2009

Intercity number theory seminar and the inaugural lecture of Gunther Cornelissen

10:00-10:50	Yiannis Petridis, Primes, prime geodesics and lattice points
10:55-11:45	Wladimir Pribitkin, The signs they are a-changin'
13:30-14:20	Gautam Chinta, Zeta functions, heat kernels and spectral asymptotics on degenerating families of discrete tori
14:25-15:15	Don Zagier, New constructions of mock modular forms
16:15-17:00	Gunther Cornelissen, Het meten van dubbelzinnigheid

Intercity number theory seminar

11:30-12:20	Yifan Yang, Construction and application of a class of modular functions Abstract. In this talk, we focus on a class of modular functions, known as the generalized Dedekind eta functions or the Siegel functions, and discuss several applications of these modular functions, including defining equations of modular curves, structure of the cuspidal rational torsion subgroup of the Jacobian J1(N), and the gonality of X1(N).
13:30-14:20	Chieh-Yu Chang, On periods and logarithms for Drinfeld modules of rank 2 Abstract. In this talk, we will present motivic methods to determine the algebraic relations among the periods and logarithms of algebraic points for rank 2 Drinfeld modules. Two major applications will be also discussed.
14:30-15:20	Shu-Yen Pan, On local theta correspondence of supercuspidal representations Abstract. The local theta correspondence asserts a one-to-one correspondence between certain irreducible admissible representations of two classical groups over a p-adic field. The preservation principle of local theta correspondence predicts the extistence of a chain of irreducible supercuspidal representations of p-adic classical groups. In the talk, we want to investigate and describe these supercuspidal representations in certain circumstance.
15:30-16:20	Jeng-Daw Yu, Ordinary crystals with logarithmic poles Abstract. We study the abstract formalism of crystals with logarithmic poles and give some properties that generalize some of the work of Deligne in the 1970's.

Intercity number theory seminar

11:00-12:00	Tom Schmidt, Mediants of Rosen fractions and Hurwitz constants of Hecke groups Abstract. In work with C. Kraaikamp and H. Nakada, we complete a program of J. Lehner using continued fractions to determine the bounds on best Diophantine approximation by the orbit of infinity under each of a family of matrix groups. Lehner began this work in the 1980s, using the continued fractions introduced by his Ph.D. student D. Rosen in the 1950s. A. Haas and C. Series determined the bounds, but using hyperbolic geometry. In fact, work of Nakada shows that it is insufficient to use the Rosen fractions --- we thus turn to a means to interpolate this approximating sequences, the mediant maps. A key point in the proof is to show that an ergodic theoretic constant (known as the Lenstra constant) is equal to a certain number theoretic constant.
13:30-14:30	Michel Dekking, Algebraic differences of random Cantor sets Abstract. The study of the algebraic difference F2 - F1 = {y - x: x ∈F1; y ∈F2} of two dynamically defined Cantor sets F1, F2, was motivated by the research of Palis and Takens in regards with the unfolding of homoclinic tangency in some one-parameter families of surface diffeomorphisms. Palis conjectured that if dimH F1 + dimH F2 > 1, then generically it should be true that F2 -F1 contains an interval. For generic dynamically generated non-linear Cantor sets this was proved in 2001 by de Moreira and Yoccoz. The problem is open for generic linear Cantor sets. In this talk I will speak about related results for random Cantor sets.
15:00-16:00	Dieter Mayer , The transfer operator approach to Selberg`s zeta function for Hecke triangle groups Gq Abstract. By using the Hurwitz-Nakada continued fractions generated by the interval map fq: Iq→Iq defined by fq (x)=-1/x-[-1/(x λq) +1/2] λq with Iq=[-λq/2,λq/2] and λq=2cos(π/q), q=3,4,... we derive a symbolic dynamics for the geodesic flow on the Hecke surfaces Gq \  H. This allows us to construct a transfer operator Lβ whose Fredholm determinant det(1-Lβ) is closely related to the Selberg zeta function for the Fuchsian group Gq. This is common work with T.Muehlenbruch and F. Stroemberg (see also J. of Modern Dynamics vol 2, No. 4, 2008, 3-49).
16:00-17:00	Charlene Kalle, Expansions and Tilings Abstract. The transformation Tx = βx (mod 1) for any real β>1 can be used to generate number expansions in base β and with integers between 0 and the floor of β as digits. If β is a certain kind of algebraic integer, then this transformation is linked to a tiling of a Euclidean space. Properties of the number expansions can be obtained from the tiling and vice versa. We will discuss the construction of this tiling and generalizations of it to a more general class of expansion generating transformations.

