite exponential moment and non zero mean vector let be the corresponding renewal measure and q a cube we obtain an asymptotic formula for x q as x which is uniform in a small cone about the mean vector this formula depends on moments of arbitrarily high order but depends only on the first and second moments of x in a region x x 1 o x 2 3

random walks in weyl chambers and crystals we use kashiwara crystal basis theory to associate a random walk w to each irreducible representation v of a simple lie algebra this is achieved by endowing the crystal attached to v with a possibly non uniform probability distribution compatible with its weight graduation we then prove that the generalized pitmann transform defined by biane bougerol and o connell for similar random walks with uniform distributions yields yet a markov chain when the representation is minuscule and the associated random walk has a drift in the weyl chamber we establish that this markov chain has the same law as w conditionned to never exit the cone of dominant weights at the heart of our proof is a quotient version of a renewal theorem that we state in the context of general random walks in a lattice

on a class of elliptic curves with rank at most two in this note we consider the elliptic curves y 2 x 3 px defined over q for primes p satisfying p 1 mod 8 and review some of their properties we then compute and list in the supplement their ranks and give when the rank is positive the generators of the group of rational points and mordell weil lattice invariant for all primes p 50000 of the form m 2 64n 2

optimal investment horizons for stocks and markets abstract the inverse statistics is the distribution of waiting times needed to achieve a predefined level of return obtained from detrended historic asset prices i simonsen m h jensen a johansen eur phys j 27 2002 583 m h jensen a johansen i simonsen physica a 234 2003 338 such a distribution typically goes through a maximum at a time coined the optimal investment horizon which defines the most likely waiting time for obtaining a given return by considering equal positive and negative levels of return we reported in m h jensen a johansen i simonsen physica a 234 2003 338 on a quantitative gain loss asymmetry most pronounced for short horizons in the present paper the inverse statistics for 2 3 of the individual stocks presently in the djia is investigated we show that this gain loss asymmetry established for the djia surprisingly is not present in the time series of the individual stocks nor their average this observation points towards some kind of collective movement of the stocks of the index synchronization

crystal graphs for representations of the q analogue of classical lie algebras the explicit form of the crystal graphs for the finite dimensional representations of the q analogue of the universal enveloping algebras of type a b c and d is given in terms of semi standard tableaux and its analogues