Joint DIAMANT / GQT seminar

Hendrik Lenstra's sixtieth birthday

A special day at the occasion of Hendrik Lenstra's sixtieth birthday will be held on April 24 in Leiden. The morning program contains two mathematical talks and is part of the workshop Counting Points on Varieties. The afternoon will feature more personal talks by friends and colleagues of Hendrik Lenstra. Everybody is welcome to attend.

Morning program in the De Sitterzaal, Oort building:

•  9:15: coffee
•  9:45: opening
•  9:50 - 10:40: Carl Pomerance (Dartmouth)
• 10:50 -11:40: Manjul Bhargava (Princeton)

Afternoon program in the Lorentzzaal (A144) Kamerlingh Onnes Gebouw, Steenschuur 25, Leiden.

• 14:00 - 14:25: Ed Schaefer (Santa Clara)
• 14:30 - 14:55: Everett Howe (San Diego)
• 15:00: coffee
• 15:30 - 15:55: Johannes Buchmann (Darmstadt)
• 16:00 - 16:25: Richard Groenewegen (London)
• 16:30: closing and drinks

Organisers: Ronald van Luijk, Bart de Smit, Peter Stevenhagen, Lenny Taelman.

09:50-10:40	Carl Pomerance, Sociable numbers Abstract. Consider iterating the function which sends a natural number to the sum of its proper divisors. A fixed point for this system, such as 6 or 28, is called perfect, while a number belonging to a cycle of length 2, such as 220 or 284, is called amicable. Known to Euclid and Pythagoras, some scholars have even found allusions to perfect and amicable numbers in the Old Testament. Sociable numbers are the natural generalization of perfect and amicable numbers to cycles of arbitrary length - they are mere youngsters, having been studied for only a century! This talk will describe the colorful history of the problem (with more connections to Hendrik Lenstra than you might imagine) and report on some recent results on the distribution of sociable numbers within the natural numbers.
10:50-11:40	Manjul Bhargava, Galois closure for rings Abstract. Abstract: We all learn the notion of "Galois closure" or "normal closure" for a finite extension of fields. But what might we mean by the "Galois closure" of an extension of rings? (joint work with Matthew Satriano; based on conversations with Hendrik Lenstra, Jean-Pierre Serre, Bart de Smit, and Kiran Kedlaya)
14:00-14:25	Ed Schaefer, I started feeling sorry for the problem.
14:30-14:55	Everett Howe, Channeling your advisor
15:30-15:55	Johannes Buchmann, Hendrik Lenstra and cryptography
16:00-16:25	Richard Groenewegen, A sense of reality

Intercity number theory seminar

11:45-12:30	Steve Meagher (Freiburg), Equations for Abelian varieties with a prescribed number of points over a finite field. Abstract. This talk is about ongoing work with Robert Carls. We describe equations for a zero-dimensional sub-variety of the moduli space of Abelian varieties whose points correspond to an Abelian variety with a given L-function. We also explain potential algorithmic applications to curves of very small genus.
13:15-14:00	Vivija Ceprkalo, Elliptic curves in Edwards form Abstract. This is a survey on the Edwards form, which is a particular type of quartic equations for describing elliptic curves.
14:15-15:00	Cecilia Salgado, On the rank of the fibres of rational elliptic surfaces. Abstract. We compare the generic and the special ranks of rational elliptic surfaces over number fields. We show that, for a big class of rational elliptic surfaces, there are infinitely many fibres with rank at least equal to the generic rank plus two. If time allows we will discuss the same type of problem for some K3 elliptic surfaces.
15:15-16:00	Marius van der Put, Painlevé differential equations Abstract. The classical work of Painlevé, Gambier, Garnier et al. on nonlinear ordinary differential equations was ended by the 1930's. New interest and new ideas in the subject came in the 1980's by Jimbo, Miwa, Ueno, Okamato and others. We present a historical survey and some new joint results with Masa-Hiko Saito.
16:15-17:00	René Pannekoek, Parametrizations over Q of cubic surfaces Abstract. Given any smooth cubic surface S defined over a number field K, it is a well-known fact that there exists a birational map f: S →P2. If we pose the additional requirement that f be defined over K, however, the assertion may no longer be true. In the 1970s, Manin and Swinnerton-Dyer formulated a necessary and sufficient criterion for S to allow a birational map to P2 over K. First, I will discuss their criterion and show that it is a pretty strong restriction on S. Also, I will give several examples of cubic surfaces in order to give some idea which cases actually occur. After this, I will elaborate on the fact that there are several cases in which a special sort of birational map can be found; I will show how these cases overlap and that they do not exhaust the class of all birationally trivial cubic surfaces. Finally, I will give examples of explicit birational and rational maps that I have been able to construct.

Intercity number theory seminar

11:15-12:15	Ben Kane, Equidistribution of Heegner points and quadratic forms Abstract. In this talk we will investigate a relationship between supersingular reduction of Heegner points and representations by quadratic forms. Using equidistribution for representations by quadratic forms, we will establish a certain corresponding equidistribution result for the reduction map.
13:00-13:55	Ted Chinburg, Lifts of group actions on curves from characteristic p to characteristic 0 Abstract. This talk is about lifts to characteristic 0 of faithful actions of a finite group G on smooth projective curves in characteristic p. One can ask for which G every such action lifts, and for which G at least one such action lifts. I will survey some recent results concerning these questions, including joint work with David Harbater and Bob Guralnick.
14:05-15:00	Ted Chinburg, Katz Gabber covers with extra automorphisms Abstract. It is an old problem to write down explicit automorphisms of order p2 of a power series ring k[[t]] over a perfect field k of characteristic p. I will describe some positive and negative results concerning this problem which are based on a classification of Katz Gabber covers of the projective line which have large automorphism groups. This is joint work with Frauke Bleher, Peter Symonds, Bjorn Poonen and Florian Pop.
15:20-16:20	Shabnam Akhtari, Representation of integers by binary forms Abstract. Suppose F(x,y) is an irreducible binary form with integral coefficients, degree n ≥3 and discriminant DF ≠0. Let h be an integer. The equation F(x,y) = h has finitely many solutions in integers x and y. I shall discuss some different approaches to the problem of counting the number of integral solutions to such equations. I will give upper bounds upon the number of solutions to the Thue equation F(x,y) = h. These upper bounds are derived by combining methods from classical analysis and geometry of numbers. The theory of linear forms in logarithms plays an essential rule in this study.

RISC/Intercity number theory seminar

11:30-12:30	Gabriele Dalla Torre, The unit-residue group of an algebraic number field Abstract. Given a positive integer m and a local field F which contains a primitive m-th root ζm of unity, it is possible to define the norm-residue symbol, a bilinear map of F* ×F* into the group ⟨ζm ⟩. This definition can be extended to ideles of any number field K which contains a primitive m-th root of unity. We will study the ``unit-residue group", a quotient of the group of unit ideles which naturally arises in this setting, with special care to the structure deriving from the norm-residue symbol. Finally, we will give some examples and a complete description in the case of quadratic number fields.
13:20-14:15	Ted Chinburg, Deformations of complexes of modules for a profinite group Abstract. This talk is about a new finiteness problem concerning deformations of complexes of Galois modules arising from arithmetic geometry. The main question is whether the associated versal deformations can be represented by bounded complexes of finitely generated modules over the versal deformation ring. This is joint work with F. Bleher, with key ideas from Luc Illusie and Ofer Gabber.
14:25-15:20	Ted Chinburg, Rationality of Euler characteristics Abstract. This talk is about characterizing those finite groups G which have the following property. Whenever G acts faithfully on a smooth projective irreducible curve C in characteristic 0, the action of G on the holomorphic differentials of C defines a character of G having only rational values. We will find all such G, using Dirichlet L-functions to determine when certain cyclotomic algebraic integers are rational. This is joint work with Amy Ksir.
15:30-16:25	Chaoping Xing, Construction of algebraic curves over finite fields from linear codes and vice versa Abstract. The interrelationship between error correcting codes and algebraic curves over finite fields with many rational points was discovered as early as the 1980s with the invention of Goppa geometric codes. In this talk, we present yet another method to show how linear codes and algebraic curves are intertwined. More precisely, by using upper bound on linear codes, we can show existence of algebraic curves over finite fields with many rational points. On the other hand, by using upper bounds on algebraic curves, we can construct good linear codes.

Intercity number theory seminar

13:00-14:00	Sander Zwegers, Mock modular forms: an introduction Abstract. The main motivation for the theory of mock modular forms comes from the desire to provide a framework to understand the mysterious and intriguing mock theta functions, defined by Ramanujan in 1920, as well as related functions. In this talk, we will describe the nature of the modularity of the original mock theta functions, formulate a general definition of mock modular forms, and consider some further examples. Time permitting, we will also consider a generalization to higher depth mock modular forms.
14:30-15:30	Oliver Lorscheid, Toroidal Eisenstein series and double Dirichlet series Abstract. A formula of Erich Hecke in an article from 1917 laid a connection between a sum of values of an Eisenstein series E(-,s) with the value ζ(s) of the zeta function ζ. We call an automorphic form toroidal if the corresponding sum (or integral in its adelic formulation) vanishes for all right translates. The importance of this definition lies in a reformulation of the Riemann hypothesis in terms of the space of toroidal automorphic forms as observed by Don Zagier. Namely, the Eisenstein series E(-,s) lies in a tempered representation if and only if s has real part 1/2, and by Hecke's formula, E(-,s) is toroidal if s is a zero of the zeta functions. In order to reverse the latter statement, non-vanishing results has to be shown for the factors occuring in Hecke's formula. In a joint work with Gunther Cornelissen, double Dirichlet series are used for this purpose. In this talk, we will introduce into the theory of (toroidal) automorphic forms and give an overview over results in this direction. Then we will explain how to use double Dirichlet series to show non-vanishing results.
16:00-17:00	Dimitar Jetchev, Global divisibility of Heegner points and Tamagawa numbers Abstract. We improve Kolyvagin's upper bound on the order of the p-primary part of the Shafarevich-Tate group of an elliptic curve of rank one over a quadratic imaginary field. In many cases, our bound is precisely the one predicted by the Birch and Swinnerton-Dyer conjectural formula.

Intercity Number Theory Seminar at the Woudschoten conference on automorphic forms.

Speakers:
10-11 Henryk Iwaniec (Rutgers): Some features of spectral summation formulas
11-12 Akshay Venkatesh (Stanford): Torsion in homology of arithmetic groups
12-13 Lunch
13-14 Emmanuel Kowalski (ETHZ): Families of Cusp Forms and L-functions

Intercity number theory seminar

	René Schoof, p-adic representations and (φ, Γ)-modules Abstract. In this lecture we discuss p-adic representations of p-adic fields. We explain Fontaine's description in terms of (φ, Γ)-modules.
14:00-15:00	Jan-Hendrik Evertse, Complexity of algebraic numbers Abstract. Given an integer b>1, any real number from (0,1) can be expressed uniquely as d1b-1+d2b-2+... with b-ary digits d1, d2, ... from {0,...,b-1}. It is generally believed that the sequence of b-ary digits of an algebraic irrational number from (0,1) should behave like a random sequence, but up to know only some weak results in this direction have been obtained. We discuss some work of Adamczewski and Bugeaud, and of Bugeaud and the speaker, on the distribution of the b-ary digits of an algebraic number.
15:15-16:15	Hendrik Lenstra, Finding the ring of integers in a number field Abstract. A classical algorithmic problem in algebraic number theory is to find the ring of integers of a given algebraic number field. The lecture is devoted to a new technique for solving this problem. It does not always work, but if it does, then it writes down the answer in one stroke.

Intercity number theory seminar

DIAMANT seminar room (HG 9.41). There will be coffee and tea at 12:00 and 14:45.

	René Schoof, Curves over finite fields
12:15-13:00	Ronald van Luijk, Unfaking the fake Selmer group Abstract. Let C be a smooth projective curve over a global field k with Jacobian J. Then the Mordell-Weil group J(k) of k-rational points on J is finitely generated. Knowing the torsion subgroup, which is usually relatively easy to find, the rank of J(k) can be read off from the size of the finite group J(k)/2J(k). This quotient injects into the so called Selmer group, which is abstractly defined as a certain subgroup of the cohomology group H1(k,J[2]). The Selmer group is finite, so the image of J(k)/2J(k) in it can be determined by deciding for each element of the Selmer group separately whether or not it is in the image of J(k)/2J(k). Unfortunately, the abstract definition of the Selmer group is not very amenable to explicit computations, which are in practice done with the fake Selmer group instead. In general the fake Selmer group is isomorphic to a quotient of the Selmer group by a subgroup of order 1 or 2. In this talk we will define all the groups just mentioned and we will introduce a new group, equally amenable to explicit computations as the fake Selmer group, that is always isomorphic to the Selmer group. This is joint work with Michael Stoll.
14:00-14:45	David Freeman, Pairing-friendly hyperelliptic curves and Weil restriction Abstract. A "pairing-friendly curve" is a curve C over a finite field Fq such that (a) the Jacobian of C has a subgroup of large prime order r, and (b) the r-th roots of unity are contained in an extension field Fqk for some small value of k. Pairing-friendly curves have found many uses in cryptography. For such applications one wants to control the extension degree k, known as the "embedding degree," while keeping the field size q as small as possible relative to the subgroup size r. We describe a construction of pairing-friendly genus 2 curves that, for certain embedding degrees k, achieves the smallest known ratio log q/log r for simple, non-supersingular abelian surfaces. The proof that these curves have the desired properties relates them to Weil restrictions of elliptic curves. We also describe some experimental results suggesting that our construction fails in certain cases. Finding alternative constructions for these cases is an open problem. This is joint work with Takakazu Satoh (Tokyo Institute of Technology).
15:00-15:45	Bart de Smit, The valuation criterion for normal bases Abstract. For a finite Galois extension L/K of local fields we consider the question whether there is an integer d so that all elements x of L of valuation d have the property that the Galois conjugates of x form a basis of L as a vector space over K. This is joint work with Lara Thomas (Lausanne) and Mathieu Florence (Paris).

Special day in honour of Wilberd van der Kallen and Joop Kolk

Intercity number theory seminar

	Christine Bachoc, Secure Message Transmission with Small Public Discussion Abstract. In the problem of Secure Message Transmission in the public discussion model (SMT-PD), a Sender wants to send a message to a Receiver privately and reliably. Sender and Receiver are connectedby n channels, up to t<n of which may be maliciously controlled by a computationally unbounded adversary, as well as one public channel, which is reliable but not private. The SMT-PD abstraction has been shown instrumental in achieving secure multi-party computation on sparse networks, where a subset of the nodes are able to realize a broadcast functionality, which plays the role of the public channel. However, the implementation of such public channel in point-to-point networks is highly costly and non-trivial, which makes minimizing the use of this resource an intrinsically compelling issue. In this talk, after a brief introductory survey, we present the first SMT-PD protocol with sublinear (i.e., logarithmic in m, the message size) communication on the public channel. In addition, the protocol incurs a private communication complexity of O(mn/(n-t)), which, as we also show, is optimal. By contrast, the best known bounds in both public and private channels were linear. Furthermore, our protocol has an optimal round complexity of (3,2), meaning three rounds, two of which must invoke the public channel. Finally, we ask the question whether some of the lower bounds on resource use for a single execution of SMT-PD can be beaten on average through amortization. In other words, if Sender and Receiver must send several messages back and forth (where later messages depend on earlier ones), can they do better than the naïve solution of repeating an SMT-PD protocol each time? We show that amortization can indeed drastically reduce the use of the public channel: it is possible to limit the total number of uses of the public channel to two, no matter how many messages are ultimately sent between two nodes. (Since two uses of the public channel are required to send any reliable communication whatsoever, this is best possible.) This is joint work with Clint Givens (UCLA) and Rafi Ostrovsky (UCLA).
11:30-12:30	Andries Brouwer, The eigenvalues of the graph on the flags of a finite building, joined when in mutual general position.
13:30-14:30	Heng Huat Chan, Class invariants Abstract. It is well known that values of the modular j-invariant function evaluated at certain imaginary quadratic integer generates the Hilbert class fields of the corresponding imaginary quadratic fields. In this talk, we will replace j-invariant by some other modular functions and examine their behavior as class invariants.
16:00-17:00	Ronald Cramer, Towers of Algebraic Function Fields in Secure Computation

Intercity number theory seminar

11:45-12:45	Marios Magioladitis, The discrete logarithm problem on isogenous hyperelliptic curves of genus 2 Abstract. In 2005, Jao, Miller, and Venkatesan proved that the DLP of elliptic curves with the same endomorhism ring is random reducible under the GRH. In this talk, we discuss a possible generalization of this result to hyperelliptic curves of genus 2 (and 3) defined over a finite field and show the difficulties involved. First, we explain the role of the endomorphism rings of the Jacobian and the polarization. Following the work of Jao, Miller and Venkatesan, we construct isogeny graphs for genus 2 curves. Specifically, we discuss the connection between isogenies and ideal classes in the Jacobian of these curves. This project is research in progress and we describe the current status of this research.
13:30-14:30	Jaap Top, Ruled quartic surfaces Abstract. In the 19th century, Cremona in a synthetic geometric way and Cayley using analytic geometric methods subdivided the ruled quartic surfaces in P3 into twelve classes. Somewhat later, K. Rohn wrote about the subject and moreover had models made of such surfaces. We explain and extend some of these classical results in more modern terms.
15:00-16:00	Robin de Jong, Logarithmic equidistribution of division points on superelliptic curves Abstract. A superelliptic curve is a curve over a number field K given by an equation yN=f(x), with suitable conditions on f and N. On such curves one has the notion of n-division points, generalising the notion of n-torsion points on elliptic curves. We discuss two results. First, the Neron-Tate height restricted to the canonical image of X in its jacobian can be written as a sum, over all places of K, of certain local logarithmic integrals over X. Second, for almost all algebraic points on X these local integrals can be computed by averaging over the n-division points of X, and letting n tend to infinity. For elliptic curves these results were shown by Everest-ní Fhlathúin and Everest-Ward.
16:05-17:05	Jorge Mozo Fernández, Results on analytic classification of germs of holomorphic foliations Abstract. We shall review the main known results concerning the analytic classification of germs of codimension one, singular holomorphic foliations in dimension two and three. In dimension two, we shall focus in the classical works of J. Martinet and J.P. Ramis, in the reduced case, and in the works of Cerveau, Moussu, Meziani, Berthier, Sad and others, in the nilpotent case. In the quasi-homogeneous case, we shall mention the work of Y. Genzmer. The state-of-art of this subject in dimension three will be explained. We shall recall the main concepts involved: reduction of singularities, existence of separatrices, and holonomy of the leaves, and how they are used in order to establish the results.

DIAMANT/EIDMA-symposium

Intercity number theory seminar

11:30-12:20	Jonathan Reynolds, Power integral points on elliptic curves. Abstract. Siegel proved that there are finitely many rational points on an elliptic curve which have an integral coordinate. I will explain why finiteness still holds when the denominator of the coordinate is an integer raised to a fixed non-trivial power. The effectiveness of this result for certain families of curves will be discussed.
13:30-14:20	Jeroen Sijsling, Equations for (1;e)-curves Abstract. A (1;e) curve is a special compact quotient of the upper half plane: it has genus 1 and one elliptic (branch) point. A Shimura curve is a quotient of the upper half plane coming from orders in certain quaternion algebras over totally real fields. Kisao Takeuchi classified all (1;e) curves that are naturally commensurable to Shimura curves: there are 72 of these. This talk discusses techniques for calculating explicit equations for Takeuchi's curves. These range from dessins d'enfants to modularity over Q and classical modular curves.
14:30-15:20	Marco Streng, Abelian surfaces admitting an (l,l)-endomorphism Abstract. We give a classification of all principally polarized abelian surfaces that admit an (l,l)-isogeny to themselves. We make the classification explicit in the simplest cases l=1 and l=2 and show how to compute all abelian surfaces that occur. This research was done during an internship of the speaker at Microsoft Research (MSR). It is joint work with Reinier Bröker (MSR, currently Brown University) and Kristin Lauter (MSR).
15:40-16:30	Gunther Cornelissen, Arithmetic equivalence for function fields Abstract. A theorem of Tate and Turner says that global function fields have the same zeta function if and only if the Jacobians of the corresponding curves are isogenous. In this talk, we discuss what happens if we replace the usual (characteristic zero) zeta function by the positive characteristic zeta function introduced by Goss, or by a "Teichmüller lift" of that (joint work with Aristides Kontogeorgis and Lotte van der Zalm, arXiv:0906.4424, doi:10.1016/j.jnt.2009.08.002).